draft - home - math › ~margalit › flp › flp.pdf · 2008-06-13 · phisms of surfaces” by...

DR

AFT

13 J

un 2

008

Thurston’s Work on Surfaces

A. Fathi, F. Laudenbach, and V. Poenaru

Translation by Djun Kim and Dan Margalit

DR

AFT

13 J

un 2

008

This translation is copyright (c) 2007 Djun M. Kim and Dan Margalit

DR

AFT

13 J

un 2

008

Table of Contents

Table of Contents iii

Expose 1. Collected theorems of Thurston on surfaces 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The space S of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 The space of functionals . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Measured Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Teichmuller space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Pseudo-Anosov Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . 7

1.5.1 Spectral properties of pseudo-Anosov diffeomorphisms . . . . 9

1.5.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 The case of the torus T 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Expose 2. Some reminders on the theory of surface diffeomorphisms 19

2.1 The space of functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 The braid groups and their computation . . . . . . . . . . . . . . . . . . 20

2.3 Diffeomorphisms of the two holed disk

(The spaces A(P 2), A′(P 2)) . . . . . . . . . . . . . . . . . . . . . . . . 24

Expose 3. Reminders on hyperbolic geometry in dimension 2

and generalities on i : S × S → R+ 31

3.1 A little hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 The Teichmuller space of the two-holed disk . . . . . . . . . . . . . . . 35

3.3 Generalities on intersection . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Systems of simple curves in M and hyperbolic isometries . . . . . . . . 53

Expose 4. The space of simple closed curves in a surface 55

4.1 The weak topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 The space of multicurves . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 An explicit parameterization of the space of multicurves . . . . . . . . 59

iii

DR

AFT

13 J

un 2

008

Expose 5. Measured Foliations 69

5.1 Definition. The Poincare recurrence theorem and the Euler–Poincare

formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1.2 Permissible singularities in the interior . . . . . . . . . . . . . . 70

5.1.3 Permissible singularities on the boundary . . . . . . . . . . . . . 70

5.1.4 Good Atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.1.5 Poincare Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1.6 The Euler–Poincare formula. . . . . . . . . . . . . . . . . . . . . 74

5.1.7 Quasitransverse curves . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Measured foliations and simple curves . . . . . . . . . . . . . . . . . . . 78

5.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.2 I∗ :MF → RS+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.4 Stability Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Curves as measured foliations . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.1 The “enlarging” procedure . . . . . . . . . . . . . . . . . . . . . 96

5.3.2 The inclusion R+ × S → MF . . . . . . . . . . . . . . . . . . . . 98

Expose 6. Measured Foliations (Continued) 99

6.1 Classification of foliations of the annulus . . . . . . . . . . . . . . . . . 101

6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1.3 Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.1.4 Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2 Classification of foliations on a pair of pants . . . . . . . . . . . . . . . . 103

6.2.1 Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2.3 Reduced good foliations . . . . . . . . . . . . . . . . . . . . . . . 106

6.2.4 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3 The arcs jaune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.3.2 Length of the arc jaune . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3.3 Definition of the arc A. . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4 Normal form of a foliation . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4.2 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

DR

AFT

13 J

un 2

008

6.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4.4 Definition of NF . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.4.5 Classification of foliations in normal form . . . . . . . . . . . . . 118

6.5 Classification of measured foliations . . . . . . . . . . . . . . . . . . . . 120

6.5.1 Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5.2 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.5.3 Proposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.6 Return to curves as functionals . . . . . . . . . . . . . . . . . . . . . . . 126

6.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.6.2 Proposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.7 Minimality of the action of π0(Diff M) on PMF . . . . . . . . . . . . . 128

6.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.7.5 Proof of Theorem 6.18 . . . . . . . . . . . . . . . . . . . . . . . . 130

6.8 Existence of complementary measured foliations . . . . . . . . . . . . . 131

Expose 7. Teichmuller Space 133

Expose 8. How Thurston compactifies Teichmuller space 147

8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.1.2 Construction of a projection q : T →MF . . . . . . . . . . . . . 148

8.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.2 The Fundamental Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3 The manifold T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3.2 Map of the neighborhood of a foliation . . . . . . . . . . . . . . 159

8.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Expose 9. Classification of surface diffeomorphisms 163

9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9.2 The reducible case: βU(µ) 6= ∅ . . . . . . . . . . . . . . . . . . . . . . . 165

9.3 Arational Measured Foliations . . . . . . . . . . . . . . . . . . . . . . . 166

9.4 Case II: (F , µ) is arational and λ = 1 . . . . . . . . . . . . . . . . . . . 171

9.5 Case III; (F , µ) is arational and λ 6= 1 . . . . . . . . . . . . . . . . . . . . 172

DR

AFT

13 J

un 2

008

9.5.1 Construction of a measured foliation F ′ . . . . . . . . . . . . . . 177

9.5.2 Construction of a “diffeomorphism” . . . . . . . . . . . . . . . . 178

9.5.3 Pseudo-Anosov “diffeomorphisms” . . . . . . . . . . . . . . . . 179

9.6 Some properties of pseudo-Anosov diffeomorphisms . . . . . . . . . . 181

Expose 10. Some dynamics of pseudo-Anosov diffeomorphisms 187

10.1 Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

10.2 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

10.3 Subshifts of finite type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

10.4 The entropy of pseudo-Anosov diffeomorphisms . . . . . . . . . . . . . 200

10.5 Construction of Markov partitions for pseudo-Anosov diffeomorphisms207

10.6 Pseudo-Anosov diffeomorphisms are Bernoulli . . . . . . . . . . . . . . 211

Expose 11. Thurston’s theory for surfaces with boundary 215

11.1 The space of curves and measured foliations . . . . . . . . . . . . . . . 215

11.2 Teichmuller space and its compactification . . . . . . . . . . . . . . . . 217

11.3 Preparation for the classification of diffeomorphisms . . . . . . . . . . 219

11.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

11.3.2 Reducible diffeomorphism . . . . . . . . . . . . . . . . . . . . . 220

11.3.3 Arational foliations . . . . . . . . . . . . . . . . . . . . . . . . . . 220

11.3.4 Case ϕ(F , µ) = (F , µ). . . . . . . . . . . . . . . . . . . . . . . . . 221

11.3.5 Case ϕ(F , µ) = (F , λµ), λ > 1. . . . . . . . . . . . . . . . . . . . 221

11.3.6 Pseudo-Anosov diffeomorphism . . . . . . . . . . . . . . . . . . 224

11.3.7 Example: a disk with 3 holes . . . . . . . . . . . . . . . . . . . . 224

11.4 Thurston’s classification; Nielsen’s theorem . . . . . . . . . . . . . . . . 225

11.5 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Expose 12. Uniqueness theorems for pseudo-Anosov diffeomorphisms 233

12.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

12.2 Reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

12.2.1 The theorem of Perron–Frobenius . . . . . . . . . . . . . . . . . 235

12.2.2 Markov partition (see Exposes 9 and 10) . . . . . . . . . . . . . . 235

12.3 Proof of Theorem 12.1: unique ergodicity . . . . . . . . . . . . . . . . . 236

12.4 Proof of Theorem 12.2 and its corollaries . . . . . . . . . . . . . . . . . . 239

12.5 Proof of Theorem III (uniqueness of pseudo-Anosov diffeomorphisms) 250

Expose 13. Construction of pseudo-Anosov diffeomorphisms 255

13.1 Generalized pseudo-Anosov diffeomorphism . . . . . . . . . . . . . . . 255

13.2 Construction by ramified cover . . . . . . . . . . . . . . . . . . . . . . . 257

13.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

13.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

DR

AFT

13 J

un 2

008

13.3 Construction by Dehn twists . . . . . . . . . . . . . . . . . . . . . . . . . 25913.3.1 Flat structure on M . . . . . . . . . . . . . . . . . . . . . . . . . . 25913.3.2 Affine homeomorphisms . . . . . . . . . . . . . . . . . . . . . . 26213.3.3 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26413.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

Expose 14. Fibrations of S1 with Pseudo-Anosov Monodromy 267

Expose 15. Presentation of the group of diffeotopies of a compact orientable sur-

face 28315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28315.2 A method for presenting G . . . . . . . . . . . . . . . . . . . . . . . . . 285

15.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28515.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28715.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

15.3 The cell complex of marked functions . . . . . . . . . . . . . . . . . . . 28815.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28815.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29115.3.3 The 1-skeleton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29315.3.4 The 2-skeleton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

15.4 The marking complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29515.4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29515.4.2 Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29615.4.3 The case of the torus T 2 . . . . . . . . . . . . . . . . . . . . . . . 299

Appendix A. The “pair of pants” decomposition of a surface. 303

Appendix B. Spines of manifolds of dimension 2 307

Appendix C. Explicit formulas for Measured Foliations 313

Appendix D. Estimates of hyperbolic distances 323D.1 Hyperbolic distance of i to z0. . . . . . . . . . . . . . . . . . . . . . . . . 323D.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324D.3 Hyperbolic translation along the imaginary axis . . . . . . . . . . . . . 325D.4 Translation along the hyperbolic geodesic of complex numbers of mod-

ulus 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325D.5 Relations between the sides of a right hyperbolic hexagon . . . . . . . 325D.6 Distance between two points at equal distance from a line . . . . . . . 327D.7 Bounding distances in pairs of pants . . . . . . . . . . . . . . . . . . . . 328

Appendix E. Errata 333

References 335

DR

AFT

13 J

un 2

008

(FLP — Expose 1: Draft – SVN Revision : 304) June 13, 2008

DR

AFT

13 J

un 2

008

June 13, 2008 (FLP — Expose 1: Draft – SVN Revision : 304) i

Foreword to First Edition

This book contains an exposition of results of William Thurston on the theory

of surfaces (measured foliations, natural compactification of Teichmuller space, and

classification of surface diffeomorphisms). Our scope is essentially that outlined in

the research announcement of Thurston, and in the notes of his Princeton course, as

written up by M. Handel and W. Floyd.

A part of this work, notably the classification of curves and of measured folia-

tions, is an elaboration of lectures made in the Seminaire d’Orsay in 1976–1977. But

we were not able to write the proofs for the remaining portions of the theory until

much later. In the Spring of 1978, at Plans-Sur-Bex, Thurston explained to us how

to see the projectification of the space of measured foliations as the boundary of Te-

ichmuller space.

The first expose enumerates the principal results, the proofs of which follow in

Exposes 2 through 13. The last two exposes present work somewhat marginal to the

theme of the classification of surface diffeomorphisms. Expose 14 (orally presented

by D. Fried and D. Sullivan) discusses nonsingular closed 1-forms on 3-dimensional

manifolds, following Thurston, in particular on the fiber bundles over S1 for which

the monodromy diffeomorphism is pseudo-Anosov. Expose 15 (orally given by A. Marin)

gives a finite presentation of the “mapping class group” following Hatcher and Thurston.

The seminar consisted also of exposes of an analytical nature (holomorphic quad-

ratic differentials, quasi-conformal mappings) given by W. Abikoff, L. Bers, and J. Hub-

bard. In the end, the two points of view were found to be more independent of each

other than was initially believed. The analytic point of view is the subject of a sepa-

rate text written by W. Abikoff (see [Abi72]).

DR

AFT

13 J

un 2

008

ii (FLP — Expose 1: Draft – SVN Revision : 304) June 13, 2008

We thank all of the active participants in the seminar; all have contributed assis-

tance in various sections: A. Douady, who, after the oral presentations, helped us to

capture the content of the lectures; M. Shub, who discussed with us the ergodic point

of view; D. Sullivan, who besides giving much advice and encouragement, strove to

make us understand how the image of a curve under iteration of a pseudo-Anosov

diffeomorphism “approaches” the foliation of the surface (it took many more months

to fully understand this “mixing”.)

Finally, we thank Mme. B. Barbichon (typography) and S. Berberi (illustrations)

for the care that they took in preparing the manuscript.

Foreword to Second Edition

The “Research announcement” “On the geometry and dynamics of diffeomor-

phisms of surfaces” by William Thurston has finally appeared in Bulletin of Amer.

Math. Soc. 19 (1988), 417–431. One can find there a list of references to later work

and to the first edition of this book. We also point out the book by S. Bleiler and A.

Casson, “Automorphisms of surfaces after Nielsen and Thurston,” Cambridge Uni-

versity Press, 1988.

We limit ourselves to a few corrections that one can find assembled in an errata

at the end of this volume.

Orsay, May 27, 1991

DR

AFT

13 J

un 2

008

June 13, 2008 (FLP — Expose 1: Draft – SVN Revision : 304) iii

Translators’ notes

We would like to thank Jayadev Athreya, Matt Bainbridge, Jason Behrstock, In-

dira Chatterji, Matt Clay, Moon Duchin, Justin Malestein, Ben McReynolds, Dylan

Thurston, and Liam Watson for reading early drafts and offering corrections and

helpful suggestions.

Djun would like to thank Dale Rolfsen, for providing support, encouragement

and many fruitful discussions, the Mathematics departments at Berkeley and McGill,

for providing library and computer facilities during visits in 1990 and 1996 respec-

tively, and Elisabeth Kim, who read early drafts of some of the exposes and provided

corrections and helpful suggestions.

Thanks are also due to Mary Ann Lacey, Ronald Ferguson, Kevin Pilgrim, Jonathan

Walden, Frances Fry, Bill Casselman, and Curt McMullen.

Djun Kim,

Vancouver, B.C.

[email protected]

Dan Margalit,

Salt Lake City, UT

[email protected]

Colophon

This book was typeset using a modified LATEX 2ǫ report style. Most figures were

produced using xfig, free software written by Supoj Sutanthavibul, Ken Yap, Brian

Smith, and Micah Beck, among others. The resulting figures were converted into en-

capsulated PostScript using transfig and then further post-processed. The drafts

were printed using dvips.

DR

AFT

13 J

un 2

008

iv (FLP — Expose 1: Draft – SVN Revision : 304) June 13, 2008

Abstract

This book is a an exposition of Thurston’s theory of surfaces (measured foliations,

compactification of Teichmuller space and classifications of diffeomorphisms).

The mathematical content is roughly the following.

For a surface M (let’s say closed, orientable, of genus g > 1), one denotes by Sthe set of isotopy classes of simple closed curves in M . For α, β ∈ S, one denotes by

i(α, β) the minimum number of geometric intersection points of α′ with β′, where α′

(resp. β′) is a simple curve in the class α (resp. β). This induces a map i∗ : S → RS+

which turns out to be injective. In fact, if one projectivizes RS+ \ 0, i∗ induces an

injection i∗ : S → P (RS+) which endows S with a nontrivial topology. Here RS

+ is

provided with the weak topology ( = product topology). Two curves α, β ∈ S are

“close” in P (RS+) if, up to a multiple, they are made up of more or less the same

strands going more or less the same direction. This has nothing to do with homotopy

theory.

The limits of curves are naturally interpreted as projective classes of “measured

foliations”, that is, foliations that have an “invariant” transverse distance, and that

have certain kinds of singularities (well-known in the theory of quadratic differen-

tials, or in smectic liquid crystals). The space of measured foliations considered in RS+

(or in P (RS+)) is denoted byMF (resp. PMF ). One shows that

MF ≃ R6g−6 and PMF ≃ S6g−7.

In P (RS+), PMF and the Teichmuller space T (M) glue together into a 6g− 6 dimen-

sional disk:

T (M) = T (M) ∪ PMF(M) = D6g−6.

The group Diff(M) acts continuously on this compactification of T (this is hence

a “natural” compactification).

Hence any φ ∈ Diff(M) has a fixed point in T (M) (Brouwer) and the analysis of

this fixed point shows that (up to isotopy) each φ is either a hyperbolic isometry, or

“Anosov-like” (the word is “pseudo-Anosov”), or else “reducible”.

Pseudo-Anosov diffeomorphisms minimize the topological entropy in their iso-

topy class. Also two pseudo-Anosov’s which are isotopic are actually conjugate.

DR

AFT

13 J

un 2

008

June 13, 2008 (FLP — Expose 1: Draft – SVN Revision : 304) v

Every diffeomorphism φ : M →M has a (finite) spectrum defined in terms of the

length of φnα′ raised to the power 1/n. A pseudo-Anosov is characterized by the fact

that the spectrum is a single value λ > 1.

There is a good method for producing many pseudo-Anosovs out of combina-

tions of Dehn twists which is explained in Expose 13.

The last two exposes are of a somewhat different character: Expose 14 is about

closed nonsingular 1-forms on 3-manifolds, and Expose 15 is about the Hatcher–

Thurston theorem on finite presentability of π0Diff(M).

DR

AFT

13 J

un 2

008

Expose 1

Collected theorems of Thurston

on surfaces

by V. Poenaru

1.1 Introduction

Thurston’s theory ([Thu67], see also [Thu], [Poe78]) is concerned with the following

three questions:

I. Describe “all” simple closed curves (not necessarily connected)

on a surface, up to isotopy.

II. Describe “all” diffeomorphisms of a surface, up to isotopy.

III. Give a natural (with respect to the action of diffeomorphisms)

boundary for Teichmuller space.

For a closed surface, there always exists a Riemannian metric of constant cur-

vature [Gau27]. Table 1.1 below summarizes the possibilities and at the same time

establishes a parallel between the geometric and the topological properties.

1

DR

AFT

13 J

un 2

008

2 (FLP — Expose 1: Draft – SVN Revision : 320) June 13, 2008

Most of Thurston’s theorems hold for any compact surface, but in what follows,

we restrict ourselves to orientable surfaces (closed, or with nonempty boundary).

Surface K (curvature) χ (Euler

characteristic)Remarks

S2, RP 2K = 1

(Elliptic geometry) χ > 0 π1 is finite, π2 6= 0.

T 2, K2

(Klein bottle)

K = 0(Euclidean geometry)

χ = 0

These are K(π, 1)’s and

their universal covering

space is R2.genus > 1 K = −1(Hyperbolic geometry)

χ < 0

Table 1.1: The three possible geometries on surfaces.

1.2 The space S of curves

Let M be a compact, connected, orientable surface. We write S(M) = S for the set of

isotopy classes of simple, closed, connected curves of M that are not homotopic to a

point or homotopic to a boundary component of M .

Remarks.

(1) The elements of S are not oriented.

(2) Since two simple closed curves that are homotopic are also isotopic [Eps66],

we may replace “isotopy classes” in the above definition with “homotopy classes”.

Consider the symmetric map

i : S × S → Z+ = {0, 1, 2, . . .}

defined in the following fashion: i(α, β) is the minimum number of intersections of

a representative for α with a representative for β. This is the geometric intersectiongeometric intersection number

number (as opposed to the algebraic intersection number).algebraic intersection number

Example. On the torus T 2, we choose two oriented generators x and y for π1(T2).

Then

DR

AFT

13 J

un 2

008

June 13, 2008 (FLP — Expose 1: Draft – SVN Revision : 320) 3

all elements of S may be represented by γ(a, b) = ax+ by, where a, b ∈ Z and (a, b) =

1; in S, we have γ(a, b) = γ(−a,−b). The following formula is easy to verify.

i (γ(a, b), γ(c, d)) =

∣∣∣∣det

(a b

c d

)∣∣∣∣

Lemma 1.1.

(1) If α ∈ S, there is a β ∈ S such that i(α, β) 6= 0.

(2) If α1 6= α2 in S, there is a β ∈ S such that i(α1, β) = 0 6= i(α2, β).

The proof is given in Expose 3.

1.2.1 The space of functionals

We consider the set RS+ of functions from S to the nonnegative reals, with the weak

topology. The usual multiplication by the positive reals defines rays in RS+. The set of rays

these rays is the projective space P (RS+); it is given the quotient topology. We have

the natural maps

S i∗−→ RS+ \ 0

π−→ P (RS+)

where i∗ is defined by i∗(α)(β) = i(α, β). By statement (1) of Lemma 1.1, i∗(S) does

not contain 0; (2) ensures the injectivity of π ◦ i∗.

Consider the completion of S, denoted S, which is the closure of π ◦ i∗(S) in

P (RS+). The elements of S are represented by sequences {(tn, αn)}, tn > 0, αn ∈ S,

such that for all β in S, the sequence of real numbers tni(αn, β) converges.

Thus, within P (RS+), the setS has a nontrivial topology. Intuitively, we may give a

meaning to the notion that “two curves γ, γ′ are close to each other”. This ‘proximity’

has nothing to do with the respective homotopy classes of the curves, but with the

fact that, up to a multiple, in every region of the surface, γ and γ′ are more or less

made up of the same number of strands, going in more or less the same direction. All

of this will be discussed in greater detail in Expose 4.

We also need to introduce the space S′ of isotopy classes of simple, closed, not

necessarily connected curves on M , whose every component “belongs” to S; but two

distinct components of the same curve are allowed to be isotopic to each other, so

that we may consider scalar multiplication: for an integer n > 0 and γ ∈ S′, nγ is

represented by n parallel curves.

As before, we define i : S′ × S → Z+, and obtain the diagram

DR

AFT

13 J

un 2

008


ε

{ε

{

Figure 1.1: p-saddles, for p = 3, p = 4.

S′ i∗−→ RS+ \ 0

π−→ P (RS+).

Clearly, i∗ respects multiplication by scalars, hence π◦i∗ is not injective on S′. But one

may easily show that π ◦ i∗ admits S as its closure (see Expose 4). In the following,

we denote by R+ × S the cone on i∗(S) in RS+.

Theorem 1.2. If M is a closed orientable surface of genus g > 1, then S is homeomorphic to

S6g−7 (this is proved in Expose 4).

Let M2g,b = #

g(S1 × S1) − ⋃

b

D2. If χ(M2g,b) < 0, then S(M2

g,b) is homeomorphic to

S6g+2b−7 (see Expose 11). Lastly, S(T 2) ≃ S1, S(D2) = S(S2) = S(S1 × [0, 1]) = ∅.

1.3 Measured Foliations

For simplicity, M will be closed. A measured foliation onM is a foliation F with sin-measured foliation

gularities (of the type of a holomorphic quadratic differential zp−2dz2, p = 3, 4, . . . )

together with a transverse measure invariant under holonomy. In the neighborhood

of a nonsingular point, there exists a chart φ : U → R2x,y such that φ−1(y = constant)

consists of the leaves of F|U . If Ui ∩ Uj is nonempty, there exist transition functions

φij of the form

φij(x, y) = (hij(x, y), cij ± y)where cij is a constant. In these charts, the transverse measure is given by |dy|.

Remark. The foliations that admit transition functions of the form (f(x, y), c+ y) are

those that are defined by a closed 1-form ω; away from singularities, y is a local root

for ω.

The singularities of F are p-saddles (p ≥ 3) as in Figure 1.1.

If γ is a simple closed curve in M , we call∫γ F the total variationtotal variation

DR

AFT

13 J

un 2

008


Figure 1.2:

of the coordinate y of p ∈ γ as p traverses γ. For α ∈ S, define

I(F , α) = infγ∈α

∫

γ

F .

One says that F1 and F2 are Whitehead equivalent if one may be transformed Whitehead equivalent

to the other by isotopies and elementary deformations of the type suggested by Fig-

ure 1.2. (Observe that these deformations allow the transfer of the transverse mea-

sure.)Denote byMF the set of equivalence classes. Define

I∗ : MF → RS+

by

I∗(F)(α) = I(F , α).

One says that F1 and F2 are m-equivalent (or Schwartz equivalent) if I∗(F1) = m-equivalent

Schwartz equivalentI∗(F2). Schwartz equivalence is an immediate consequence of Whitehead equiva-

lence.

Theorem 1.3. The map I∗ injects MF into RS+; I∗(MF) ∪ 0 = R+ × S, and if g > 1,

this set is homeomorphic with R6g−6. In particular, Schwartz equivalence is the same thing

as Whitehead equivalence.

The proof of this theorem is dealt with in Exposes 5 and 6. What is more, since

I∗(MF) misses 0, the theorem says that in P (RS+) we have S = π ◦ I∗(MF). This

gives a nice geometric representation of the functionals in R+ × S.

1.4 Teichmuller space

We will consider a surface M with χ(M) < 0. Consider the spaceH of all metrics on

M with constant curvature K = −1, such that every component

DR

AFT

13 J

un 2

008


of the boundary of M is a geodesic. Let Diff0(M) be the group of diffeomorphisms

isotopic to the identity, with the C∞ topology. As we shall see later, this acts freely

and continuously onH. The orbit space under this action, equipped with the quotient

topology, is called the Teichmuller space T (M) = T . If M is orientable, there isTeichmuller space

another definition in terms of complex structures on M . The equivalence of the two

definitions is a consequence of the uniformization theorem [Wey55].

Remarks. Consider a fixed M , together with another surface Xρ = X with a hy-

perbolic metric ρ. If φ : M → X is a diffeomorphism, the pair (X,φ) is called a

“Teichmuller surface”. Two Teichmuller surfaces (X,φ), (X ′, φ′) are said to be equivalentTeichmuller surface

equivalent if there is an isometry f : X → X ′ such that φ and f ◦ φ′ are isotopic.

It is convenient to identify the points of T with equivalence classes of Teichmuller

surfaces.

We remark here that two diffeomorphisms ofM are homotopic if and only if they

are isotopic (see [Eps66]).

If M is closed, of genus g > 1, a classical theorem of Teichmuller theory asserts

that

T (M) ≃ R6g−6.

This result, due to Fricke and Klein, will be proven in Expose 7.

Further, we have

T (M2g,b) ≃ R6g−6+2b.

For all θ ∈ T and α ∈ S, we define

ℓ(θ, α) = infγ∈α

(θ(γ))

where θ(γ) denotes the length of γ computed in the metric of θ, which is prescribed

up to isotopy on M .

The metric being fixed, the infimum is attained for a unique geodesic. From the

above formula, we obtain the map

ℓ∗ : T → RS+;

it can be easily seen that the image of the map misses I∗(MF)∪0. The group π0(Diff(M))

acts on Teichmuller space as well as on S, and thus on RS+; the map ℓ∗ is clearly equiv-

ariant.

DR

AFT

13 J

un 2

008


In Expose 7, we prove the theorem below.

Theorem 1.4. The map ℓ∗ is a homeomorphism onto its image.

It is thus possible to put a natural topology on T ∪S; we consider the topological

space ℓ∗(T ) ∪ I∗(MF), in which the rays in I∗(MF) are identified to points, and we

take the quotient topology.

In Expose 8, we prove (for the case where M has no boundary) the following:

Theorem 1.5.

1. The topological space T ∪ S is homeomorphic to D6g−6, if M is closed, of genus

g > 1; it is homeomorphic to D6g−6+2b if M has Euler characteristic < 0 and b boundary

components.

2. The canonical map T ∪ S → P (RS+) is an embedding.

The space T ∪ S , denoted T , is the Thurston compactification of Teichmuller Thurston compactification

space. It follows immediately from the definitions that for any diffeomorphism φ of

M , the natural action of φ on T is continuous.

If φ is a diffeomorphism of M , and [φ] denotes the homeomorphism induced by

φ on T , then [φ] has a fixed point, by the Brouwer Fixed Point Theorem.

(i) If [φ] has a fixed point in T , then φ is isotopic to an isometry φ′ in a hyperbolic

metric; in particular, φ′ is periodic.

(ii) If [φ] fixes a point in S , there is a foliation F such that φ(F) is Whitehead

equivalent to λF , λ ∈ R+, where λF has the same underlying foliation as F , with a

transverse measure λ times that for F .

This cursory analysis will be made more precise in what follows.

1.5 Pseudo-Anosov Diffeomorphisms

We begin with a very elementary example. Let φ ∈ Diff+(T 2). Up to isotopy, φ is in

SL2(Z). There are three distinct possibilities for the eigenvalues λ1 and λ2 of φ, as

follows:

(a) λ1 and λ2 are complex (λ1 = λ2, λ1 6= λ2, |λ1| = |λ2| = 1). In this case, φ is of

finite order.

DR

AFT

13 J

un 2

008


(b) λ1 = λ2 = 1 (respectively, λ1 = λ2 = −1). Up to a change of coordinates,

φ =

(1 a

0 1

),

[respectively, φ =

(1 −a0 1

)];

which is a “Dehn twist”. In either case, φ leaves invariant a simple curve.Dehn twist

(c) λ1 and λ2 are distinct irrationals. Then φ is an Anosov diffeomorphism.Anosov

reference miss-

ing

This analysis is generalized by Thurston to any compact surface.

Theorem 1.6. Any diffeomorphism φ on M is isotopic to a diffeomorphism φ′ satisfying one

of the following three conditions:

(i) φ′ fixes an element of T and is of finite order.

(ii) φ′ is “reducible”, in the sense that it preserves a simple curve (representing an element

of S′); in this case, one pursues the analysis of φ′ by cutting M open along this curve.

(iii) There exists λ > 1 and two transverse measured foliations FS and FU such that

φ′(FS) =1

λFS ; and

φ′(FU ) = λFU .

These equalities indicate that the underlying foliations are equal.Aside from the obvious things, to say that FS and FU are transverse means that

their singularities are the same, and that in a neighbourhood of the singularities the

configuration is similar to that in Figure 1.3. A diffeomorphism that satisfies condi-

tion (iii) is called pseudo-Anosov.pseudo-Anosov

FU

FS

Figure 1.3: Pseudo-Anosov Singularities

Theorem 1.6 is proved in Expose 9. In order to apply this theorem inductively,

we need to extend the theory to the case with boundary. This is done in Expose 11.

DR

AFT

13 J

un 2

008


In Expose 12, we show that, for a pseudo-Anosov φ, FS and FU represent the

only fixed points of [φ] in T , and two homotopic pseudo-Anosov diffeomorphisms

are conjugate by a diffeomorphism isotopic to the identity.

The key to these theorems is the following “mixing” property that the pseudo-

Anosov diffeomorphism φ possesses: for all α, β ∈ S:

limn→∞

i(φnα, β)

λn= I(FS , α)I(FU , β)

1.5.1 Spectral properties of pseudo-Anosov diffeomorphisms

For θ ∈ T and α ∈ S, we defined in Section 1.4 the positive number ℓ(θ, α). Diffeo-

morphisms have eigenvalues in the following sense:

Theorem 1.7. Let φ ∈ Diff(M2). There exists a finite family of algebraic integers λ1, . . . , λk ≥1 such that, for every α ∈ S, there exists λj satisfying: for all θ ∈ T , we have

limn→∞

ℓ(θ, φnα)1/n = λj .

Furthermore, φ is pseudo-Anosov if and only if k = 1 and λ1 > 1; in this case λ1 = λ (see

Exposes 9 and 11).

1.5.2 Entropy

On a compact metric space X with a continuous map f : X → X , we may define the

topological entropy h(f) (see Expose 10). If φ is a pseudo-Anosov diffeomorphism, topological entropy

one proves that h(φ) = log(λ). Moreover, φ possesses an obvious invariant measure

and h(φ) is its metric entropy [Sin76]. metric entropy

Theorem 1.8. A pseudo-Anosov diffeomorphism minimizes the topological entropy in its

isotopy class.

The list of Thurston’s results is much longer, but we end this overview here to

come to the heart of the subject.

1.6 The case of the torus T 2

This case is particularly simple and is treated separately. On the torus T 2, consider

the three elements e1, e2, e3 of S(T 2), shown in Figure 1.4. Let these be oriented for

the time being.

Let x1 and x2 be the canonical generators e1 and e2 given the orientations shown

in Figure 1.4.

DR

AFT

13 J

un 2

008


3

e

e2

e

1

Figure 1.4: The torus T 2

If γ is a simple oriented curve, γ = mx1 + nx2, we find

i(e1, γ) = |n|, i(e2, γ) = |m|, i(e3, γ) = |n−m|.

These three numbers determine γ in S, but the first two are not sufficient. They

form a “degenerate triangle”, in the sense that any one of them is equal to the sum of

the other two.

We now consider the standard simplex with barycentric coordinates X1, X2, X3

(where Xi ≥ 0,∑Xi = 1). The simplex decomposes into the four regions indicated

in Figure 1.5.

X1

triangle inequalities X1 ≥ X2 +X3

X2 ≥ X1 +X3 X3 ≥ X1 +X2

X2 X3

Figure 1.5:

Let (≤ ∇) be the domain where the triangle inequality holds; the boundary ∂(≤∇) corresponds to degenerate triangles. We think of the standard simplex as being

in R3+, and we denote by cone(∂(≤ ∇)) the cone of half-lines that start at 0 and pass

through ∂(≤ ∇).

DR

AFT

13 J

un 2

008


To each γ ∈ S, we associate the numbers

Xj =i(ej , γ)

3∑

j=1

i(ej , γ)

, j = 1, 2, 3;

a simple calculation shows that we can thus identify with S the set of rational points

of ∂(≤ ∇).

Lemma 1.9. Let β ∈ S. There exists a continuous function

Φβ : cone(∂(≤ ∇))→ R+,

DR

AFT

13 J

un 2

008


homogeneous of degree 1 (for multiplication by positive scalars), such that, for all α ∈ S,

i(α, β) = Φβ (i(α, e1), i(α, e2), i(α, e3)) .

Proof. We can give an explicit construction as follows. Suppose that β is represented

by mx1 + nx2, n,m ∈ Z, (m,n) = 1. (The only ambiguity is that −mx1 − nx2 also

corresponds to β.) On the surface of the subcone X3 = X1 +X2, we set

Φβ(X1, X2, X3) =

∣∣∣∣det

(X2 −X1

m n

)∣∣∣∣

On the other two faces, we have

Φβ(X1, X2, X3) =

∣∣∣∣det

(X2 X1

m n

)∣∣∣∣

At the intersection of these faces, these formulas agree and Φβ has the stated property.

Remark. Φβ is piecewise linear, a property that we may recover from the other “ex-

plicit formulas” of the theory.

Consider now a sequence (λn, αn) with λn ∈ R+, αn ∈ S, such that, for all β ∈ S,

the sequence λni(αn, β) converges. Denote by lim(λn, αn) the functional

lim(λn, αn)(β) = limλni(αn, β).

Since Φβ is homogeneous, we have

lim(λn, αn)(β) = Φβ(lim(λn, αn)(e1), lim(λn, αn)(e2), lim(λn, αn)(e3)).

This implies that the bijection of R+ × S, regarded as part of RS+, onto the rational

rays of cone(∂(≤ ∇)) extends to a homogeneous homeomorphism:

R+ × S ≃ cone(∂(≤ ∇)) ≃ R2

Thus, in P (RS+), we have S ≃ S1.

Consider a measured foliation F of T 2. One can show that F has no singularities

and that it is transversely orientable (this is a consequence of a simple Euler–Poincare

type formula); with this measured foliation we identify a closed nonsingular 1-form.

This form is then isotopic to a unique linear form (a 1-form with constant coefficients

in the canonical coordinates

DR

AFT

13 J

un 2

008


on T 2) [Ste69].

If ω is linear, every curve γ = mx1 + nx2 is transverse to ω, or else contained in a

leaf; thus ∣∣∣∣∫

γ

ω

∣∣∣∣2

= I(ω, γ);

ω is determined up to sign by I(ω, e1), I(ω, e2), and I(ω, e2). Lemma 1.10 is now clear:

Lemma 1.10. Let F be a measured foliation on T 2.

1. I(F , e1), I(F , e2), and I(F , e3) form a degenerate triangle.

2. If β ∈ S, we have:

I(F , β) = Φβ(I(F , e1), I(F , e2), I(F , e3))

where Φβ is the function from Lemma 1.9.

The first point is clear from Figure 1.6

2

1e

ee

3

I(F , e3) =

I(F , e1) + I(F , e2)I(F , e2) =

I(F , e1) + I(F , e3)I(F , e1) =

I(F , e2) + I(F , e3)

Figure 1.6: Proof of Lemma 1.10(1)

As a consequence, in P (RS+), we have π ◦ I∗(MF) = S.

In Section 1.4, we defined Teichmuller space in the context of χ < 0. For T 2, one

may give an analogous definition, by considering the flat metrics (K = 0) such that

Area(T 2) = 1. [This normalization condition is useless in the hyperbolic case, where

the volume of an object is determined by its shape.]

Remark 1. Instead of this normalization, one may work with flat metrics up to scal-

ing.

DR

AFT

13 J

un 2

008


On the other hand, if T 2 is given a complex structure, its universal cover T 2 is

isomorphic to C and the group of automorphisms of C, z 7→ αz + β, α, β ∈ C, coin-

cides with the group of orientation preserving maps of R2 preserving the

DR

AFT

13 J

un 2

008


Euclidean metric up to a scalar. From this, one easily deduces the equivalence of our

definition of T with the classical definition: “the set of complex marked structures on

T 2, up to isotopy.”

Remark 2. A flat structure on T 2 has an underlying affine structure. If we fix two gen-

erators e1, e2, for π1(T2), the affine structure underlying the metric ρ is determined

in the following way: the data of all the geodesics of the class ei, which are parallel

closed curves, as well as the numbers

dist(∆

∆′ )

/dist(

∆′

∆′′ ) ∈ R+

corresponding to three geodesics of the ei system. It is easy to see that all these affine

structures on T 2 are isotopic to each other. Thus we may always represent an element

of T by a flat metric ρ whose underlying affine structure is the canonical structure

(this choice will always be made in what follows). In other words, the usual straight

lines are the geodesics for ρ.

To ρ ∈ T , we may associate (X1, X2, X3), Xj = ρ(ej)/∑k ρ(ek), where ρ(ej) is

the length of the geodesic ej in the metric ρ.

Lemma 1.11. The above map is a homeomorphism T → int(≤ ∇).

Proof. It is clear that (X1, X2, X3) satisfies the triangle inequality. Let ∆ be a trian-

gle in R2; every assignment of lengths to the sides satisfying the triangle inequality

determines a flat metric on R2 compatible with the affine structure; this is invariant

under the group of translations, hence induces a metric on T 2. This shows surjectiv-

ity. For injectivity, we note that two flat metrics with standard affine structures giving

the same lengths to the sides of ∆ are identical. The topology is left for the reader.

In other words, the composition

T ℓ∗−→RS+

proj.−→ R(e1,e2,e3)+

is a homeomorphism of T onto its image. To see that ℓ∗ is a homeomorphism onto

its image, note that the length of a given line segment depends continuously on the

lengths assigned to e1, e2, e3 (classical trigonometry!)

We have

ℓ∗(T )⋂I∗(MF) = ∅.

Indeed, if ω is a differential form, there exists a sequence γn of simple closed curves

such that∫γnω → 0; if αn denotes the class of γn in S, we have

DR

AFT

13 J

un 2

008


I∗(ω)(αn) → 0, while for a given metric the lengths of the closed geodesics do not

approach zero.

To prove the analogue of Theorem 1.5 for the torus T 2, it remains to prove the

following lemma.

Lemma 1.12. Let ρn be a sequence of flat metrics (normalized to the canonical affine struc-

ture), λn a sequence of positive reals, and ω a linear form. Suppose that, for j = 1, 2, 3,

λnρn(ej)→∣∣∣∣∣

∫

ej

ω

∣∣∣∣∣ .

Then for all closed geodesics α,

λnρn(α)→∣∣∣∣∫

α

ω

∣∣∣∣ .

Proof. Let ρ′n denote the metric λnρn. We treat the case where ω is on the face X3 =

X1 +X2 of cone(∂(≤ ∇)) (Figure 1.7) and∫eiω 6= 0 for i = 1, 2.

2

1

e3

e

e

Figure 1.7:

We orient ej , j = 1, 2, 3, so that∫ejω ≥ 0. Now let θn be the measure of the angle

between e1 and e2 in the metric ρ′n.We have:

[ρ′n(e3)]2 = [ρ′n(e1)]

2 + [ρ′n(e2)]2 + 2ρ′n(e1)ρ

′n(e2) cos θn.

The hypothesis then implies that cos θn tends to 1. If α is a linear segment, α = a1e1 +

a2e2, where a1, a2 ∈ R, we have:

[ρ′n(α)]2 = a21[ρ

′n(e1)]

2 + a22[ρ

′n(e2)]

2 + 2a1a2ρ′n(e1)ρ

′n(e2) cos θn.

Thus,

[ρ′n(α)]2 →[a1

∫

e1

ω + a2

∫

e2

ω

]2=

[∫

α

ω

]2

DR

AFT

13 J

un 2

008


For T 2 the analysis of Theorem 1.6 is trivial. As for Theorem 1.7, it reduces in the

case of the torus to a spectral property well known in linear algebra.

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008

Expose 2

Some reminders on the theory of

surface diffeomorphisms

2.1 The space of functionals

Let M = M2 be a compact, connected manifold of dimension 2. We will consider the

group of diffeomorphisms of M2, denoted Diff(M2). If A ⊂ M2, we will denote by

G(M,A) the space of homotopy equivalences Mf−→M , such that f |A = id, with the

topology of uniform convergence.

Theorem 2.1 (Smale). Diff(D2, rel ∂D2) is contractible (indicated as Diff(D2, rel∂D2) ≃∗).

For a proof, see [Cer68], [Sma59].

Theorem 2.2 ([Cer68]). The following natural inclusions are homotopy equivalences:

O(3) → Diff(S2) → G(S2)

SO(3) → Diff(RP 2) → G(RP 2)

is this in the er-

rata?These cases being settled, we may assume that M is a K(π1(M), 1); We consider

∗ ∈M and the fibration:G(M, ∗) → G(M)yev(∗)

M

By standard methods of obstruction theory, one proves the following theorem.

19

DR

AFT

13 J

un 2

008


Theorem 2.3.

πi(G(M, ∗)) =

{Aut(π1(M, ∗)) if i = 0,

0 if i > 0

Therefore, the homotopy exact sequence of our fibration reduces to

1→ π1(G(M))→ π1(M)∂−→Aut(π1(M))→ π0(G(M))→ 1

One may easily verify the following facts:

1) If x ∈ π1(M), then ∂(x) is the inner automorphism corresponding to x.

2) π1(G(M)) is the center of π1(M). This group is trivial except in the following

exceptional cases: the torus, π1(G(T 2)) = Z⊕ Z; the Klein bottle, π1(G(K2)) = Z.

3) π0(G(M)) = Aut(π1(M))/Inn(π1(M)), where Inn(π1(M)) is the group of inner

automorphisms of π1(M).

2.2 The braid groups and their computation

See [Bir75] for more details.

Let X be a topological space, n a positive integer and Pn(X) = Xn−∆, where ∆

is the “big diagonal” of Xn, which is the set of n-tuples (x1, . . . , xn) of points of X ,big diagonal

such that for some i 6= j, xi = xj . The symmetric group Sym(n) acts freely on Pn(X)

and by definition, Bn(X) = Pn(X)/Sym(n). One thus has a regular covering

Sym(n) −→ Pn(X)yBn(X).

By definition, π1(Pn(X)) = “the pure braid group ofX on n strands”, and π1(Bn(X)) =

“the braid group of X on n strands”.

Henceforth, X = R2, and we write:

π1(Pn(R2)) = Pn (the pure braid group on nstrands)

π1(Bn(R2)) = Bn (the braid group on n strands)

We have an obvious exact sequence:

1→ Pn → Bn → Sym(n)→ 1.

An element of Bn may be represented in the following manner: fix once

DR

AFT

13 J

un 2

008


Figure 2.1: A element of Bn.

and for all a set of n distinct points x1, . . . , xn in intD2. Then an element of Bn is a

system of arcs in D2 × I , going from (x1, . . . , xn) × 0 to (x1, . . . , xn) × 1, transverse

to every horizontal slice D2 × t. The arcs do not meet ∂D2 × I , and everything is

defined up to isotopy (leaving invariant the boundary of the cylinder and respecting

the projection D2 × I → I).

With this representation, the law of composition in Bn is the same as for cobor-

disms and the pure braids are those for which the arc leaving xi × 0 ends at xi × 1.

Figure 2.1 represents an element of Bn.

Theorem 2.4 (Fadell–Neuwirth). The map Pn(R2) → Pn−1(R2) that “forgets” xn is a

fibration with fibre R2 − (n− 1 points).

Corollary 2.5.

Pn(R2) ≃ K(Pn, 1)

Bn(R2) ≃ K(Bn, 1)

Remark. The theorem of Fadell–Neuwirth gives us a short exact sequence

1→ Fn−1 → Pn → Pn−1 → 1

which is split, so Pn is determined by Pn−1 and the action of Pn−1 on the free group

Fn−1.

We will now give a presentation of the group Bn. In R2, consider the coordinates

(x, y) and, for p = (p1, . . . , pn) ∈ Bn(R2), arrange the indices so that

x(p1) ≤ x(p2) ≤ · · · ≤ x(pn).

By definition, Mi ⊂ Bn(R2) is the set of p such that x(pi) = x(pi+1), as in Figure 2.2.

DR

AFT

13 J

un 2

008


i+1

pi−1

pi

pi+2

p

Figure 2.2: Transverse orientation of Mi −⋃j 6=i

Mj

We note the following things:

(1) Mi −⋃j 6=i

Mj is a submanifold of codimension 1 of Bn(R2) endowed with a

canonical transverse orientation, defined as in Figure 2.2. If the numbering is such

that y(pi+1) > y(pi), a displacement of pi+1 along the positive normal pushes pi+1

until x(pi+1) > x(pi).

(2) Mi −⋃j 6=i

Mj is connected.

(3) Bn(R2)−⋃

i

Mi is contractible.

These remarks imply that the simple loops ai, based in Bn(R2)−⋃

i

Mi and such

that ai crosses Mi exactly once (and does not cross any other stratum), in the positive

direction, generateBn. One may find the relations among the ai by considering what

happens in a neighbourhood of the stratum of codimension 2, where Mi and Mj

meet.

Case 1. |i− j| ≥ 2. At the level of R2, a point of Mi ∩Mj is as in Figure 2.3. One

may move independently along the dashed horizontal arrows, which give us a small

square, transverse to Mi ∩Mj in Bn(R2), as shown in Figure 2.4. We have drawn the

correct orientations of the transversals to the strata Mi and Mj .

j+1

p

p p

p

i+1

i j

j

Mi

Mj

ai

aj

ai

a

Figure 2.3: Figure 2.4:

This gives us the relation aiaj = ajai.

DR

AFT

13 J

un 2

008


Case 2. |i − j| = 1. At the level of R2, we have Figure 2.5, and at the level of

Bn(R2), Figure 2.6, from which we may read off the relation:

aiai+1ai = ai+1aiai+1.

II

I

II

i

ai+1

ai ai+1

ai+1

ai

I

a

Figure 2.5: Figure 2.6:

We therefore have found the following theorem.

Theorem 2.6 (E. Artin). Bn admits generators a1, a2, . . . , an−1 and the relations:

[ai, aj] = 1 (|i− j| > 1)

aiai+1ai = ai+1aiai+1

Corollary 2.7. B3 = π1(S3 − the trefoil knot).

[The explanation of this “coincidence” is this: Bn(R2) may be identified with the

set of complex monic polynomials of degree n, having distinct roots. Thus Bn =

π1(Cn − the discriminant locus) . . . ].

The generator ai is the following braid:

1 2 · · · i i+ 1 · · · n

DR

AFT

13 J

un 2

008


and the relation aiai+1ai = ai+1aiai+1 may be visualized as follows:

i i+ 1 i+ 2 i i+ 1 i+ 2

aiai+1ai ai+1aiai+1

In particular, B2 ≈ Z and the generator a1 is

. Similarly, P2 ≈ Z is generated by

and the natural inclusion P2 → B2 is multiplication by 2: Z×2−→Z.

2.3 Diffeomorphisms of the two holed disk

(The spaces A(P 2), A′(P 2))

Let K ⊂ intD2 be a finite set of cardinality k. We introduce the following notation:

Diff(D2, rel(K, ∂)) =

{diffeomorphismsD2 ϕ−→D2, such that ϕ|K∪∂D2 = id},Diff(D2,K, rel ∂) =

{diffeomorphismsD2 ψ−→D2, such that ψ(K) = K,ψ|∂D2 = id}.

DR

AFT

13 J

un 2

008


We have a natural action of Diff(D2, rel ∂) on Bk(int D2), and on Pk(int D2),

which gives us two fibrations:

Diff(D2,K, rel ∂) → Diff(D2, rel ∂)→ Bk(int D2)

and

Diff(D2, rel(K, ∂)) → Diff(D2, rel ∂)→ Pk(int D2).

Applying the theorem of Smale:

Diff(D2, rel ∂) ≃ ∗,

and we have the following corollary.

Corollary 2.8.

1) Each connected component of Diff(D2, rel(K, ∂)) and of Diff(D2,K, rel∂) is contractible.

2) We have the canonical isomorphisms:

Pk = π0(Diff(D2, rel(K, ∂))), (2.1)

Bk = π0(Diff(D2,K, rel ∂)). (2.2)

We will now consider the manifold (with boundary) P 2, which is the disk with

two holes, or the “pair of pants” (see Figure 2.7.)

∂3P²

∂2 P²

∂1P²

Figure 2.7: The “pair of pants” P 2

fix figure 2.7

Remark. Let Diff(P 2, ∂2, ∂3, rel ∂1) =

{ϕ ∈ Diff(P 2) : ϕ|∂1P 2 = id, ϕ(∂2P2) = ∂2P

2, ϕ(∂3P2) = ∂3P

2}.

Then Diff(P 2, ∂2, ∂3, rel ∂1) is manifestly of the same homotopy type as Diff(P 2, rel(K, ∂)).

DR

AFT

13 J

un 2

008


Proposition 2.9. π0(Diff(P 2, rel ∂)) = Z⊕ Z⊕ Z.

Proof. Considering the 1-jets of the diffeomorphisms at the two points ofK , we have

a fibrationDiff(P 2, rel ∂P 2) → Diff(D2, rel(K, ∂))y

S1 × S1

DR

AFT

13 J

un 2

008


from which we get the exact sequence:

0→ π1(S1 × S1)→ π0(Diff(P 2, rel ∂P 2))→ P2 → 0.

One may verify that this sequence splits, that the extension is central, and that the

action of P2 on π1(S1 × S1) is trivial, which gives the stated result.

We now consider

Diff+(P 2, ∂1, ∂2, ∂3) =

{orientation preserving diffeomorphisms ϕ : P 2 → P 2such that ϕ(∂iP2) = ∂iP

2}.

Proposition 2.10. Diff+(P 2, ∂1, ∂2, ∂3) is contractible.

Proof. By restriction of an element ϕ ∈ Diff+(P 2, ∂1, ∂2, ∂3) to ∂1P2 = ∂D2, we have

a fibration:Diff(P 2, ∂2, ∂3, rel ∂1)︸︷︷︸ → Diff+(P 2, ∂1, ∂2, ∂3)

P2 = K(Z, 0)yrestriction

Diff+S1 = K(Z, 1)

One verifies also that the arrow

π1(Diff+S1)

∂−→π0(Diff(P 2, ∂2, ∂3, rel ∂1)) = P2

is an isomorphism, which gives the result.

Now letN2 be a compact surface, with (unspecified) nonempty boundary. Define

A(N2) = the set of isotopy classes of arcs I ⊂ N2, with ∂I ⊂ ∂N2, each end free to

move on the respective connected component of ∂N2, and representing the nontrivial

elements of π1(N2, ∂N2), and let

A′(N2) = {the same as above but with several pairwise disjoint arcs}.

Corollary 2.11. A(P 2) consists of exactly six elements, classified by the connected compo-

nents of ∂P 2 in which the endpoints of the respective arcs fall.

Proof. Let τ and τ ′ be two representatives of elements of A(P 2) with their endpoints

in the same connected component of ∂P 2. We may easily check that there is an ori-

entation preserving diffeomorphism

(P 2, τ)ψ−→(P 2, τ ′).

Since π0(Diff+(P 2, ∂1, ∂2, ∂3)) = 0, this diffeomorphism is isotopic to the identity,

which

DR

AFT

13 J

un 2

008


gives the result. The six models are given in Figure 2.8.

2∂

1P

∂2P

2

∂3P

2

Figure 2.8: The six models for A(P 2).

Now let A′ be the set of ordered triples (a1, a2, a3), where ai ≥ 0, ai ∈ Z+, and∑i ai ≡ 0 (mod 2). If τ ∈ A′(P 2), we associate to it

i(τ) = (i(τ, ∂1), i(τ, ∂2), i(τ, ∂3)) ∈ A′

where i(τ, γ) is the number of points τ has in common with γ. For convenience, we

adjoin ∅ ∈ A′(P 2), with i(∅) = (0, 0, 0).

Theorem 2.12. The map A′(P 2)i−→A′ is a bijection.

Proof. We begin by constructing a map A′ τ−→A′(P 2) such that i(τ(a1, a2, a3)) =

(a1, a2, a3). If (a1, a2, a3) 6= 0, then the point with barycentric coordinates(

a1∑ai,a2∑ai,a3∑ai

)

falls in one of the four regions of Figure 2.9.

X1

a1 ≥ a2 + a3 region of triangle inequality

a2 ≥ a1 + a3 a3 ≥ a1 + a2

(≤∇)

X2 X3

Figure 2.9:

DR

AFT

13 J

un 2

008


If (a1, a2, a3) satisfies the triangle inequalities, we consider the nonnegative integers

x12 =1

2(a1 + a2 − a3), x23 =

1

2(a2 + a3 − a1), x31 =

1

2(a3 + a1 − a2)

and we define τ(a1, a2, a3) to be the element of A′(P 2) which consists of xij = xjisegments of the type τij , for i 6= j.

If a1 ≥ a2 + a3, we set

x11 =1

2(a1 − a2 − a3), x12 = a2, x13 = a3,

and we define τ(a1, a2, a3) as in Figure 2.10.

∂

∂

∂

x12

x11

x13

1

32

Figure 2.10:

The other cases are treated in a similar manner. One may verify that on ∂(≤ ∇)

the different definitions agree and that i ◦ τ is the identity. Thus i is surjective.

We now remark that the compatible pairs of elements of A(P 2) are exactly those

that are joined by a segment in Figure 2.11.

The four triangles in Figure 2.11 correspond canonically with the four triangles of

Figure 2.9. More precisely, let xαβ be the number of segments of type ταβ that appear

in τ ∈ A′(P 2). We have the following four mutually exclusive situations:

DR

AFT

13 J

un 2

008


23 τ

τ 12

τ 11τ 13

22

τ 33

τ

Figure 2.11:

1) xαα = 0 for α = 1, 2, 3, which implies that i(τ) ∈ (≤ ∇).

2) x11 6= 0, which implies that a1 > a2 + a3.

..............

Suppose now that τ1, τ2 ∈ A′(P 2) and that i(τ1) = i(τ2) ∈ A′. We have previously

deduced that τ1 and τ2 are in the same one of the four situations described above;

by a linear algebra calculation on the a1, a2, a3 that are (by definition) the same for

τ1 and τ2, we conclude that the xαβ are also the same. We still have to prove that

if τ1, τ2 ∈ A′(P 2) are such that all their xαβ are equal, then τ1 = τ2. This is already

proven if∑

α≤β xαβ = 1. The proof of the general case is an induction on∑

α≤βxαβ .

We leave the details to the reader. We have thus proven that i is injective.

Remark. Let τ ∈ A′(P 2). There does not exist a nontrivial element of π0(Diff+(P 2, ∂1, ∂2, ∂3, τ)).

In particular, for a given τ , one may not interchange the different connected compo-

nents of τ among themselves.

DR

AFT

13 J

un 2

008

Expose 3

Reminders on hyperbolic

geometry in dimension 2

and generalities on

i : S × S → R+

by V. Poenaru

3.1 A little hyperbolic geometry

Consider a compact surface M , with a Riemannian metric of curvature −1, whose

boundary, if nonempty, is geodesic. The universal cover M is isometric to a domain

in the hyperbolic plane H2, possibly bounded by geodesics in H2.

Lemma 3.1. Let α and β be distinct geodesic arcs in M , with the same endpoints. Then the

closed curve α ∪ β is not homotopic to a point.

Proof. If α ∪ β were homotopic to a point, it would lift to a closed curve in M . But

two distinct geodesics in H2 cannot meet in more than one point.

31

DR

AFT

13 J

un 2

008


This property of H2 follows for example from the Gauss–Bonnet formula: for a disk

D with a Riemannian metric so that the boundary is a geodesic polygon, we always

have ∫ ∫

D

K = 2π −∑

(exterior angles),

where K denotes the curvature.

Lemma 3.2. Let V be a compact Riemannian manifold with totally geodesic boundary. In

every (free) homotopy class of maps S1 → V there is a geodesic immersion whose length is a

lower bound for the length of any loop in its homotopy class.

Proof. We take a homotopy class α ∈ [S1, V ], a number ǫ > 0, and an integer N ; we

set L = Nǫ. We choose ǫ to be smaller than the injectivity radius of the exponential

map and N large enough so that α contains at least one curve of length ≤ L.

Let I(α, ǫ,N) be the space of continuous maps S1 →M in the class α, composed

of at least N geodesic arcs of length ≤ ǫ each. This space, with the compact-open

topology, is compact. The length function y is continuous. Let φ be a curve that realizes

the minimum length in I(α, ǫ,N). It is easy to check that φ is in fact smooth (if ∂V 6= ∅,the hypothesis that ∂V is totally geodesic intervenes here).

To see that the length of φ is a lower bound for the class α, it suffices to remark

that, if C is a rectifiable curve in α, of length ≤ L, there exists a curve belonging to

I(α, ǫ,N) of length less than or equal to that of C.

Remark. Without compactness, with only the hypothesis that the metric is complete,

we see that each element of π1(V, x0) can be realized by a closed geodesic that, in

general, is not smooth at x0.

Lemma 3.3. For every nontrivial covering transformation T of M over M , there exists a

unique geodesic invariant under T . It is the lift of the closed smooth geodesic in M that is in

the free homotopy class given by the element α of π1(M,x0) corresponding to T .

Proof.

1) Existence. Here is a proof that does not use the curvature assumption.

DR

AFT

13 J

un 2

008


We take as a model for M the set of continuous paths {φ : [0, 1] → M | φ(0) = x0}subject to the relation of homotopy with endpoints fixed. The projection p : M → M

is given by φ 7→ φ(1). The constant path defines the basepoint of M . Let ψ ∈ M with

p(ψ) = y, and let χ be a path in M such that χ(0) = y; the lift of χ in M starting from

ψ is a one parameter family of paths in M , obtained by truncating the path ψ ∗ χ;

this family begins with ψ and ends with ψ ∗ χ itself. The left action of π1(M,x0) on

M is defined as follows: for α ∈ π1(M,x0), which we represent by a loop φ, and for

ψ ∈ M , we set Tα(ψ) = φ ∗ ψ.

Now, consider the element α for which T = Tα. By Lemma 3.2, its free homotopy

class contains a smooth closed geodesic g1. Let x1 be a point of the image of g1 and λ

a path joining x0 to x1; this is chosen so that λ ∗ g1 ∗ λ−1 belongs to α. If λ ∗ g1 is the

lift of λ ∗ g1 starting from the base point of M , we have widetilde not

wide enough

in following

display

λ ∗ g1(1) = ˜λ ∗ g1 ∗ λ−1 ∗ λ(1) = Tα(λ(1)).

Then, if we take in M the image of λ ∗ g1 and all of its translates by Tαn , n ∈ Z, we

construct a connected component of p−1(λ ∗ g1), consisting of a geodesic g of M and

of segments lifting λ, as in Figure 3.1. By construction, g is invariant under Tα.

Figure 3.1:

A second proof of existence that utilizes the fact thatM is a compact surface with

a hyperbolic structure is the following.

The transformation Tα is an isometry of H2. As Tα does not have any fixed points,

DR

AFT

13 J

un 2

008


it cannot be elliptic. On the other hand, if Tα is a parabolic isometry of H2 (having

a unique fixed point on the circle at infinity), then for all ǫ > 0 there is an x ∈ H2

such that d(x, Tα(x)) < ǫ. This implies the existence of closed geodesics of arbitrarily

short length on M , which is forbidden by compactness. Hence Tα is hyperbolic (two

fixed points on the circle at infinity); the geodesic g of H2, which joins them, is hence

invariant under Tα. Thus g/Tα is a smooth closed geodesic in the same free homotopy

class as α.

2) Uniqueness. Let g1 and g2 be two distinct geodesics in M , invariant under T .

If g1 ∩ g2 is nonempty, the intersection consists of a unique point, which must be

invariant under T ; but this is impossible.

Hence g1 ∩ g2 = ∅. Let x ∈ g1; at x, we drop a perpendicular to g2; we denote

this geodesic segment by δ. We note that Tδ ∩ δ = ∅, otherwise we have a geodesic

triangle where the sum of the (interior) angles is > π.

Now g1, g2, δ, and T (δ) form a quadrilateral in which the sum of the interior

angles is 2π (see Figure 3.2), but this is impossible by the Gauss–Bonnet formula (or

by elementary reasoning).

Figure 3.2:

Lemma 3.4. Let α be a nontrivial element of π1(M,x0). Then there exists a unique smooth

closed geodesic in the homotopy class of α.

Proof. Existence is already ensured by Lemma 3.2. Suppose that g1 and g′1 are two

such geodesics. The “existence” part of the preceding proof provides two distinct

geodesics in M , invariant under Tα.

But the “uniqueness” part of the preceding lemma tells us precisely that this is

impossible (use that π1(M,x0) does not have torsion).

DR

AFT

13 J

un 2

008


3.2 The Teichmuller space of the two-holed disk

The pair of pants P 2 (the two-holed disk) is the fundamental “building block” in the

theory of surfaces. We recall from Expose 2 that Diff+(P2, ∂1, ∂2, ∂3) is contractible;

in particular a diffeomorphism, that preserves orientation and sends each boundary

component to itself, is isotopic to the identity.

∂3P²

∂2 P²

∂1P²

Figure 3.3: The pair of pants P 2

If ρ is a metric of curvature −1 on P 2, for which every boundary component is

geodesic, we say that (P 2, ρ) is a P 2-Teichmuller surface. By definition, two surfaces

(P 2, ρ) and (P 2, ρ′) are equivalent if there is a diffeomorphism φ of P 2, isotopic to

the identity, such that φ∗ρ = ρ′. Since Diff+(P2, ∂1, ∂2, ∂3) is connected, the set of

equivalence classes—which by definition is the Teichmuller space T (P 2) of P 2—is T (P2)

identified with the quotient H(P 2)/Diff+, where H(P 2) is the space of Riemannian

metrics of curvature−1 for which the boundary is geodesic:

T (P 2) = H(P 2)/Diff+.

We endow H(P 2) with the C∞ topology and T (P 2) with the quotient topology.

There is a natural continuous map

L : H(P 2)→ (R∗+)3 = { triples of numbers > 0}

defined by:

L(ρ) = (ℓρ(∂1P2), ℓρ(∂2P

2), ℓρ(∂3P2)),

DR

AFT

13 J

un 2

008


where ℓρ denotes the length in the metric ρ. This induces a map which we denote by

the same letter:

L : T (P 2)→ (R∗+)3.

Theorem 3.5. The mapL : T (P 2)→ (R∗+)3 is a homeomorphism. Moreover,L : H(P 2)→

(R∗+)3 admits continuous local sections.

The classification of the P 2-Teichmuller surfaces reduces to the classification of

right hyperbolic hexagons because a hyperbolic pair of pants may be obtained by

gluing two isometric hexagons as indicated in Lemma 3.7. In addition, an “abstract”

hyperbolic hexagonX , where every angle is right and where each boundary compo-

nent is geodesic, is isometric to a hexagon in the hyperbolic plane H2; to see this, we

use X as a fundamental domain, and use symmetries about the sides of X to con-

struct a complete, simply connected hyperbolic manifold Y ; by a classical theorem

(Hadamard-Cartan, [CE75]), Y is isometric to H2. We are interested therefore on the

set Hex of (outright) isometry classes of hexagons in H2, where the angles are right,

the sides are geodesic, and one vertex is distinguished. We write a1, b1, a2, b2, a3, b3for the sides, starting from the base vertex and turning clockwise (see Figure 3.4).

Figure 3.4:

Lemma 3.6. The lengths ℓ(a1), ℓ(a2), and ℓ(a3) establish a bijection from Hex to (R∗+)3.

Proof. 1) Existence. Let ℓ1, ℓ2, ℓ3 > 0. We want to construct a hexagon X in H2 such

that ℓ(ai) = ℓi for i = 1, 2, 3.

We start by fixing three geodesics G,G′, G′′ as in Figure 3.5; G and G′′ are a dis-

tance ℓ1 apart. Let x ∈ G and let Lx be the perpendicular to G starting at x; if x is

sufficiently far from x0, then Lx never meets G′′ (we suggest that the reader sketch

the picture in the Poincare model). Let x(ℓ1) be the point of G closest to x0 that satis-

fies

DR

AFT

13 J

un 2

008


Lx(ℓ1) ∩G′′ = ∅.We set f(ℓ1) = d(x0, x(ℓ1)).

Figure 3.5:

We perform the construction in Figure 3.6, which is determined up to isotopy by

the numbers ℓ1, ℓ3, and λ.

Figure 3.6:

Let µ(λ) be the distance from G′′1 to G′′

2 ; this is a continuous function of the

DR

AFT

13 J

un 2

008


length λ, with µ(0) = 0 and µ(+∞) = +∞ (to vary λ, we utilize the fact that there

exists a one parameter group of isometries of H2, leaving G invariant); µ takes every

positive value. This proves the existence of X .

2) Uniqueness. As we have just seen, the data of three consecutive sides of a

hexagon determines it completely.

Thus, if the right hexagons X and X ′ in Figure 3.2 satisfy ℓi = ℓ(ai) = ℓ(a′i) and

are not isometric, then the lengths ℓ(b3) and ℓ(b′3) are not equal; say that ℓ(b′3) > ℓ(b3).

X

a1

a2

a3

b1 b2

b3

X

a′1

a′2

a′3

b′1 b′2

b′3

ℓi = ℓ(ai) = ℓ(a′i)

Figure 3.7:

It is a small exercise in hyperbolic geometry to see that there exists a (unique)

perpendicular from b3 to a2 inX . This decomposes the lengths of b3 and a2 as shown

in Figure 3.2: ℓ(b3) = α+ β; ℓ(a2) = γ + δ.

DR

AFT

13 J

un 2

008


α β

γ δ

a1a3

b1 b2

Figure 3.8:

InX ′, we erect perpendiculars to b′3 at distances α and β from the two endpoints,

as in Figure 3.2. In this figure, all of the angles marked by a box are equal to π/2; the

others are not necessarily right.

Figure 3.2 gives a contradiction, since we have γ + δ > γ + δ.

DR

AFT

13 J

un 2

008


α β

γ + δ

δ

a′1a′3

Figure 3.9:

Remark. (1) The uniqueness which we have just proven may be interpreted in the

following way: ℓ(a1) and ℓ(a2) being fixed, the function ℓ(b3) → ℓ(a2) is monotone;

or, the function λ 7→ µ(λ) (Figure 3.6) is a homeomorphism of R+.

(2) Referring to the notation of Figure 3.4, we may parametrize the set Hex by

(ℓ(a1), ℓ(a2), ℓ(a3)) or by (ℓ(b1), ℓ(b2), ℓ(b3)). The transition from one set of coordinates to

DR

AFT

13 J

un 2

008


the other is by means of a homeomorphism of (R∗+)3. [Indeed, we have just seen the tran-

sition from (ℓ(a1), ℓ(a2), ℓ(a3)) to (ℓ(a1), ℓ(b1), ℓ(a3)) is done by a homeomorphism of

(R∗+)3. Then, we may easily verify that the same thing is true for the transition from

(ℓ(a1), ℓ(b3), ℓ(a3)) to (ℓ(b3), ℓ(a2), ℓ(b2)), etc . . . ]

(3) In Figure 3.6, we see that, ℓ1 = ℓ(a1) and ℓ3 = ℓ(a3) being fixed, if µ = ℓ(a2)

tends to 0, then ℓ(b1) and ℓ(b2) tend to +∞.

The classification of right hexagons leads to a classification of pairs of pants, be-

cause every P 2-Teichmuller surface is the double of a hexagon, as indicated precisely

in the statement of Lemma 3.7.

Lemma 3.7. Suppose a P 2-Teichmuller surface is given.

(1) There exists a unique simple geodesic gij of P 2 that joins ∂iP2 to ∂jP

2 and that is

perpendicular to both of them. The arcs g12, g13, and g23 are mutually disjoint (Figure 3.2).

(2) The endpoints of g12 and g13 cut ∂1P2 into segments of equal length. The same is

true for ∂2P2 and ∂3P

2.

g12 g13g23

∂1P2

∂2P2

∂3P2

Figure 3.10:

DR

AFT

13 J

un 2

008


Proof. (1) A path of shortest length joining ∂iP2 to ∂jP

2 meets the boundary at right

angles at its endpoints (the first variation formula [CE75]). We deduce right away

that it is a simple arc. For uniqueness, we remark that the homotopy class is fixed by

the condition that the path be simple; by an argument using negative curvature as in

Lemma 3.3, we obtain (1).

(2) The arcs g12, g13, and g23 cut P 2 into two right hexagons. These are isometric

since they have three equal sides.

Proof of Theorem 3.5

1) Surjectivity. Given ℓ1, ℓ2, ℓ3 > 0, we may construct a unique right hexagon X

with ℓ(ai) = ℓi/2 for i = 1, 2, 3 (Lemma 3.6). To form the pair of pants, we take two

copies of X which we glue together along b1, b2, and b3. Thus, we have ℓ(∂iP2) =

2ℓ(ai) = ℓi. This shows the surjectivity of L.

2) Uniqueness. Let ρ′, ρ′′ ∈ H(P 2), such that ℓi = ℓρ′(∂iP2) = ℓρ′′(∂iP

2), for i =

1, 2, 3. We are going to prove that there exists f ∈ Diff+(P2, ∂1, ∂2, ∂3) that takes ρ′ to

ρ′′.

By Lemma 3.7, (P 2, ρ′) = X ′1∪X ′

2 and (P 2, ρ′′) = X ′′1 ∪X ′′

2 , whereX ′1, X

′2, X

′′1 , X

′′2

are right hexagons, parametrized by (ℓ1/2, ℓ2/2, ℓ3/2). Hence, there exists a direct

isometry of X ′1 to X ′′

1 and X ′2 to X ′′

2 ; the desired f is the “union” of these two isome-

tries.

3) Continuity. We just saw that the continuous map

L : T (P 2)→ (R∗+)3

is bijective. To prove thatL−1 is continuous, it suffices to show thatL : H(P 2)→ (R∗+)3

admits continuous local sections. It will be more convenient to change coordinates in

(R∗+)3, passing from the lengths of the boundary curves to the lengths ℓ12, ℓ23, ℓ13 of

the geodesics g12, g23, g13 (Figure 3.2). This gives a new continuous map:

Λ: H(P 2)→ (R∗+)3,

and it will suffice to prove that Λ has continuous local sections.

We begin with a few preliminaries. Let E be the portion of R2 that is the union of

E0 = {−1 ≤ y ≤ 1, x = 0} and E1 = {−1 ≤ y ≤ 0, 0 ≤ x ≤ 1}.

We define C∞(E) as the set of functions f : E → R such that

DR

AFT

13 J

un 2

008


f |E0 ∈ C∞(E0) and f |E1 ∈ C∞(E1). We have a natural topology on C∞(E) coming

from the C∞ topologies of C∞(E0) and C∞(E1).

Lemma 3.8. There is a continuous map ǫ : C∞(E)→ C∞(R2) such that

ǫ(f)|E = f.

Proof. Let f ∈ C∞(E). By applying a result of Seeley [See64], we may extend the

normal derivative of f |E0∩E1 to all of E0. This gives us a first extension of C∞(E) in

the Whitney C∞ jets on E (we use the fact that E0 and E1 are in regular position).

Then, we apply the Whitney extension theorem [Mal66].

By definition, a truncated hexagon is a set composed from the boundary of a C∞

hexagon of R2 and the collar neighbourhoods of three alternating sides (Figure 3.11).

The C∞ structure of the truncated hexagon Z is locally (where there could be

problems) like that ofE. From Lemma 3.8 and some classical arguments, we establish

the following lemma.

Figure 3.11: Z = truncated hexagon

Figure 3.12:

DR

AFT

13 J

un 2

008


Lemma 3.9. Let Emb(Z,R2) be the set of C∞ embeddings of Z in R2, with the C∞ topol-

ogy. If φ : (Rn, 0) → Emb(Z,R2) is a germ of a C∞ function, we may lift φ to a germ

Φ: (Rn, 0)→ Diff(R2) such that Φ(0) = Id and φ(t) = Φ(t)(φ(0)).

Now let ℓ0 = (ℓ012, ℓ023, ℓ

013) ∈ (R∗

+)3 and let X(ℓ0) be a right hyperbolic hexagon

in H2 parametrized by ℓ0. Let G1 and G2 be two geodesics forming two consecutive

sides of X(ℓ0). For ℓ near ℓ0 in (R∗+)3, we consider the hexagonX(ℓ) lying onG1∪G2

like X(ℓ0) (Figure 3.12). For each ℓ, the double of X(ℓ) along the “marked” sides

(those whose lengths are the parameters ℓij) is a hyperbolic manifold, denoted by

2X(ℓ); it is diffeomorphic to P 2.

The question is to find a diffeomorphism ψ(ℓ) : 2X(ℓ)→ 2X(ℓ0), so that the met-

ric ρ(ℓ), the image of the natural metric of 2X(ℓ) under ψ(ℓ), depends continuously

on ℓ as an element of H(2X(ℓ0)).

For small fixed ǫ > 0 (independent of ℓ), we consider in X(ℓ) the geodesic collars

of radius ǫ along the marked sides; we thus associate to X(ℓ) a truncated hexagon

Z(ℓ). Every rectangle of Z(ℓ) is foliated on one hand by the geodesics orthogonal

to the sides of the hexagon, and on the other by the trajectories orthogonal to these

geodesics. It is easy to construct a germ of a continuous function

φ : ((R∗+)3), ℓ0)→ Emb(Z(ℓ0),R2)

such that:

1. φ(ℓ0) is the standard embedding;

2. φ(ℓ)[Z(ℓ0)] = Z(ℓ);

3. φ(ℓ) respects the names of the marked sides and the foliations of the rectangles.

By Lemma 3.9, there exists a germ

ψ : ((R∗+)3, ℓ0)→ Emb(X(ℓ0),R2)

such that ψ(ℓ)|Z(ℓ0) = φ(ℓ). Condition 3 ensures then that 2ψ(ℓ) is a diffeomorphism

of the doubles 2X(ℓ0)→ 2X(ℓ). On the other hand, the construction ensures that the

metric on X(ℓ0), obtained from the natural metric on X(ℓ) via ψ(ℓ), depends contin-

uously on ℓ. Therefore ψ(ℓ) = [2ψ(ℓ)]−1 has all of the required properties.

DR

AFT

13 J

un 2

008


3.3 Generalities on the geometric intersection of simple

curves and on i : S × S → R+

In what follows, M is an orientable surface of genus g > 1. For practicality, we only

explain the case whereM is closed; the adaptations to the case of nonempty boundary

are left to the reader. We consider the set S of isotopy classes of simple curves in M

that are not homotopic to a point. For α, β ∈ S, we define i(α, β) as the minimal

number of intersection points of a representative for α with a representative for β.

We are led to a map

i∗ : S → RS+.

Throughout this expose, we shall often use the following theorem due to D. Ep-

stein [Eps66]: Let f0 : S1 → M be a two-sided embedding (i.e. with trivial normal fibre)

that is not the boundary of a disk; if f1 is an embedding homotopic to f0, then f0 and f1 are

isotopic. [With a base point, the same thing is true if additionally f0 is not the boundary of a

Mobius band.]

In the same article, one finds the relative version: If N is a surface with boundary

and if A,B are two embedded arcs with ∂A = ∂B = A ∩ ∂N = B ∩ ∂N , homotopic with

endpoints fixed, A and B are isotopic rel ∂.

We will also use the following two facts, which may be read in [Eps66].

Every simple curve homotopic to a point in a surface is the boundary of a disk (this is a

consequence of the Jordan–Schonflies theorem).

A two-sided embedding of the circle in a surface is never homotopic to a k-fold cover of a

two-sided simple curve, for k > 1.

Proposition 3.10. Let α′0 and α′

1 be two transverse simple curves in M , not homotopic to

a point. We suppose that their isotopy classes α0 and α1 are distinct. Then the following

conditions are equivalent.

(1) card(α′0 ∩ α′

1) = i(α0, α1).

(2) No simple closed curve formed from an arc of α′0 and an arc of α′

1 is homotopic to a

point in M .

(3) If α0 and α1 are connected components of p−1(α′0) and p−1(α′

1), respectively, in the

universal covering p : M →M ; then we have card(α0 ∩ α1) ≤ 1.

(4) There exists on M a Riemannian metric ρ of curvature −1, such that α′0 and α′

1 are

geodesics.

DR

AFT

13 J

un 2

008


Proof. The reader will notice that the following implications are immediate.

(1) =⇒ (2); indeed, a simple closed curve γ of α′0∪α′

1 that is homotopic to a point

in M is the boundary of a disk D; furthermore, γ is the union of an arc of α′0 and an

arc of α′1; through the disk, we may make an isotopy of α′

1 that decreases its number

of intersection points with α′0.

(3) =⇒ (2) by the theory of covering spaces.

(4) =⇒ (2) and (3) by Lemma 3.1.

Lemma 3.11. If card(α′0 ∩ α′

1) > i(α0, α1), there exist two distinct points q1 and q2 of

α′0 ∩ α′

1 and two (not necessarily simple) paths Γ0,Γ1 joining q1 to q2, respectively, on α′0

and α′1, such that the singular loop Γ0∗Γ−1

1 is homotopic to a point inM . Hence (3) =⇒ (1).

Proof. By hypothesis, there exists a homotopy ht : S1 → M , for t ∈ [0, 1], such that

h0 parametrizes α′0 and such that h1(S

1) satisfies

card(h1(S1) ∩ α′

1) < card(α′0 ∩ α′

1).

We may suppose that the isotopy ht is in general position with respect to α′1; that is

h : S1 × [0, 1] → M is transverse to α′1. Thus h−1(α′

1) is a submanifold of dimension

1 transverse to the boundary that consists of four types of connected components,

shown in Figure 3.13.

IVI II III

Figure 3.13:

The points q1, q2, q3, . . . of this figure are exactly the preimages, under the em-

bedding h0, of the points of intersection α′0 ∩ α′

1. By assumption, there exists at least

one component Γ1 of type I; we obtain Γ0 by choosing the arc q1q2 of S1 × {0} that is

homotopic to Γ1, with endpoints fixed, in S1 × [0, 1].

Lemma 3.12. (2) =⇒ (3)

Proof. If the components α0 and α1 intersect each other in more than one point in M ,

DR

AFT

13 J

un 2

008


it is easy to find an embedded disk ∆ in M where the boundary is the union of an arc

of α0 and an arc of α1. On ∆, we see p−1(α′0 ∪ α′

1) as in Figure 3.14, where p−1(α′0) is

dashed and p−1(α′1) is drawn as a solid line.

Figure 3.14:

We may find a (minimal) disk δ where the boundary is also the union of a dashed

arc and a solid arc and whose interior does not meet p−1(α′0 ∪ α′

1). The immersion p

embeds the boundary of δ by the minimality. Now, we may check that p embeds δ: an

immersion in codimension 0 that embeds the boundary and where the interior does

not meet the boundary is an embedding (the number of points for the fibre is locally

constant).

Hence, we have proved the equivalence of conditions (1), (2), and (3) of Proposi-

tion 3.10.

It remains to prove (1) =⇒ (4). This follows immediately from Proposition 3.13

and Theorem 3.16 below.

Proposition 3.13. Let α′0, α

′′0 , and α′

1 be three simple curves in M , each not homotopic to a

point. We suppose

1. α′0 and α′′

0 belong to the same isotopy class α0, which is distinct from the isotopy class

α1 of α′1; and

2. card(α′0 ∩ α′

1) = card(α′′0 ∩ α′

1) = i(α0, α1).

Then there exists an ambient isotopy of the pair (M,α′1) that pushes α′

0 onto α′′0 .

Extension. The same proof shows that the proposition remains valid if α′1 is a simple

arc representing a nontrivial element of π1(M,∂M).

DR

AFT

13 J

un 2

008


Proof. Let h : S1 × [0, 1] → M be a map that is transverse to α′1, whose restriction

h|S1 × {0} (respectively h|S1 × {1}) parametrizes α′0 (respectively α′′

0 ).

Lemma 3.14. The closed components of h−1(α′1) are homotopic to a point in S1 × [0, 1].

Proof. Let γ be a component of h−1(α′1), not homotopic to a point in S1 × [0, 1]; then

γ is isotopic to the boundary. Let d be the degree of h : γ → α′1. We cannot have d = 0,

since this would imply that α′0 is homotopic to a point.

We cannot have |d| > 1; this would imply that a nontrivial multiple of α′1 is freely

homotopic to an embedded curve, namely, α′0. This is known to be impossible (see

the reference to Epstein cited at the beginning of this section). If |d| = 1, this means

α′0 is homotopic to α′

1, which we have excluded.

Conclusion of the proof of Proposition 3.13. From here, the components of h−1(α′1)

are of types I, II, II, IV, as in Figure 3.13. In fact, by the second hypothesis, types I and

IV do not exist. As π2(M,α′1) = 0, it is easy to kill the components of type III. If after

this, h−1(α′1) is empty, we conclude that α′

0 and α′′0 are homotopic (hence isotopic) in

M − α′1 and we have the conclusion of the proposition by extending the isotopy to

have support in M − α′1. Otherwise, there remain components of type II, which we

may draw as vertical. However, in general, the map h thus constructed is singular

and does not give an isotopy.

Let s1, . . . , sn be the points of h−1(α′1) ∩ S1 × {0}; these cut the circle into in-

tervals I1, . . . , In and, if h−1(α′1) = {s1} × [0, 1] ∪ · · · ∪ {sn} × [0, 1], we may think

of h|Ik×[0,1] as a proper homotopy (i.e., the boundary moves within the boundary)

between two embedded arcs of the surface N obtained by cutting M along α′1. We

remark that, by hypothesis 2, for all k, h|Ik×{0} represents a nontrivial element of

π1(N, ∂N). Proposition 3.13 is then obtained by applying to each arc Lemma 3.15

below, which generalizes the relative version of the result of Epstein already cited.

Lemma 3.15. Let N be a surface with boundary, and γ0, γ1 two properly embedded arcs in

N . Let h : [0, 1]× [0, 1]→ N be a proper homotopy between these two arcs: h(t, 0) and h(t, 1)

parametrize γ0 and γ1, respectively, and h(0, u) and h(1, u) belong to ∂N for all u.

Then h is deformable, rel [0, 1] × {0, 1}, to an isotopy from γ0 to γ1. Furthermore, if

h(0, u) = h(0, 0) for all u, or if h(1, u) = h(1, 0) for all u, then the deformation

DR

AFT

13 J

un 2

008


may be made through maps with the same properties.

Proof. As usual in these situations, the lemma is clear if γ0 and γ1 do not intersect

except at their endpoints; indeed, γ0 and γ1 bound a disk in N , through which the

required isotopy is done; the isotopy is a deformation of the initial homotopy, for N

is an Eilenberg–MacLane space.In the case where they do intersect, we consider the universal covering p : N →

N ; consider one component γ0 of p−1(γ0) and the union Γ1 of all components of

p−1(γ1). If we have taken care to begin with an initial isotopy that fixes the endpoints

of γ0, to make card(γ0∩γ1) as small as possible, then, by the equivalence (1) ⇐⇒ (3)

in Proposition 3.10, γ0 meets every component of Γ1 in at most one point.Let γ1 be any component of Γ1; we denote by γi(0) and γi(1) the endpoints of

γi. If γ0 and γ1 meet (somewhere other than at their endpoints), we have the con-

figurations of Figure 3.3. In this figure, the endpoints of the arcs belong to distinct

components of ∂N , unless explicitly indicated otherwise.

II

III IV

I

γ0

γ1

γ1(0)

γ0(0)

γ0(1)

γ0(1)

γ1(1)γ1(1)

∆

∆

∆

Figure 3.15:

Configuration I is excluded; indeed, this configuration contradicts the existence

of a proper separating homotopy. Similarly, II is excluded in the case where h(0, u) is

DR

AFT

13 J

un 2

008


fixed. By the same argument, configurations III and IV are excluded, if in addition,

h(1, u) is fixed.

Thus, in the case where the endpoints are fixed, the lemma is totally proven.

DR

AFT

13 J

un 2

008


Let us analyse the case where the origin γ0(0) is fixed; then we only have con-

figurations III and IV. We see in N a triangle ∆. Up to changing components of γ1,

we may suppose that int ∆ ∩ Γ1 = ∅. Therefore, p|∆ is an embedding; there is an

isotopy of γ0 supported in a neighbourhood of p(∆), that kills at least one point of

intersection with γ1. We continue in this manner until int γ0∩ Γ1 = ∅. The case where

the two endpoints are free is treated similarly.

Theorem 3.16. If a surface M is endowed with a metric of curvature −1, each simple curve

not homotopic to a point is isotopic to a simple geodesic. Moreover, two simple geodesics meet

in the minimal number of points of intersection in their isotopy classes.

Proof. The second part of the theorem follows from the implication (3) =⇒ (1) in

Proposition 3.10.

Let f : S1 → M be an embedding not homotopic to a point; by Lemma 3.4, f is

homotopic to a geodesic immersion g. Let p : M → M be the universal covering. Let

f0, f1 : R → M be two proper embeddings with distinct images above f ; let g0 and

g1 be the geodesic maps to which they are homotopic. By Lemma 3.1, g0 and g1 are

embeddings that have at most one point in common. We show that g0 and g1 do not

meet.

If M is regarded as the interior of the Poincare disk D2, for i = 0, 1, gi has two

limit points. Since the homotopy from gi to fi is obtained by lifting a homotopy in

M , the hyperbolic distance from gi(x) to fi(x) is uniformly bounded for x ∈ R. We

know that in a neighbourhood of infinity, the Euclidean ds2 is infinitesimally small

compared with the hyperbolic ds2; hence as x → ±∞, the Euclidean distance from

gi(x) to fi(x) tends to zero. Thus fi has the same limit points on ∂D2 as gi. Thus, if g0and g1 have a common point, then by an algebraic intersection argument (or by the

Jordan curve theorem), f0 and f1 must meet again. This is impossible, since f is an

embedding.

Thus we have proved that the image of g is a simple curve covered by g a certain

number of times. To see that g as an embedding, we apply the result of Epstein cited

in the beginning of the section.

We can give an application of the theorem that illustrates condition (3) of Propo-

sition 3.10.

Corollary 3.17. Let α′0 and α′

1 be two simple curves that intersect transversely.

DR

AFT

13 J

un 2

008


We suppose that they have components αi (i = 0, 1) of p−1(α′i) in the universal cover satis-

fying card(α0 ∩ α1) =∞. Then the classes α0 and α1 are equal.

Proof. By the hypothesis of transversality, we have card(α′0 ∩ α′

1) < ∞. Therefore

there are points ∗ ∈ α′0 ∩ α′

1 and x, y ∈ α0 ∩ α1, such that x 6= y, p(x) = p(y) = ∗. We

orient each arc αi from x to y and each arc α′i as α′

i. Consider α0, α1 as elements of

π1(M, ∗). The segment from x to y on α0 (respectively α1) covers α′0 k times (respec-

tively α′1 l times). We therefore have in π1(M, ∗) the equality:

αk0 = αl1.

Now, we give M a metric of curvature −1. If gi denotes the (unique) geodesic of Minvariant under Tαi , we see that Tαk

0= Tαl

1leaves invariant g0 and g1. Thus g0 = g1,

p(g0) = p(g1) and α′0, α

′1 are (freely) homotopic to the same geodesic in M .

From the equivalence (1) ⇐⇒ (2) of Proposition 3.10, we deduce the following

fact. Let α′, β′, γ′ be three simple arcs 6∼ 0 inM , with α′ ∩γ′ = β′ ∩γ′ = ∅; if card(α′ ∩β′) is minimal in M − γ′, then card(α′ ∩ β′) is also minimal in M . This criterion will

be used below.

We recall from Expose 1 that P (RS+) is the “projective” space associated to RS

+

and that

π : RS+ − {0} → P (RS

+)

is the natural projection.

Proposition 3.18. 1. The image of i∗ is contained in RS+ − {0}.

2. The map π ◦ i∗ (in particular i∗) is injective.

Proof. It suffices to prove that if α1 6= α2 ∈ S, there exists β ∈ S such that

i(α1, β) = 0 6= i(α2, β).

If i(α1, α2) 6= 0, it suffices to take β = α1. If i(α1, α2) = 0, there exist simple

curves α′1 ∈ α1 and α′

2 ∈ α2 such that α′1 ∩ α′

2 = ∅. By cutting m along α′1, we obtain

a surface N containing α′2 in its interior.

As α′2 is not isotopic to α′

1, there exists in N a curve β′ that cannot be separated

from α′2 in N . If α′

2 does not separate N , we take β′ with

DR

AFT

13 J

un 2

008


card(β′ ∩ α′2) = 1. If α′

2 separates N into N1 and N2, we take β′ = I1 ∪ I2 where Ij is

an arc representing a nontrivial element of π1(Nj , α′2); this is possible because neither

N1 nor N2 is an annulus or a disk.

If β is the isotopy class of β′ inM , we have, by Proposition 3.10, that i(α2, β) 6= 0.

3.4 Systems of simple curves in M and hyperbolic iso-

metries

Consider a system of distinct elementsα1, . . . , αk ∈ S, with the property that i(αl, αq) ≤1. We define the complex Γ(α1, . . . , αk) having as vertices the α1, . . . , αk; the ver-

tices αl, αq are joined by an edge if i(αl, αq) = 1. I will henceforth suppose that

Γ(α1, . . . , αk) is a tree.

Lemma 3.19. Under the conditions above, let α′j , α

′′j ∈ αj be such that card(α′

l, α′q) =

card(α′′l , α

′′q ) = i(αl, αq). Then there exists a diffeomorphism of M , isotopic to the identity,

that transforms ∪α′j into ∪α′′

j .

Proof. For k = 2, this is Proposition 3.13. For a proof by induction, we suppose that

α′j = α′′

j for j ≤ l, l ≥ 2, the indexing being compatible with the tree structure. Let

p, q be such that p ≤ l < q, and i(αp, αq) = 1. Let N be the manifold obtained by

cutting M along the arcs α′j , where j ≤ l, j 6= p. Then α′

p is cut into one or more arcs

in N ; let I be one such arc that meets α′q (α′

q is a closed curve in N since Γ is a tree);

as card(α′q ∩ I) = 1, the arc I represents a nontrivial element of π1(N, ∂N).

We prove that α′′q intersects the same arc I (and not some possibly different com-

ponent of α′p ∩ N ). Otherwise, for some j 6= p such that j ≤ l and i(αj, αp) = 1, we

have αj = αq [look at the preimage of α′j in the domain of the homotopy from α′

q to

α′′q inM ; one of these components is necessarily parallel to the boundary of the annu-

lus]. The extension of Proposition 3.13 is now applicable: we have, in N , an isotopy

that pushes α′′q onto α′

q and that leaves α′p ∩N alone.

Application. Let ρ be a metric of curvature −1 on the surface M . We consider the

simple curves α′1, . . . , α

′k as in Figure 3.16 (here, we argue with M closed); M − ∪α′

j

is a cell. Let α′′j be the geodesic, in the metric ρ, of

DR

AFT

13 J

un 2

008


the isotopy class of α′j ; we verify that card(α′′

l ∩α′′q ) = card(α′

l ∩α′q). By Lemma 3.19,

M − ∪α′′j is a cell. In particular, Figure 3.16 is realizable by geodesics.

2

3

β

ββ

β4

1

Figure 3.16:

Theorem 3.20. Let ρ be a metric of curvature −1 on a compact surface M . The group

I(M,ρ) of isometries of ρ is finite and any isometry isotopic to the identity is the identity.

Proof. We begin by considering the set MM of all maps M → M , with the topology

of pointwise convergence. By the Tychonov theorem, MM is compact. We remark, in

addition, that on I(M,ρ) the topology of pointwise convergence and the topology of

uniform convergence coincide. [Indeed, if we consider a finite setX inM , sufficiently

dense, an isometry is completely characterized by what it does on X . . . .] We remark

that I(M,ρ) is closed in MM .

Moreover, we claim that an isometry isotopic to the identity is equal to the iden-

tity. Indeed, let φ be such an isometry; the action of φ on S is trivial; by the uniqueness

of geodesics in a given isotopy class α ∈ S in hyperbolic geometry, the geodesic gαof the class α ∈ S is invariant: φ(gα) = gα. We immediately deduce that φ is the

identity on the system of geodesics in Figure 3.16. Hence, φ is the identity on the

complementary cell.

Thus, I(M,ρ) is discrete. But a closed discrete set in a compact space is finite.

Corollary 3.21. Let f ∈ Diff(M) and let T (f) be the natural action of f on the Teichmuller

space of M (see Expose 7). If T (f) has a fixed point, there is a periodic diffeomorphism of M

isotopic to f .

DR

AFT

13 J

un 2

008

Expose 4

The space of simple closed

curves in a surfaceby V. Poenaru

4.1 The weak topology

LetM be a closed orientable surface of genus g ≥ 2. Denote by S the space of isotopy

(= homotopy) classes of unoriented simple closed curves that are not homotopic to a

point in M . We have already seen (Section 3.3) that the composite map

S i∗−→RS+ − {0}

π−→P (RS+)

is injective. The map i∗ extends to a map that we will denote by the same symbol

i∗ : R+ × S → RS+,

given by the formula

i∗(λ, α)(β) = λi(α, β) where λ ∈ R+ and α, β ∈ S

Remark. If i∗(R+ × S) denotes the closure of i∗(R+ × S) in RS+, we have

π(i∗(R+ × S)− {0}

)= π ◦ i∗(S).

This is a general fact about cones.

55

DR

AFT

13 J

un 2

008


Proposition 4.1. In P (RS+), the set π ◦ i∗(S) is precompact.

For the proof, we begin by choosing on M a metric ρ of curvature −1, and we

denote by ℓ(α) the ρ-length of the unique geodesic belonging to the class of α ∈ S.

Lemma 4.2. There exists a constant C = C(M,ρ) such that for all α, β ∈ S, we have

i(α, β) ≤ Cℓ(α)ℓ(β).

Proof. If α = β, we have i(α, β) = 0 and the inequality is clear. Let us suppose

therefore that α 6= β. Let ǫ be a positive number smaller than the injectivity radius

of the exponential map. The geodesic gα in the isotopy class α may be covered by

fewer than ( ℓ(α)ǫ + 1) small arcs, each of which is contained in a geodesic disk. The

same holds for gβ . By the definition of injectivity radius, a small arc of gα intersects a

small arc of gβ in at most one point; therefore, in a small arc of gα, there are at most

( ℓ(β)ǫ + 1) points of intersection with gβ . We therefore find

i(α, β) = card(gα ∩ gβ) ≤(ℓ(α)

ǫ+ 1

)(ℓ(β)

ǫ+ 1

).

As ℓ(α) > ǫ, the desired inequality is clear.

On M , we now consider the system of elements β1, . . . , β2g+1 ∈ S represented in

Figure 4.1. In Section 3.4, we saw that such a system may be realized by geodesics.

2

3

β

ββ

β4

1

Figure 4.1:

DR

AFT

13 J

un 2

008


Lemma 4.3. There exists a constant c such that for all α ∈ S,

∑

j

i(α, βj) ≥ cℓ(α).

Proof. The system {gβj} decomposes M into a number of simply connected regions.

In each of these, the length of a geodesic arc is bounded, say by L; thus, we have the

desired result by taking c = 1/L.

Proof. [Proof of Proposition 4.1] For a fixed constant C, consider the subset S(C) ⊂ why is the ar-

gument to slash-

proof not work-

ing here?

RS+, defined by

S(C) ={f ∈ RS+ | ∀ β ∈ S, f(β) ≤ Cℓ(β)

}.

By the Tychonov theorem, S(C) is compact. Now taking C to be the constant of

Lemma 4.2, consider S0 ⊂ S(C), which is the closure in RS+ of the set of functionals of

type i∗(α)/ℓ(α). By Lemma 4.3, we see that S0 ⊂ RS+ − {0}. Moreover, S0 is compact;

thus π(S0) is compact. By Lemma 4.2, we have the inclusion π ◦ i∗(S) ⊂ π(S0); this

gives the compactness of π ◦ i∗(S).

4.2 The space of multicurves

As S is difficult to study, we introduce a space that is larger and easier to study. Let

S′ = S′(M) be the space of isotopy classes of closed submanifolds of dimension 1

(not oriented and not necessarily connected) where no component is homotopic to

a point. An element of S′ is called a multicurve. As in the case of simple curves, we

define i(α, β) for α ∈ S′ and β ∈ S, as well as i∗ : S′ → RS+ and π ◦ i∗ : S′ → P (RS

+).

The minimal intersection between a multicurve and a simple curve is the sum of the

minimal intersections with the different components.

Remark. By the same reasoning as in Section 3.3, we prove that i∗ is injective and that

two elements α1 and α2 of S′ have the same image under π ◦ i∗ if and only if they are

integer multiples of the same α0 ∈ S′ (there is indeed a natural map N× S′ → S′).

Theorem 4.4. In P (RS+), we have

π ◦ i∗(S) = π ◦ i∗(S′).

DR

AFT

13 J

un 2

008


Applying Theorem 4.4, we obtain the following.

Corollary 4.5. In RS+, we have

i∗(S′) ⊂ i∗(R+ × S).

Proof. [Proof of Theorem 4.4] It suffices to show that π◦i∗(S) is dense in π◦i∗(S′). Let

α ∈ S′ be represented by a union of pairwise disjoint simple curves α1, α2, . . . , αk.

We may choose a simple connected curve γ such that card(γ ∩αj) is equal to i(γ, αj)

and nonzero for all j. Let n1, n2, . . . , nk be positive integers. We shall construct an

element Γ(n1, . . . , nk) of S. Each arc of γ that crosses a small tubular neighbourhood

of αj is replaced by an arc with the same endpoints making nj positive turns (see

Figure 4.2, for nj = 2.)

Figure 4.2:

We obtain by this construction a curve Γ(n1, . . . , nk), that is well-defined up to

isotopy. We prove in the appendix that for β ∈ S, we have the inequality∣∣∣i(Γ(n1, . . . , nk), β)−

∑

j

nji(γ, αj)i(αj , β)∣∣∣ ≤ i(γ, β).

For any n, set nj = n∏

ℓ 6=ji(γ, αℓ), and denote the resulting curve Γ(n1, . . . , nk) by

Γ(n); we have∣∣∣i(Γ(n), β)− n

∏

j

i(γ, αj)[∑

j

i(αj , β)]∣∣∣ ≤ i(γ, β).

In other words, when we projectivize, the contributions of γ to the intersection be-

come negligible as n tends to infinity. Thus the sequence π◦i∗(Γ(n)) tends to π◦i∗(α).

DR

AFT

13 J

un 2

008


4.3 An explicit parameterization of the space of multic-

urves

Recall that P 2 denotes the standard pair of pants; the boundary curves are numbered

∂1P2, ∂2P

2, ∂3P2. In Section 2.3, we classified the “multi-arcs” of P 2. An element τ

of A′(P 2), the space of multi-arcs, is completely characterized by the three integers

mj = i(τ, ∂jP2), (j = 1, 2, 3); a triplet of integers that are not all zero describes a

multi-arc exactly when m1 +m2 +m3 is even.

In each class of A′(P 2), we choose once and for all a representative, which we

shall call canonical, as designated in Figure 4.3. For each τ ∈ A′(P 2) and each ∂jP2, canonical

we choose an arc xj , a connected component of ∂jP2− τ , as in Figure 4.3. This choice label of fig 4.3 is

messed up/not

there

is uniquely defined, since (P 2, τ) does not admit any nontrivial orientation preserv-

ing automorphisms.

DR

AFT

13 J

un 2

008


For each model τ , we chose an “arc jaune” J1 = J1(τ) that has the following

properties.

1. J1 is a simple arc joining ∂1P2 to itself and that cuts P 2 into two regions, one of

which contains ∂2P2, the other ∂3P

2;

2. J1 has one endpoint in the arc x1(τ).

3. J1 has minimal intersection with τ .

Similarly, we construct arcs J2 and J3.

Remark. In Expose 6, we will classify the measured foliations on P 2. The models in

Figure 4.3 are the “discrete models” for these foliations, where we do not see any

nonsingular leaves.

For the classification of multicurves, we follow a procedure analogous to that

which we will follow in the classification of measured foliations: for example the

technique of the arcs jaune, which is used to recover the way the pairs of pants are

glued together to form the surface.

To parameterize S′, we make a number of choices.

I) We choose 3g−3 mutually disjoint simple curvesK1,K2, . . . ,K3g−3 that cutM into

2g − 2 regions diffeomorphic to pairs of pants. We take these Ki to have a connected

complement inM ; in this way, the pairs of pants Rj are embedded inM ; that is, each

Ki belongs to two distinct pairs of pants.

II) For each Kj , we choose two simple curves K ′j and K ′′

j as in Figure 4.4 (this is

possible because of the preceding condition). K ′j and K ′′

j differ by a positive twist

along Kj (this only depends on the orientation of the surface and not on that of Kj).

DR

AFT

13 J

un 2

008


III) We give each Kj a tubular neighbourhood Kj × [−1, 1]; these are taken to be

pairwise disjoint; the complement of their union is a number of pairwise disjoint

pairs of pants R′1, R

′2, . . . , R

′2g−2.

IV) Each R′j is parameterized by P 2, via diffeomorphisms φj , that are fixed (not only

up to isotopy).

We consider in R9g−9+ the cone

B = {(mi, si, ti) | i = 1, . . . , 3g − 3; mi, si, ti ≥ 0, (mi, si, ti) ∈ ∂(≤ ∇)}.

It is homeomorphic to R6g−6 (the cone on ∂(≤ ∇) is homeomorphic to R2). We will

construct a classification map Φ: S′ → B.

Let β ∈ S′; we start by defining mj(β) = i(β,Kj). Knowledge of these integers

determines the model of the pairs of pants R′k: the corresponding model for P 2 is

carried by the diffeomorphism φk . If the representative β0 of β is chosen to have

minimal intersection with the boundary of all of the pants R′k, then β0|R′

k is isotopic

to the model. We therefore choose β0 equal to the model in all of the pairs of pants R′k;

we say that this representative is in normal form. Note that if β0 has a component normal form

isotopic to Kj , this component is contained in the annulus Kj × [−1, 1].

Lemma 4.6. The normal form of β is “unique”. Precisely, if β0 and β1 are two representatives

of β in normal form, then, for all j = 1, . . . , 2g− 2, β0 ∩Kj × [−1, 1] and β1 ∩Kj × [−1, 1]

are isotopic relative to the boundary.

Proof. We need an extension of Proposition 3.13 to the case that one of the curves

is a multicurve; the proof is analogous. It suffices to show this: if γ0 is a component

of β0 and if γ1 is the corresponding component of β1, then there exists an isotopy of

M that pushes γ0 onto γ1 and that leaves invariant all of the curves Kj × {−1} and

Kj ×{+1}, j = 1, 2, . . . , 3g− 3. Actually, the proof of the cited proposition only lacks

the following improvement: if γ0 ∩ Kj × {±1} = γ1 ∩ Kj × {±1} = ∅, then γ0 is

isotopic to γ1 in M −Kj × {±1}. This assertion is true by the “classical” arguments

(Lemma 3.14) except possibly if γ0 is isotopic to Kj . but then, because of the “normal

form” condition, there is nothing to prove.

Now, in the discussion above, we may replace γ0 (resp. γ1) by the collection γ0

(resp. γ0) of all of the components of β0 (resp. β1) parallel to γ0 (resp. γ1). We may

thus construct a normal form β′0 with the following properties.

DR

AFT

13 J

un 2

008


1. β′0 and β0 are isotopic by an isotopy that respects the curves Kj × {±1};

2. The collection γ′0, corresponding to γ0, coincides with γ1.

Now let δ0 be a curve of β′0− γ1 and let δ1 be the corresponding curve of β1− γ1. If

δ0 is not parallel to γ1, δ0 and δ1 are isotopic inM− γ1. Again by the same arguments,

we then find that there exists an isotopy of M , constant on γ1 and respecting the

curves Kj × {±1}, that pushes δ0 onto δ1. We continue in this way with the rest. In

the end, β0 and β1 are isotopic by an isotopy that respects all of the curvesKj×{±1}.Now, we claim that the above isotopy may be chosen to be constant in all of

the small pairs of pants R′j , which will prove the lemma. It is clear if β0 ∩ R′

j is

empty. Otherwise, it follows from the fact that the loops of Diff(P 2, ∂1, ∂2, ∂3) are

each homotopic to a point (see Expose 2).

The above lemma will be essential for the classification.

The models β0 ∩ R′ℓ are equipped with their “arcs jaune”. Consider the curve Kj

and the two adjacent pairs of pants R1 and R2. In the small pairs of pants R′1 and

R′2, we have the two arcs jaune J1 and J2 emanating from the respective boundaries

parallel to Kj . There exists in Kj × [−1,+1] simple arcs Sj , S′j , Tj , and T ′

j such that

J1 ∪Sj ∪J2 ∪S′j is isotopic to K ′

j and J1 ∪ Tj ∪J2 ∪T ′j is isotopic to K ′′

j . If we impose

the condition that ∂Sj = ∂Tj and ∂S′j = ∂T ′

j , Sj ∩ S′j = ∅, Tj ∩ T ′

j = ∅, then Sj ∪ S′j

(resp. Tj ∪ T ′j) is unique up to isotopy relative to the boundary. Moreover, Tj ∪ T ′

j is

obtained from Sj ∪ S′j by a positive twist in the annulus.

Since the endpoints of these arcs are not in β0, this gives a way to put the arcs

into minimal position with β0. Once this is done, we set

sj(β) = card(β0 ∩ Sj) and

tj(β) = card(β0 ∩ Tj).

Lemma 4.7. For each j, the triple (mj(β), sj(β), tj(β)) belongs to the boundary ∂(≤ ∇) of

the triangle inequality. [Compare with the classification theorem for S′(T 2) in Expose 1.]

Proof. The proof is given by Figure 4.5.

DR

AFT

13 J

un 2

008


Let B0 ⊂ B be the set of points 6= 0, with integer coordinates, also satisfying the

following additional condition: ifKj1 ,Kj2 ,Kj3 , are on the boundary of the same pair

of pants, then mj1 +mj2 +mj3 is even.

Theorem 4.8. The map Φ: S′ → B is a bijection of S′ onto B0.

Remark. By an analogous procedure, we will classify the measured foliations and

Teichmuller structures. In reality, as we will explain, Theorem 4.8 above is strictly

contained within the classification theorem for measured foliations. But the simplic-

ity of the means implemented here makes it worth including this particular case [in

particular, for foliations, one only obtains uniqueness of the normal form only after

a long, roundabout argument].

Proof. The image is obviously contained in B0. On the other hand, we have a recipe

for making a multicurve β from the element {mj , sj, tj | j = 1, . . . , 3g − 3} of B0. By

Expose 2, the coefficients mj determine the arcs in the small pairs of pants R′k. With

these, come the “arcs jaune” and hence, for each j, we get the arcs Sj and Tj in the

annulus Kj × [−1,+1].

If mj = 0, sj = tj indicates the number of curves of β parallel to Kj . If mj 6= 0,

we already have mj points on Kj × {−1} and on Kj × {+1}; the coefficients sj

DR

AFT

13 J

un 2

008


and tj completely determine the way in which these are joined. It remains to ver-

ify that the multicurve constructed in this way has the property of having minimal

intersection with each Kj , i.e., that i(β,Kj) = mj ; for this we use the criterion of

Proposition 3.10.

As soon as Sj and Tj are fixed, β0 ∩ Kj × [−1,+1] is determined up to isotopy

relative to the boundary by sj and tj . The injectivity of Φ follows.

Remark. The members of the seminar do not know how to detect which coefficients

give a simple curve.

Obviously, Φ is homogeneous (of degree 1) with respect to multiplication by an

integer scalar. We may thus extend Φ by homogeneity to Φ: R+ × S → B.

Corollary 4.9. The map Φ: R∗+ × S → B is injective.

Proof. If not, there exists α0 and α1 ∈ S and a scalar λ > 0 such that Φ(α0) = λΦ(α1).

It is very easy to see that λ is rational. Thus, we have integers n0 and n1 such that

Φ(n0α0) = Φ(n1α1). By Theorem 4.8, we have n0α0 = n1α1. It follows immediately

that α0 = α1.

Problem. To show directly that Φ(R+ × S) is dense in B. This is plausible since the

(positive) cone on B0 is dense in B. Of course, this is a consequence of the following

theorem which is the “discrete” version of the theorem on foliations and which will

not be proved until Expose 6.

Theorem 4.10. There exists a closed cone C in RS+ and a continuous map θC : C → B,

positively homogeneous of degree 1, that makes the following diagram commute.

R+ × S (resp. S′) -i∗ C ⊂ RS+

@@@R

��

Φ θC

B

Furthermore, θC induces a homeomorphism of i∗(R+ × S′) onto B.

Consequences.

1. Φ(R+ × S) is dense in B. (Use Theorem 4.4 and the fact

DR

AFT

13 J

un 2

008


that Φ(S′) is a “net” by the continuity and the homogeneity of θC .)

2. The space π ◦ i∗(S) is homeomorphic to S6g−7.

Remark. The existence of θC implies that the coefficients of sj(β) and tj(β) are given

by formulas that are continuous and homogeneous of degree 1 as functions of the

i(β, α), α ∈ S. We will give these explicit formulas in the framework of measured

foliations; they make it possible to interpolate continuous values of the variables.

On the other hand, as Φ is injective for all α ∈ S, there exists a map ψα : B0 → Nsuch that for all β ∈ S′, we have

i(β, α) = ψα(Φ(β)).

It seems very difficult to make these last formulas explicit.

DR

AFT

13 J

un 2

008


23

1

m m

Arcs Jaunes

xj

Legend:

m 1

32

∂

∂

∂

DR

AFT

13 J

un 2

008


Figure 4.4:

Sj Kj × {+1} Tj

Kj × {−1} β0

mj = sj + tj tj = mj + sj sj = mj + tj

Figure 4.5: The annulus Kj × [−1, 1] is cut along Sj .

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008

Expose 5

Measured Foliations

by A. Fathi and F. Laudenbach

5.1 Definition. The Poincare recurrence theorem and the

Euler–Poincare formula.

5.1.1 Definition.

Let M be a surface1 and F a foliation of M with isolated singularities. By a transverse

invariant measure, we mean a measure µ defined on each arc transverse to the foliation

and satisfying the following invariance property:

Let α, β : [0, 1] → M be two arcs transverse to F , isotopic through trans-

verse arcs whose endpoints remain in the same leaf; then, µ(α) = µ(β).

If the arc passes through a singularity, the transversality pertains to all points of the

arc belonging to a regular leaf.

N.B. In what follows, we restrict ourselves to the case where the measure is regular

with respect to Lebesgue measure: every regular point admits a smooth chart (x, y)

where the foliation is defined by dy and the measure on each transverse arc is induced

by dy.

1The theory can be done for nonorientable surfaces. For simplicity, we assume M to be orientable.

69

DR

AFT

13 J

un 2

008


5.1.2 Permissible singularities in the interior

For each integer k > 1, we consider the singularity of the holomorphic quadratic

form zk dz2. We consider

Im√zk dz2 = rk/2

(r cos

(2 + k

2θ

)dθ + sin

(2 + k

2θ

)dr

)

which is a form of degree 1, well-defined up to sign. It thus defines a measured folia-

tion where the origin is an isolated singularity, and the separatrices are the half-linesseparatrices

r ≥ 0,2 + k

2θ = 0 modπ.

As models for the singularity we choose a compact domain containing the origin,

bounded by arcs transverse to the foliation (faces) and arcs contained in the leavesfaces

of F (sides).sides

Remark. Let ω be a closed differential form of degree 1 on M (∂M = ∅), whose

singularities are “Morse” (genericity property). Suppose in addition that ω does not

have a center (critical point of index 0 or 2); then ω defines a measured foliation. It

is easy to see that a measured foliation is defined by a closed form if and only if it is

transversely orientable in the complement of the singularities.

5.1.3 Permissible singularities on the boundary

The regular points of the boundary are those where the boundary is transverse to the

foliation as well as those that have a neighbourhood where the boundary is a leaf.

The singular points admit for a chart the image of the aforementioned models in

the upper half-plane if k is even, or on the half-plane of negative real part if k is odd.

Finally, in this entire work, being given a measured foliation (F , µ) on the man-

ifold M , each point of M has a neighborhood that is the domain of a chart that is

foliated isomorphically to one of the models of Figure 5.1.

DR

AFT

13 J

un 2

008


Singlar points on the boundary

point (k=1) pointInterior singular point

Interior singularInterior regular

Regular points on the boundary

Figure 5.1:

N.B. In the chart of a singular point, it will be convenient that the separatrices belong

to different plaques. Therefore, in M , all leaves are diffeomorphic to intervals of R or a plaque is a

horizontal line

of a foliation

in a chart; if

it contains a

singularity, then

it is at an end-

point. footnote

or glossary?

to S1.

5.1.4 Good Atlas

If M is compact, there exists a constant ǫ0 and two finite covers {Uj}j∈J , {Vj}j∈J , by

domains of charts, satisfying:

1. M =⋃j∈J (intUj);

2. For each j ∈ J , Uj ⊂ Vj and the faces of Uj are contained in the faces of Vj (see

Figure 5.2);

DR

AFT

13 J

un 2

008


Figure 5.2:

3. Every point of the sides of Uj is a distance greater than ǫ0 from the sides of

Vj (all distances are measured along trajectories in the sense of the invariant

measure µ);

4. Each singular point belongs to only one chart Uj ;

5. The intersection of two charts Uj1 and Uj2 (resp. Vj1 and Vj2 ) is a rectangle:

To satisfy this condition, we chose a line field transverse to the foliation on the

complement of the singularities and we insist that the charts are small enough that

their faces are tangent to this line field.

5.1.5 Poincare Recurrence

Theorem 5.1 (Poincare Recurrence). Let M be a compact surface equipped with a mea-

sured foliation (F , µ). Let α be an arc (∼= [0, 1]) of ∂M , transverse to F at all points of int α,

and let x be one of its endpoints. Then the leaf Lx leaving from x either goes to a singular

point or to the boundary.

Proof. We will use the atlas of Section 5.1.4. We suppose that Lx does not reach a

singularity, and we truncate α so that we have µ(α) = ǫ < ǫ0, and that, for every

y ∈ α, the leaf Ly does not end in a singularity. We claim that if Lx does not meet

DR

AFT

13 J

un 2

008


the boundary again, then we have an injective immersion Φ: α × R+ → M , where

Φ({y} × R+) = Ly for each y ∈ α.

Indeed, if P is a plaque of Lx in Ui, it is in the boundary of a strip of Vi, of width

ǫ that does not contain any singularities by the hypothesis on α. If two plaques of Lxoverlap, the strips in question glue together by the properties of the atlas. Hence φ is

an immersion. Injectivity holds by the facts that Φ−1(α) = α×{0} and that for every

point of the image of Φ, only one leaf passes through.

Let z be a point of recurrence of the leaf Lx; if z ∈ Ui there exist infinitely

many strips of size ǫ, components of Im Φ ∩ Vi. But two distinct bands are disjoint—

impossible.

Corollary 5.2. If a leaf L of F is not closed in M − sing(F ), and if α is an arc transverse to

F intersecting L, then α ∩ L is infinite.

Proof. It suffices to show that it is impossible that α ∩ L is one endpoint of α. For

this, we cut M along intα to obtain M ′, equipped

DR

AFT

13 J

un 2

008


with the induced foliation F ′. If C is the curve of ∂M ′ arising from α, then along C

the foliation F ′ gives the configuration of Figure 5.3 with two singularities s1 and s2corresponding to the endpoints of α; to L corresponds two leaves Lg and Ld of F ′

emanating from s1.

1

s2

L dL g

s

Figure 5.3:

By the theorem, Lg (resp. Ld) reaches a singularity of F ′ or the boundary ofM ′. If

this boundary isC, by the hypotheses on α, we conclude thatLd = Lg, which implies

that L is closed—contradiction. Otherwise, considering M ′ contained in M , Lg and

Ld reach singularities of F or the boundary of M ; thus L is closed (contradiction).

5.1.6 The Euler–Poincare formula.

Let M be a compact surface equipped with a foliation F having the singularities

allowed in Sections 5.1.2 and 5.1.3. We recall that each component of the boundary is

(A) either transverse to F ,

(B) or a cycle of leaves (a finite union of leaves and singular points).

To each singularity s, we associate an integer Ps:{Ps = number of separatrices, if s ∈ intM or if s ∈ ∂M (case (B)),

Ps = number of separatrices + 1 if s ∈ ∂M (case (A)).

DR

AFT

13 J

un 2

008


Then the following holds.

Formula 5.1 (Euler–Poincare Formula).

2χ(M) =∑

singF(2− Ps).

Proof. We begin by reducing to the case where ∂M does not contain singularities fol-

lowing the procedure shown in Figure 5.4. By pushing each singularity of the bound-

ary

DR

AFT

13 J

un 2

008


into the interior as shown, we preserve the integer Ps.

(B)

(A)

Figure 5.4:

Denote by Σ′ the set of singular points with an odd number of separatrices, Σ′′

the set of singular points whose number of separatrices is even and Σ = Σ′ ∪Σ′′. We

have an orientation homomorphism of the tangent bundle of F :

π1(M − Σ)→ Z/2.

This defines a 2-sheeted covering that extends over Σ′′ and is branched over Σ′. We

therefore have a branched covering p : M → M , where M is equipped with an sin-

gular orientable foliation F , which we may think of as generated by a vector field X ;

if s is a singularity of F , then Ps is an even integer and the index of X at s is−Ps2

+1.

Since there are no singularities on the boundary, we have:

χ(M) =∑

sing eX

index =∑

sing eF

(−Ps2

+ 1),

or

χ(M) = 2χ(M)− card Σ′ and∑

sing eF

1 = 2 card Σ′′ + card Σ′.

Finally, if p(s) ∈ Σ′′, Ps = Pp(s), but s has a “twin”; if p(s) ∈ Σ′, Ps = 2Pp(s). By

regrouping the equalities one has the desired formula.

N.B. In the computations, we must not forget that Ps ≥ 3.

DR

AFT

13 J

un 2

008


5.1.7 Quasitransverse curves

We say that a curve γ is quasitransverse to F if each connected component of γ −quasitransverse

sing F is either a leaf or is transverse to F . Further, in a neighborhood of a singular-

ity, we insist that a transverse arc is not in a sector adjacent to an arc contained in a

leaf (Figure 5.5), and two transverse arcs are in two distinct sectors.

DR

AFT

13 J

un 2

008


leaf

Figure 5.5:

Proposition 5.3. There does not exist a disk D with corners, with ∂D = α ∪ β, where α is

an arc contained in a leaf and β is a quasitransverse arc.

Proof. Suppose that such a disk exists. Let N ∼= D2, the double of D along β; N

is endowed with a foliation with permissible singularities. But χ(N) > 0, which

contradicts the Euler–Poincare formula.

Remark. In the same way, we see that an immersed closed curve quasitransverse to

F is not homotopic to a point.

5.2 Measured foliations and simple curves

5.2.1 Notation

MF(M), or simply MF , when there is no ambiguity, denotes the set of measured

foliations with permissible singularities, on a given compact surface M , subject to

the following two equivalence relations.

• isotopy

• Whitehead operations:

DR

AFT

13 J

un 2

008


(1)

(We specify that if the two singularities are on the boundary, we only contract if the

connecting leaf is on the boundary.)

(2)

Recall that S denotes the set of homotopy classes (= isotopy classes) of simple

closed curves that are piecewise C∞, not homotopic to a point and not isotopic to a

curve of the boundary.

5.2.2 I∗ :MF → RS+

Let (F , µ) be a measured foliation and γ a closed curve. We set µ(γ) = sup(∑µ(αi))

where α1, . . . , αk are the arcs of γ, mutually disjoint and transverse to F , and where

the sup is taken over all the sums of this type. In other words, µ(γ) is the total varia-

tion of the y coordinate along γ in an atlas that defines the measured foliation. This

quantity is also denoted by Thurston as∫γ F . Let σ be an element of S; we set:

I(F , µ;σ) = infγ∈σ

µ(γ).

This is clearly an isotopy invariant; moreover, if (F , µ) and (F ′, µ′) differ from each

other by a Whitehead operation, then, for each curve γ ∈ σ and each ǫ > 0, there

exists γ′ ∈ σ such that |µ(γ)− µ′(γ′)| < ǫ (Figure 5.6).

DR

AFT

13 J

un 2

008


γ

γ ’

Figure 5.6:

This suffices to ensure that the above formula defines a function:

I∗ : MF → RS+

〈I∗(F , µ), σ〉 = I(F , µ;σ).

5.2.3

Proposition 5.4. If γ is quasitransverse to F , then

µ(γ) = I(F , µ;σ)

where σ is the homotopy class of γ.

Proof. Let γ′ ∈ σ; if γ and γ′ are disjoint, γ and γ′ bound an annulus A. By Poincare

Recurrence (Theorem 5.1), almost every leaf entering A at a point of γ meets the

boundary again; by index considerations (Proposition 5.3), it cannot meet γ again.

Hence µ(γ) ≤ µ(γ′).If γ and γ′ have points in common, we proceed as follows. We begin by putting γ′

in general position with respect to γ, in the sense that γ′−γ is a finite number of open

intervals; this is done by an approximation that changes the measure by an arbitrarily

small amount. Since γ and γ′ are homotopic, there exists an arc α′ in γ′ and an arc α

in γ such that intα ∩ intα′ = ∅, and α∪α′ bounds a diskD (Proposition 3.10). Almost

every leaf entering D at a point of α meets the boundary again. Thus µ(α) < µ(α′).

If γ′ = α′ ∪ β′, we may form γ′′ = α′′ ∪ β′′, with α′′ = α and β′′ = β′. We have

µ(γ′′) ≤ µ(γ′) and π0(γ′′ − γ) < π0(γ

′ − γ). Thus, by induction on π0(γ′ − γ), we

prove that µ(γ′) ≤ µ(γ).check direction

of last inequalityTo determine the classes of S that contain quasitransverse curves, we require the

following lemma about the holonomy map.

DR

AFT

13 J

un 2

008


5.2.4 Stability Lemma

Let γ be an arc (arc = compact arc) in a leaf, and let α, β be two disjoint transverse

arcs each leaving from an endpoint of γ, both on the same side. Denote by Lt the

leaf passing through α(t); α(0) and β(0) are the endpoints of γ in L0. We choose the

parameterization in such a way that

µ([α(0), α(t)]) = µ([β(0), β(t)]) = t.

There is a germ for the holonomy map

hγ : (α, α(0))→ (β, β(0))

characterized by the following property: hγ is continuous and if hγ(α(t)) is defined,

we

DR

AFT

13 J

un 2

008


have hγ(α(t)) ⊂ Lt; the invariance of the measure µ implies that hγ is an isometry,

that is to say, that hγ(α(t)) = β(t); we denote by {γt} the continuous family of arcs,

such that γ0 = γ, γt ⊂ Lt, and γt joins α(t) to β(t).

Lemma 5.5 (Stability Lemma). If hγ is defined on the open interval [α(0), α(t0)), then

the points α(t0) and β(t0) are joinable by an arc γt0 that is contained in a union of a finite

number of leaves and singular points and that is the limit of the arcs γt, where t ∈ [0, t0).

t

γ0

( )t0

( )t0

αβ

γ

γ =

Figure 5.7:

Furthermore, there exists an immersion H : [0, 1] × [0, t0] → M that is C∞ on the

interior and such that we have H([0, 1]× {t}) = γt for all t ∈ [0, t0].The only obstructions to prolonging hγ beyond α(t0) are:

• α(t0) (resp. β(t0)) is an endpoint of α (resp. β);

• γ(t0) contains a singularity.

Proof. We use the good atlas of Section 5.1.4, and the notations Uj , Vj , ǫ0. We may

clearly reduce to the case where t0 < ǫ0, where the arc [α(0), α(t0)] is contained in

a chart Vj0 , and where the arc [β(0), β(t0)] is contained in a chart Vj1 . We then cover

γ0 by the charts Uj0 = U0, U1, . . . , Un = Uj1 ; the numbering is chosen in such a way

that there is for each i a plaque P 0i of Ui, contained in Ui ∩ γ0, satisfying P 0

i ∩P 0j = ∅,

except when |j − i| = 1; of course, the labelling can repeat the same chart several

times.Consider the union X0 =

⋃{P t0 | t ∈ [0, t0]} of plaques of V0 that intersect

[α(0), α(t0)]; the eventual singularity of V0 can only be found on the plaque P t00 ; oth-

erwise the holonomy map would not be defined on [α(0), α(t0)). If we pass to the

chart V1, we find an intersection X0 ∩ V1 that is a rectangle of width t0, by the prop-

erties of a good atlas. We construct the union X1 of plaques of V1 that meet X0 ∩ V1

and we continue in this way for the rest.

DR

AFT

13 J

un 2

008


Remark. 1) The lemma requires the invariant measure; Figure 5.8 is a counterexam-

ple in the case where there is a nontrivial holonomy:

Figure 5.8:

2) The lemma remains true if γ0 passes through singularities whose separatrices

are on sides opposite of α and β.

5.2.5

Corollary 5.6. We suppose thatM is not the torus T 2. Let γ be a cycle of leaves (i.e. a simple

closed curve that is a union of leaves and singularities); either γ passes through singularities

and there exist separatrices on both sides of γ, or γ belongs to a “maximal annulus” A whose

interior leaves are cycles. In the latter case, a component of ∂A that is not in ∂M is a singular

cycle.

5.2.6

Proposition 5.7. Let γ be a simple closed (connected) curve in the surface M and (F , µ) a

measured foliation. If γ separates M into two components, M = M1 ∪γ M2, we denote by

Σi (i = 1, 2) a “spine” of Mi (i.e., a 1-complex onto which Mi deformation retracts).

1. If I(F , µ; [γ]) 6= 0, there exists (F ′, µ′) equivalent to (F , µ), such that γ is transverse

to F ′ and avoids the singularities.

DR

AFT

13 J

un 2

008


2. If I(F , µ; [γ]) = 0, there exists (F ′, µ′) equivalent to (F , µ) satisfying one or the other

of the following two (nonexclusive) conditions:

(a) γ is a cycle of leaves of F ′;

(b) γ separates and, for some i ∈ {1, 2}, Σi is an invariant set of F ′.define invariant

set: union of

leaves connect-

ing singularites?

invariant under

holonomy map?

This situation only occurs if the set of connections between the singularities has cycles.

Remarks. 1) If we do not allow modification of F , we obtain only the

DR

AFT

13 J

un 2

008


much weaker result that γ is homotopic to an immersion quasitransverse toF . More-

over, this immersion is a limit of embeddings.

2) Figure 5.9 illustrates the situation of case 2(b) of the proposition. The foliation of

the surface of genus 2 is obtained by “enlarging” the curve C (see Section 5.3).

singular leavesC

γ

Figure 5.9:

The first three paragraphs of the proof of Proposition 5.7 constitute a proof of

the following criterion, which will be useful later. Note that, by Proposition 5.4, to

prove the lemma, one only needs to show that if there is no disk D as in the second

statement of the lemma, then γ is isotopic to a quasitransverse curve of the same

length.

Lemma 5.8 (Minimality Criterion). The following two assertions are equivalent:

1. µ(γ) > I(F , µ; [γ]);

2. There exist two points x0 and x1 of γ belonging to the same leaf L so that

x0 ∪ x1 = ∂c,where c is an arc of L,

= ∂c′,where c′ is an arc of γ, and

c ∪ c′ = ∂D,where D is a 2-disk.

Proof. [Proof of Proposition 5.7] We may suppose that γ = α1 ∗ β1 ∗ · · ·αk ∗ βk ∗ fix proof label

· · · ∗ αn ∗ βn, where the arcs αi are transverse to F and where the arcs βj , possibly

reduced to a point, are in a finite union of leaves and singular points; the labeling is

DR

AFT

13 J

un 2

008


cyclic. If we do not begin with such a decomposition, we either obtain one in each

chart by an isometric isotopy, or there exists a chart in which the 2nd conclusion of

the minimality criterion is visible and a length reducing modification leads to a finite

decomposition.

Now, each βk is in one of the configurations shown in Figure 5.10. In configura-

tion 1, we can apply the Stability Lemma (Lemma 5.5); in configurations 2 and 3, βkcontains at least one singularity, and Lemma 5.5 is not applicable; in configuration 4,

βk does not contain

DR

AFT

13 J

un 2

008


any singularities.

(4)

βk

α k+1α k

βk

(1) (2)

(3)

Figure 5.10:

In configuration 1, we see a disk as in the second conclusion of the Minimality

Criterion. We claim that, if for all k, βk is not in configuration 1, then γ is isotopic to a

quasitransverse curve of the same length; that is, µ(γ) is minimal by Proposition 5.4, and

so the claim proves the Minimality Criterion. To prove the claim, we replace each

configuration of type 4 by a transversal; each configuration of type 2 or 3 is modified

as in Figure 5.11.

Figure 5.11:

Let us now continue the proof of the proposition. As in the proof above, the βk of

type 4 are replaced by transverse arcs, and those of types 2 and 3 may be supposed

to have singularities as endpoints.

DR

AFT

13 J

un 2

008


At this point, either γ is a cycle of leaves (conclusion 2(a) of the proposition), or

it is possible to shrink each full leaf contained in γ to a point, thus eliminating arcs of

type 2 and 3 (first arrow of Figure 5.12). By then performing Whitehead operations

as in the second arrow of Figure 5.12, we reduce to the situation where all of the arcs

βk are of type 1. From there, the induction is done on

DR

AFT

13 J

un 2

008


the number of arcs of γ contained in a leaf. If there are none, γ is transverse to the

foliation (conclusion 1).

Figure 5.12:

Otherwise, consider β1, which is in configuration 1 by assumption. Applying the

Stability Lemma (Lemma 5.5) to the arcs β1, α0, and α1, we obtain an immersion

h of a rectangle R. The induced foliation F = h−1(F) has all of its singularities in

the same arc λ of the boundary. We denote by β1, . . . , βm the arcs of γ = h−1(γ)

that are in the leaves of F (horizontal arcs). Let us say that β1 is the arc closest to

the singularities (in the sense of the transverse measure); then the component of γ

that contains β1 bounds a subrectangle R′ that is minimal; we see that h|int R′ is an

embedding disjoint from γ.

R’

^

1

(F)−1

h

R

λ

β

Figure 5.13:

If R′ does not contain any singularities of F , a neighborhood of h(R′) is the sup-

port for an isotopy of γ that gets rid of β1; even if h(β1) = β1, the application of this

isotopy leads to a situation where, in the new rectangleR associated with β1, the new

γ has fewer horizontal arcs.

DR

AFT

13 J

un 2

008


If R′ has a singularity, then, because of the transverse measure,

DR

AFT

13 J

un 2

008


it is easy to see that h(β1) is an arc βk distinct from β1 (if not the width of R′ would

be the same as that of R).

By the above reasoning, perhaps after cyclically relabeling the arcs, we may sup-

pose that:

1. h|R−λ is an embedding,

2. (h|int(R)) ∩ γ is empty,

3. h(λ) ∩ βk is empty for all k.

We first brush aside the following simple cases A and B, where there are visible

operations—isotopy and Whitehead operations—that reduce the number of arcs of

γ contained in a leaf.

(A) λ does not contain a singularity. See Figure 5.14.

R

λ

Figure 5.14:

(B) λ contains singularities andR is embedded. The isotopy acrossR replaces β1 with

an arc of type 2. We then perform the procedure from the beginning of the proof.

We may now assume that R is not embedded, and so h(λ) has double points.

Viewed as a singular chain, λ is written as a composition:

λ = µ0 ∗ λ1 ∗ · · · ∗ λq ∗ µ1

where µ0 (resp. µ1) is an arc of a leaf joining a point of α1 (resp. α2) to a singularity

and where λi (1 ≤ i ≤ q) is an arc of a leaf joining two singularities;

DR

AFT

13 J

un 2

008


some of these arcs may be a single point and several may belong to the same leaf.

However, λ has in R an approximation that is an embedded arc only meeting α1

and α2 at its endpoints; because of this, each leaf carries at least two arcs of λ. More

precisely, neither µ0 nor µ1 may belong to the same leaf as a λj ; if µ0 ∩ µ1 is not

reduced to one of their endpoints, then α1 = α2 (i.e., γ = α1 ∗ β1) and we have the

configuration of Figure 5.15.

0

µ 1α 1

β1

µ

Figure 5.15:

We will say that λj is simple if, for every j′ 6= j, λj does not cover the same leafsimple

as λ′j . We say that µ0 and µ1 are simple if one does not have the configuration of

Figure 5.15 (µ0 and µ1 are both simple or both not simple).

Denote by Λ the 1-dimensional complex

q⋃

i=0

λi ; this is an invariant set of the foli-

ation F . If M is closed, each Whitehead operation of Λ lifts to a Whitehead operation

of F (the terminology for foliations was chosen because of this remark.)

planar triangleshift

Lemma 5.9. (M is assumed closed) If one of the arcs λi, µ0, or µ1 is simple, there exists a

foliation F ′, equivalent to F , and equal to F in the complement of a neighbourhood of Λ, and

for which the limit arc λ′ of the domain of deformation of R′ of β1 has fewer double simplices

(edges or vertices).

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008


Proof. We slide the simple arc on its predecessor or on its successor. Figure 5.16

exhibits this operation when µ0 is simple.

R’0µ1

R

λ’

µ

Figure 5.16:

If the lemma is applicable, we reduce by induction to case B; otherwise, we find

ourselves in the following situation.

(C) All of the arcs λi, µ0 and µ1 are doubled.

In this case, the closure of R in the surface, is a regular neighborhood of the

complex Λ and γ is its boundary; we thus have conclusion 2(b) of the proposition.

In the case where M is closed, the proof of the proposition is completed by in-

duction on the number of segments of the decomposition of γ. The case of surfaces

with boundary is analogous, but one must pay attention to the Whitehead operations

permitted.

Remark. The preceding proposition does not admit a reasonable generalization to

the case of a system with k embedded curves γ1, . . . , γk, except if I(F , µ; [γ1]) 6=

DR

AFT

13 J

un 2

008


0, . . . , I(F , µ; [γk−1]) 6= 0, and I(F , µ; [γk]) is possibly nonzero.

DR

AFT

13 J

un 2

008


5.3 Curves as measured foliations

5.3.1 The “enlarging” procedure

Let M0 be a submanifold of dimension 2 in M such that M −M0 does not have any

contractible components. Let Σ be a “spine” of M −M0; by hypothesis, none of its

components are contractible; thus, perhaps after collapsing the 1-simplices that have

a free vertex, each singularity of Σ has three branches leaving from it.

We may construct a surjective map j : M0 →M such that:

• j is a (piecewise differentiable) immersion,

• j|int M0 is a diffeomorphism onto M − Σ,

• j(∂M0 − ∂M) = Σ,

• j is the identity outside of a small collar neighborhood of ∂M0 − ∂M .

Σ

M 0

Figure 5.17:

Let F0 be a measured foliation on M0 such that each component of ∂M0 − ∂M is

an invariant set. We may then define F = j∗(F0), which is a measured foliation on

M satisfying:

• Σ is an invariant set of F ,

• j|int(M0) conjugatesF0|int(M0) andF|(M−Σ) as measured foliations. We say that

F is obtained from F0 by enlarging M0.

DR

AFT

13 J

un 2

008


We remark that if Σ′ is another spine of M −M0, then Σ′ is obtained from Σ by

Whitehead operations and isotopies (see Appendix A). We conclude that the class of

F only depends on that of F0. We have therefore defined a map

MF(M0, ∂M0 − ∂M)→MF(M)

DR

AFT

13 J

un 2

008


for which the domain is the subset of MF(M0) consisting of the foliations where

every component of ∂M0 − ∂M is an invariant set.

Lemma 5.10. Let µ0 and µ1 be transverse invariant measures for F0 and F . Let γ be a

simple curve in M . Then I(F , µ; [γ]) = inf µ0(γ′ ∩M0), where γ′ is isotopic to γ.

Proof. This follows from the following remark: for each curve C, there exists a curve

C′, isotopic to C, such that C′ ∩M0 = j−1(C).

5.3.2 The inclusion R+ × S → MFLet C ∈ S, λ ∈ R∗

+. Consider a tubular neighborhood M0 of C which we foliate

by circles parallel to C; we equip this with an invariant transverse measure µ0 such

that the width of the annulus M0 is λ; this measured foliation of M0 is unique up to

isotopy. We denote by Fλ,C a foliation obtained from the latter by enlarging and by µ

its transverse measure.

Proposition 5.11. Let γ be a simple curve in M . Then we have

I(Fλ,C , µ; [γ]) = λi(C, γ).

Proof. Let α be a component of γ ∩M0. If α goes from one boundary of M0 to the

other, then µ0(α) ≥ λ. By isotopy, we deform α until it is transverse to the foliation;

then α∩C = 1 point and µ0(α) = λ. If α does not touch a component of the boundary,

then γ is isotopic to γ′ whose intersection withM0 has one less component. Applying

the preceding lemma, we have the inequality:

I(Fλ,C , µ; [γ]) ≥ λi(C, γ).

The equality is obtained by considering the case where γ has minimal intersection

with C, for then, we have

µ0(γ) = λi(C, γ).

The preceding proposition gives the commutativity of the diagram below:

R∗+ × S

i⋆

MF

I⋆

RS+

As i∗ is injective (by Proposition 3.18), R∗+ × S →MF is also an injection.

DR

AFT

13 J

un 2

008

Expose 6

Measured Foliations

(Continued)

by A. Fathi

The goal of this expose is to classify measured foliations on a compact surface (here,

it will be closed and orientable). Like in the case of curves, we will reduce to the

case of foliations on pairs of pants. To do this, we choose curves K1, . . . ,K3g−3, that

decompose the surface into pairs of pants.

If (F , µ) is a foliation such that, for all j, I(F , µ; [Kj ]) 6= 0, we perform isotopies

and Whitehead operations in order to reduce to the case where the Kj are transverse

to F . Such a foliation is classified by the measures of the curves Kj and by the twists

about these curves, which themselves are expressed according to measures

99

DR

AFT

13 J

un 2

008


of certain curves (see Appendix C).

In the case where the lengths of some of the Ki are zero, we can hope to modify

F so that these curves will all be cycles of leaves, and then to classify the foliations

as above. This unfortunately is not always possible. Here is an example on a surface

M of genus 3.

K 3

K 4

K5

K6

K 1

K 2

Figure 6.1:

MAKE CURVES

MORE BOLD In the example, the curves drawn in bold are singular leaves, and all other leaves

are curves isotopic to K6. It is impossible to modify this foliation (by Whitehead

operations and isotopies) so that the curves K3 and K4 become cycles of leaves. This

foliation is obtained in fact by starting with a foliation on an annulus A around K6

and collapsing M −A onto a spine. In this operation, the two pairs of pants that

touch K1 are collapsed onto the union of three closed curves.

We are thus obliged to take this kind of phenomenon into account. That is why we

have introduced the operation of enlargement (see the preceding expose): a foliation

F of a surface M is obtained by enlargement of a foliation F0 of a subsurface M0

(with boundary) if F is the image of F0 under a map M →M obtained by extending

a collapse of M −M0 onto a spine. Such a foliation is essentially “carried” by M0

since the transverse lengths of curves contained in M −M0 are zero.

Using this operation, we can find canonical (“normal”) forms of foliations and

proceed to the classification.

DR

AFT

13 J

un 2

008


6.1 Classification of foliations of the annulus

6.1.1

By the Euler–Poincare formula, a measured foliation on S1× [0, 1] does not have any

singularities. If S1 × {0} is a leaf, then, by the Stability Lemma, all the leaves are

closed curves. If S1 × {0} is transverse to the foliation, then all the leaves go from

one boundary to the other. Thus, if (θ, x) denotes the coordinates of S1 × [0, 1], all

measured foliations of the annulus are isotopic to those associated to λdθ or to λdx,

λ ∈ R⋆.

6.1.2

We want to give a classification of measured foliations of the annulus A modulo

the action of the group Diff0(A rel ∂A) of diffeomorphisms that are isotopic to the

identity relative to the boundary. We choose once and for all an arc γ joining the

two boundary components and an arc γ differing from γ by a twist in the positive

direction (Figure 6.2).

1

γT0

γ

T

Figure 6.2:

DR

AFT

13 J

un 2

008


If (F , µ) is a measured foliation on A, we set

m = µ(S1 × {0}) = µ(S1 × {1}) = I(F , µ; [S1 × {0}]);s = inf{µ(γ′) : γ′ isotopic to γ with endpoints fixed};t = inf{µ(γ′) : γ′ isotopic to γ with endpoints fixed}.

6.1.3 Lemma

Lemma 6.1. A triple (m, s, t) of three positive numbers is associated to a measured foliation

of A if and only if (m, s, t) belongs to ∂(≤ ∇) (boundary of the triangle inequalities).

Proof.

DR

AFT

13 J

un 2

008


We consider the triangle T1 that is hatched in Figure 6.2. Having done an isotopy

such that γ and γ are transverse to the foliation, any leaf entering through one edge

of the triangle leaves through one of the others. In this situation, it is clear that the

three measures of the edges form a triple belonging to ∂(≤ ∇). It is clear also that

this condition is the only one that needs to be satisfied so that a triple is associated to

a measured foliation.

If m = 0, then each curve of the boundary is a leaf and, with the coordinates

(θ, x), the foliation is isotopic (rel ∂A) to t dx = s dx. This case being excluded, the

foliation is transverse to the boundary.

6.1.4 Proposition

Proposition 6.2. Let F and F ′ be two measured foliations that are transverse to the bound-

ary of the annulus and that coincide on the boundary (we mean equality of the induced mea-

sures; in particular, m(F) = m(F ′)). Then F and F ′ are isotopic by an isotopy which is

constant on the boundary if and only if (s, t)(F) = (s, t)(F ′).

Proof.

Only the sufficiency is nontrivial. We deform F and F ′ until

s = µ(γ) = µ′(γ)

t = µ(γ) = µ′(γ).

Then γ and γ are transverse to the two foliations, unless one of these arcs is a leaf. In

the case of transversality, a second isotopy makes the measures induced on γ (resp.

γ) coincide. Then the foliations coincide on the boundary of each of the triangles T0

and T1. We know that such data on the boundary of a disk has a unique extension

(for example, by Theorem 2.1 in Expose 2).

6.2 Classification of foliations on a pair of pants

We denote the pair of pants, or, disk with two holes, by P 2.

DR

AFT

13 J

un 2

008


1

γ2

γ3

γ

Figure 6.3: P 2 = ’pair of pants’ = disk with two holes

6.2.1 Lemma

Lemma 6.3. For a foliation of P 2 with permissible singularities (Expose 5) either there only

one singularity (4 separatrices) or there are two singularities, each with 3 separatrices.

The lemma follows from the Euler–Poincare formula.

6.2.2 Definition

We say that F is a good foliation of P 2 if no component of ∂P 2 is a smooth leaf of F .

Lemma 6.4. Let F be a measured foliation of P 2. Then:

1o Every leaf is closed in the complement of the singularities.

2o If, further, F is a good foliation, there are no cycles of leaves interior to P 2.

Proof.

1o We suppose that L is a non-closed leaf of F in P 2− sing(F); it enjoys Poincare

Recurrence and so one can find an arc β on L and a transversal α so that α ∪ β is a

simple closed curve. We have two possible configurations (Figure 6.4).

By the corollary of the Poincare Recurrence theorem (Corollary 5.2 in Expose 5),

L intersects α infinitely many times; since α ∪ β disconnects P 2, configuration II re-

duces to configuration I. We can then approximate α ∪ β by a closed curve γ that is

transverse to F and that intersects L infinitely many times; by Proposition 5.3 (Ex-

pose 5) γ does not bound a disk; therefore γ, together with some component γ1 of

∂P 2, bounds an annulus. Each leaf that intersects γ also intersects γ1. This implies

that L cannot intersect γ infinitely many times, a contradiction.

DR

AFT

13 J

un 2

008


(I)

β

β

α

β

β

α (II)

Figure 6.4:

2o If γ is an interior cycle, γ ∪ γ1 bounds an annulus A; in the neighborhood of γ

in A the leaves are smooth and closed and, by the Stability Lemma (Lemma 5.5), γ1

is a smooth closed leaf, which is a contradiction.

DR

AFT

13 J

un 2

008


Corollary 6.5. Every leaf of a good foliation either goes

• from the boundary to the boundary,

• from the boundary to a singularity,

• or from a singularity to a singularity.

6.2.3 Reduced good foliations

A reduced good foliation is a good foliation of P 2 satisfying the following conditions:

(i) if a component of the boundary is a transversal, it contains no singu-

larities,

(ii) the singularities of the boundary are simple (3 separatrices),

(iii) there are no connections between two singularities where at least one

is interior.

LetMF0(P2) be the set of equivalence classes of good measured foliations of P 2.

Lemma 6.6. In each class ofMF0(P2), there exists a unique reduced good foliation up to

isotopy.

Proof.

We secure property (i) immediately. Then, by Section 6.2.2, if a foliation admits

two simple singularities connected by two distinct arcs α1, α2, we have: α1 ∪ α2 is a

component of the boundary. That is to say, for a non-reduced foliation, there is, up to

isotopy, only one way to reduce.

6.2.4 Classification

Theorem 6.7. The functionMF0(P2) → R3

+, which to a good measured foliation (F , µ)

associates the triple (m1,m2,m3) = (µ(γ1), µ(γ2), µ(γ3)), induces a bijection ofMF0(P2)

onto R3+ − {0}.

Proof.

We begin by describing a right inverse; the construction depends on the position

of the triple with respect to the triangle inequality; to each simplex, we associate one

topological configuration. These are given below for the 6 types of simplices.

DR

AFT

13 J

un 2

008


(4)(1)

(3) (5)

(6)

(2)

Figure 6.5:

(5)

(1) (2) (3)

(4) (6)

Figure 6.6:

We remark that if we decompose these figures along the separatrices, we obtain

foliated rectangles where the widths ( = largest measure of a transversal) are deter-

mined by the triple. For example in configuration (1), the widths of the 3 rectangles

DR

AFT

13 J

un 2

008


are:

a12 =1

2(m1 +m2 −m3)

a13 =1

2(m1 +m3 −m2)

a23 =1

2(m2 +m3 −m1).

DR

AFT

13 J

un 2

008


In configuration (3), we have the formulas:

a11 = 12 [m1 − (m2 +m3)]

a12 = m2

a13 = m3.

It is easy to see that up to renumbering, these figures represent all the possibilities

up to isotopy for the separatrices of a reduced foliation; all the other configurations

are prohibited by the Euler–Poincare formula. We deduce right away that two folia-

tions giving the same triple are isotopic.

6.2.5

We consider now the case where some curves of the boundary are smooth leaves. We

have the following general proposition.

Proposition 6.8. Let F be a measured foliation of P 2. Then F is obtained by enlargement of

a foliation F0 of a submanifold P0 having the following properties:

Each connected component C of P0 is:

1o Either a pair of pants and the foliation F0|C is a good foliation;

2o Or a collar neighborhood of a curve of ∂P 2 and in this case F0|C is a foliation by

circles.

Proof.

If each component of the boundary is not smooth, we take P0 = P 2. Otherwise,

we consider a smooth leaf γ1 in ∂P 2. We consider the maximal “annulus” A associ-

ated to γ1 by The Stability Lemma (Lemma 5.5 of Expose 5). If A = P 2, we take for

P0 a collar neighborhood of γ1 foliated by circles, where the F0-width is the F -width

of A.

If A 6= P 2, there exists a leaf L of F in the interior of P 2 that belongs to the

topological frontier of A; the leaf L goes from a singularity s0 to a singularity s1.

If s0 = s1, L is a cycle of leaves forming a Jordan curve which bounds a true

annulus A′ foliated by circles. The domain P 2 −A′, which is a pair of pants, possibly

pinched if s0 belongs to ∂P 2, is foliated with fewer smooth leaves in its boundary.

We kill the other smooth components in the same fashion.

DR

AFT

13 J

un 2

008


If s0 6= s1, the singularities are simple (Figure 6.7) and some other leaf L′ leaves

s0 in the frontier of A. If L′ returns to s0, L′ is an embedded cycle. Otherwise L′ goes

to s1 and L ∪ L′ is a Jordan cycle. In either case, we continue as above.

L’

1s0

L

s

Figure 6.7:

6.3 The arcs jaune

6.3.1

We will need a technical ingredient which will allow us to give coordinates for a

measured foliation, using its image on each piece of a pair of pants decomposition of

the given surface.

For each component C of the boundary of the pair of pants P 2, and for each type

of good foliation on P 2, we choose a quasitransverse arc that has endpoints in C, and

that is essential (not homotopic to an arc of the boundary). We call it the arc jaune

(plural: arcs jaune). For C = γ1 and for each type of good foliation, we choose the arc

indicated in bold in the figures below.

DR

AFT

13 J

un 2

008


3

Generic case:

m > m + m(or m > m + m )

12 3

1 2

Figure 6.8:

MAKE THE

BOLD MORE

BOLD. ALSO

THERE IS A

MISSING PIC-

TURE.

+)(<m1 m2 m 3 m1 m2 m 3

same measure

, ,( ) >

Figure 6.9:

DR

AFT

13 J

un 2

008


3

m 3m1m2 = +m 3m1m2 = +

m1 m2 m 3

( )<1 2 3Case where ( m , m , m )

)(or

= +

with m = 0 , m = 0 , m = 0 :1 2

Figure 6.10:

:

m2 m 3 m2 m 3 m 3 = 0m2 = 0(or )m2m 3

m1

= > > 0(or )>

Case where = 0

Figure 6.11:

DR

AFT

13 J

un 2

008


same measure

Case wherem = 0 2 (or m = 0)3 :

m = 0 m > m > 0(or m > m > 0)(or m = 0)

3

2

1 3

1 2

same measure

Figure 6.12:

m = m1 3

(or m = m )1 2

0 < m < m1 3

(or 0 < m < m )1 2

Figure 6.13:

6.3.2 Length of the arc jaune

We remark that a arc jaune realizes the minimum of the length of the essential arcs

going from γ1 to γ1 (by quasitransversality). Its length (in the sense of the transverse

measure) is given by the formula

l1 = sup

(m2 +m3 −m1

2, 0

)+ sup

(m2 −m1 −m3

2, 0

)+ sup

(m3 −m1 −m2

2, 0

)

The proof is done by examining all the cases of the figure.

DR

AFT

13 J

un 2

008


6.3.3 Definition of the arc A.

The arc jaune demarcates two arcs on γ1, with one of them being possibly reduced

to a point. We denote by A the one that is the same side as γ2 with respect to the arc

jaune (one would have to denote A by A12, but there will not be any ambiguity later).

Its length a is given by the formula:

a = sup

(m2 +m1 −m3

2, 0

)− sup

(m2 −m1 −m3

2, 0

).

6.4 Normal form of a foliation

6.4.1

Let M be a closed surface of genus g ≥ 2; let K1, . . . ,K3g−3 be a family of curves that

separate M into pairs of pants. We denote by {Rj} the 2g − 2 pairs of pants; each Rjis the closure of one of the components of M − ∪Ki.

We will need the following technical condition: each Rj must be the image of an

embedded pair of pants; in other words, each Ki does not lie in the same pair of pants

on its two sides. With this condition, we say that we have a permissible decomposition

of M .

K 1 R1

K2

K 3R2

Figure 6.14:

For each i = 1, . . . , 3g− 3, we choose a tubular neighborhood Ki × [−1,+1] ⊂MofKi; if i 6= j, the neighborhoods Ki andKj are disjoint. We denote by {R′

j}1≤j≤2g−2

the connected components of M − ∪(Ki × (−1,+1)).

DR

AFT

13 J

un 2

008


K i

R’j

K [−1,+1]i

Figure 6.15:

6.4.2 Definition.

LetM0 be a compact submanifold ofM of dimension 2 andF0 a measured foliation of

M0. We say that (M0,F0) is in normal form (with respect to the data of the preceding normal form

paragraph) if the conditions below are satisfied.

1) Each component of ∂M0 is a cycle of leaves ;

2) M0 ∩R′j is empty or equal to R′

j ; in the latter case F0|R′j

is a good foliation;

3) M0 ∩Ki × (−1,+1) is equal to one of the following:

• ∅

• Ki × (−1,+1); in this case, F0 is transverse to the circles Ki × {t}, t ∈ [−1,+1];

we also remark that in this case M0 contains the pairs of pants adjacent to the

annulus;

• Ki × [−1/2,+1/2]; in this case the foliation has the Ki × {t}, t ∈ [−1/2,+1/2]

for leaves.

Let F be a measured foliation of M . We say that (M0,F0) is a normal form of F , if

(M0,F0) is in normal form and if F is obtained by enlargement of (M0,F0).

6.4.3

Proposition 6.9. Every measured foliation on M has a normal form.

Proof. Let (F , µ) be a measured foliation on M . Up to change of numbering, we can

suppose that:

DR

AFT

13 J

un 2

008


I(F , µ; [K1]) 6= 0, . . . , I(F , µ; [Kl]) 6= 0

and that I(F , µ; [Kl+1]) = · · · = I(F , µ; [K3g−3]) = 0.

Then, by Proposition 5.7 (Expose 5), by changing F in its class,

DR

AFT

13 J

un 2

008


we obtain that F is transverse to Ki × {t} for all t ∈ [−1, 1] and i = 1, . . . , l.

Let M ′ be the complement of the annuli Ki × (−1, 1), i = 1, . . . , l. We have an

induced measured foliation (F ′, µ), transverse to the boundary; the curves Ki, i ≥l+ 1, are contained in intM ′.

For i ≥ l + 1, we have I(F ′, µ; [Ki]) = 0. In fact, in the contrary case, there

exists F ′′, a measured foliation of M ′ equivalent to F ′, coinciding with F ′ near the

boundary, and transverse to Ki. But then, we have

I(F , µ; [Ki]) = I(F ′′, µ; [Ki]) 6= 0,

which is prohibited by the numbering.

Applying Proposition 5.7 of Expose 5 to F ′, M ′, and Kl+1, up to changing F ′ in

its class, we can say that there exists a homotopy ft : Kl+1 → M ′ such that f0 =

(Kl+1 → M ′) and that f1(Kl+1) is an invariant set Σ1 of F ′. When we apply the

proposition toKl+2, we do Whitehead operations on F ′ which induce shifts of Σ1, so

thatKl+1 continues to be homotopic onto an invariant set, etc. Then we can construct

F ′ and a homotopy ft : Kl+1 ∪ · · · ∪K3g−3 → M ′ such that f0 is inclusion and that

the image of f1 is an invariant set Σ of F ′.

Let N ⊂M ′ be a regular neighborhood of Σ; then F ′ is obtained by enlargement

of a foliation F ′1 on M ′

1 = M ′ −N . By construction, each component of N is not

a disk. As M ′1 has a measured foliation, each component of M ′

1 is not a disk. We

conclude that each component of ∂M ′1 is not homotopic to a point.

We can suppose (by engulfing) that intN contains all theKi, i ≥ l+1. Indeed, the

singular map f1 is close to an immersion f ′1; there exists a “Whitney disk” ∆ through

which one can do a homotopy whose effect is to decrease the number of double

points of f ′1. But, by what we have just seen about ∂N , as soon as ∂∆ is contained

in N , ∆ is contained in N . By induction on the number of double points, we see that

one can deform, by homotopy, f ′1 to an embedding, where the image is contained in

N and which, by the work of Epstein (see the references of Expose 3), will be isotopic

to f0.

We see that Rj ∩M ′1 is made of at most one pair of pants (“concentric toRj”) and

of a certain number of annuli parallel to the components of the boundary of Rj . The

boundary of a component of Rj ∩M ′1 can be of one of two types:

• Ki × {±1} for i ≤ l; transverse curve to F ′1;

• cycle of leaves of F ′1, parallel to one of the Ki, i ≥ l + 1.

DR

AFT

13 J

un 2

008


Then a component of M ′1 ∩ Rj , which is an annulus, can only be foliated by circles.

But a component of M ′1 ∩ Rj which is a pair of pants may not carry a good foliation;

we apply Section 6.2.5, which allows us to replace (M ′1,F ′

1) by (M ′0,F ′

0) satisfying:

• F ′0 = F ′

1 in a neighborhood of ∂M ′;

• (M ′1,F ′

1) is obtained by enlargement of (M ′0,F ′

0);

• if V is a component of Rj ∩M ′0 which is a pair of pants, F ′

0|V is a

good foliation.

By an obvious isotopy of (M ′0,F ′

0), making for example the V above coincide

with R′j , we obtain the desired normal form.

6.4.4 Definition of NFTwo pairs (M0,F0) and (M ′

0,F ′0) are equivalent if M0 = M ′

0 and F ′0 can be obtained

from F0 by a finite sequence of elementary operations of the following types:

• Whitehead operation with support in one of the R′j ;

• isotopy with support in Ki × [−1,+1] ∩M0;

• isotopy with support in Rj .

The set of equivalence classes is denotedNF (orNF(M)). Enlargement induces

a function NF → MF which is surjective by the preceding proposition; we will see

later that it is in fact bijective.

6.4.5 Classification of foliations in normal form

We refer here to the permissible decomposition of M given in Section 6.4.1:

M =

2g−2⋃

j=1

R′j

∪

(3g−3⋃

i=1

Ki × [−1,+1]

).

For each R′j , we choose once and for all a diffeomorphism to the standard pair of

pants, respecting the orientation. Further, for each i = 1, . . . , 3g−3, we choose curves

K ′i and K ′′

i ; if Rj1 and Rj2 are the two pair of pants (distinct, because the decomposi-

tion is permissible) that touchKi, thenK ′i is an essential simple closed curve 1 which

is not parallel to any of the curves of ∂Rj1 ∪ ∂Rj2 . The curve K ′′i is obtained from K ′

i

by a positive twist about Ki (see Figure 6.16).

1If the two sides of Ki were to belong to the same pair of pants, K ′

i would not be embeddable.

DR

AFT

13 J

un 2

008


i

K’i

K’’iK

Figure 6.16:

We set B = {(mi, si, ti) : i = 1, . . . , 3g − 3, mi, si, ti ≥ 0, (mi, si, ti) ∈ ∂(≤ ∇)}This is a cone in R9g−9

+ which is homeomorphic to R6g−6. We are going to construct a

function:

NF → B − 0.

Let (M0,F0) be the representative of an element of NF ; without changing its class,

we can suppose that F0|R′j

is in canonical form for all j such that M0 ∩ R′j 6= ∅. The

invariant mi is the measure of Ki, and is zero if Ki ∩M0 = ∅. The invariants si and tidepend on the form of the induced foliation on the annulus Ki × [−1,+1]; there are

three cases.

1st case. Ki × [−1,+1]∩M0 ⊂ Ki × {−1} ∪Ki × {+1}. We then set si = ti = 0.

2nd case. Ki × [−1/2,+1/2] = Ki × (−1,+1) ∩ M0. In this case, the annulus is

foliated by circles; then si = ti is the width of the annulus, that is to say the measure

of a transversal. Note that, in the two first cases,mi = 0; thus it is clear that (mi, si, ti)

belongs to ∂(≤ ∇).

3rd case. M0 ∩Ki× [−1,+1] = Ki× [−1,+1]. In this case, the foliation is transverse

to the circles Ki ×{x} for all x ∈ [−1,+1], and M0 contains the two pairs of pants R′k

and R′l adjacent to Ki × [−1,+1]. We have an arc jaune Jk and an arc jaune Jl. There

exist then two arcs Si and S′i in Ki × [−1,+1] such that Jk ∪ Si ∪ Jl ∪ S′

i is a closed

curve homotopic to K ′i; their homotopy classes with endpoints fixed in Ki× [−1,+1]

are completely determined. We choose for Si

DR

AFT

13 J

un 2

008


and S′i arcs of minimal length. With an orientation convention, we distinguish Si and

S′i. We set

si = µ0(Si), s′i = µ0(S′i)

where µ0 is the measure accompanying the foliation F0.

In the same way, we construct arcs Ti and T ′i such that Jk∪Ti∪Jl∪T ′

i is homotopic

to K ′′i . We set

ti = µ0(Ti), t′i = µ0(T′i ).

In short, the invariants (mi, si, ti) are in this case the invariants classifying the in-

duced foliation on the annulus Ki × [−1,+1], in the sense of the classification of

Section 6.1. In particular, we have: (mi, si, ti) ∈ ∂(∇ ≤).

It is very easy to see that the invariants mi, si, ti only depend on the class of

(M0,F0) in NF .

Lemma 6.10. The image ofNF in B does not contain 0.

Proof.

Let (M0,F0) be given. As M0 is not empty, we have one of the following situa-

tions:

1) For some i, M0 ∩Ki × (−1,+1) = Ki × [−1/2,+1/2]; then (si, ti) 6= 0.

2) For some j, R′j is contained in M0; as the induced foliation is a good foliation, one

of the curves of the boundary has a nonzero measure.

Being given that which has been said of the classification of the measured folia-

tions on the annulus and the pair of pants, we can leave as an exercise the details of

the following proposition.

Proposition 6.11. The function constructed above

NF → B − 0

is a bijection.

6.5 Classification of measured foliations

We always consider a closed orientable surface M of genus g > 1. We return to the

other cases in Expose 11.

6.5.1 Proposition

Proposition 6.12. There exists a continuous function θ : I⋆(MF)→ B, positively

DR

AFT

13 J

un 2

008


homogeneous of degree 1 (i.e. θ(λx) = λθ(x), for λ > 0), which makes the following diagram

commutative:

NF MF I⋆I⋆(MF)

θ

⊂ RS+

B

Proof.

SinceNF →MF is a surjection, it suffices to show that, for a foliation in normal

form (M0,F0), the invariants mi, si, ti only depend on the measures of simple closed

curves. It is immediately clear that mi = I(F0, µ0; [Ki]).We will show in the appendix that si and ti are determined by I(F0, µ0; [K

′i]) and

I(F0, µ0; [K′′i ]), via homogeneous continuous formulas.

Since NF → B is an injection, we immediately draw the following corollaries.

6.5.2 Theorem

Theorem 6.13. Two measured foliations (F , µ) and (F ′, µ′) on a surface M are Whitehead

equivalent if and only if, for all simple curves γ of M , we have

I(F , µ; [γ]) = I(F ′, µ′; [γ]).

6.5.3 Proposition.

The enlargement function NF →MF is a bijection.

6.5.4

Now, we can identifyMF with its image via I⋆, to provideMF with the topology

induced by RS+ and to complete toMF =MF ∪ 0.

Theorem 6.14. The function θ is a homeomorphism ofMF onto B ≈ R6g−6, and is posi-

tively homogeneous of degree 1. Consequently, PMF is homeomorphic to S6g−7.

Proof. We already know that the classifying function θ is a continuous bijection. If

one shows that MF is a topological manifold, then Invariance of Domain implies

that θ is also open; the theorem is then proven.To prove thatMF , with the topology of RS

+, is a topological manifold, we use the

following lemmas.

Lemma 6.15 (change of decomposition). Let K be a permissible decomposition of M into

pairs of pants and (M0,F0, µ0) a

DR

AFT

13 J

un 2

008


measured foliation in normal form with respect to K. There exists another permissible decom-

position K, by a system of curves Ki, i = 1, . . . , 3g − 3, such that mi = I(F0, µ0; [Ki]) is

nonzero for each i.

N.B. It is not said that (F0, µ0) is in normal form with respect to this decomposition.

Proof.

We suppose at first that, for all i, we have

I(F0, µ0; [Ki]) = 0.

In particular, the support of M0 is concentrated in the annuli Ki × [−1/2,+1/2]. We

look at one such i and the two pairs of pants R′k and R′

ℓ that touch Ki × [−1,+1]

(Figure 6.17).

K’i

K i [−1,+1]

R’k

R’l

M0

Ki[−1,+1]

YX

Z T

Figure 6.17:

If we are in the situation suggested by the figure, where neither the pair (X,Z)

nor the pair (Y, T ) bound an annulus, we replace Ki by K ′i; this gives a permissible

decomposition where I(F0, µ0; [K′i]) 6= 0. On the contrary, if (X,Z) bounds an an-

nulus, we construct the simple curve K ′′′i , which is obtained from K ′

i by a half twist

along Ki and we replaceKi by K ′′′i (Figure 6.18). The obtained decomposition is per-

missible because the pair (Y, Z) does not bound an annulus (otherwise (Y,X) would

border one and K would not be permissible).

DR

AFT

13 J

un 2

008


T

R’k

R’l

K’’’i

YX

Z

Figure 6.18:

We are reduced to the following situation:

• if mi = 0, Ki avoids M0 or is a cycle of leaves of F0;

• if mi 6= 0, Ki ∩M0 is transverse to F0;

• at least one of the mi is not zero.

Let us say then that we have a pair of pants R, bounded by K1 ∪K2 ∪K3, with

m1 = 0 and m2 6= 0. For the enlargement of the foliation induced on R, we have the

three possibilities of Figure 6.19.

(by changing the letters K and K ,

1

K’1

K 2 K3

3

2

we obtain the case: m > m > 0)3

3 2

2

K

m > m > 0

Figure 6.19:

DR

AFT

13 J

un 2

008


As before, we construct K ′1 (or K ′′′

1 ) which is transverse to F0 and which gives a

new permissible decomposition where we get m1 6= 0.

Lemma 6.16. Let (F0, µ0) be a measured foliation in regular position with respect to a

permissible decompositionK. We suppose that, for all i = 1, . . . , 3g − 3, we have

m0i = mi(F0, µ0) 6= 0.

Then the function θ−1 : B−0→ I⋆(MF) ⊂ RS+ is continuous at the point with coordinates

(m0i , s

0i , t

0i )i=1,...,3g−3.

Remarks. 1) This proves that I⋆(MF) is a topological manifold in a neighborhood

of (F0, µ0); therefore, if we apply Lemma 6.15, I⋆(MF) is a topological manifold

globally.

2) Lemma 6.16 would be trivial if one could lay out explicit formulas which, for all

γ ∈ S, express I(F , µ; γ) as a function of (m, s, t)(F , µ) and of (m, s, t)(γ).

Proof. We denote by E the set of measured foliations transverse to all the curves Ki

of the decomposition K, without the equivalence relation. If B0 = {(mi, si, ti)i ∈ B :

mi 6= 0, for all i = 1, . . . , 3g − 3}, there exists a section of θ, call it σ : B0 → E, with

the following properties:

1) A foliation in the image of σ is in normal form with respect to K and, for all i,

F|Ki×[−1,+1] varies continuously in the sense of the topology of 1-forms.

2) If α is an arc of R′j , going from the boundary to the boundary, transversely to

(F0, µ0) = σ((m0i , s

0i , t

0i )i), thenα is transverse to (F , µ) = σ((mi, si, ti)i), for (mi, si, ti)

close enough to (m0i , s

0i , t

0i ); further, µ(α) varies continuously.

3) Let α0 ∗ β0 be an arc of R′j , going from the boundary to the boundary, where α0

is transverse to F0, where β0 is in a leaf and where α0 ∗ β0 is quasitransverse to F0.

Then, for (mi, si, ti) close enough to (m0i , s

0i , t

0i ), there exists an arc α ∗ β ⊂ R′

j going

from the boundary to the boundary such that:can we get rid of

the big dots?• a) α ∗ β is C0-close to α0 ∗ β0;

• b) α ⋔ F , where (F , µ) = σ((mi, si, ti)i);

• c) the gluing of α and β is quasitransverse to F ;

• d) µ(β) and |µ(α ∗ β) − µ0(α0)| are small.

DR

AFT

13 J

un 2

008


4) Same condition for arcs of the form β0 ∗ α0 ∗ β′0.

[In a certain sense, these conditions say that σ is continuous. But is there a good topol-

ogy on E? ].

We will be satisfied with a brief outline for the existence of σ. Since we only

define σ on B0, we will only use in each pair of pants the models (1), (2), and (3)

of Section 6.2.4. As long as we stay in the interior of the fundamental triangle, we

can “continuously” vary the actual realizations of these models as well as the corre-

sponding “arcs jaune”. This makes it possibile to reglue the pieces in order to obtain

a section σ, continuous in the topology of vector fields (outside of the singularities).

Figure 6.20 illustrates point 3.

β0

α 0

1 2 3(in a neighborhood of "m = m + m ")

α

β

Figure 6.20:

Knowing σ, we easily finish the proof of Lemma 6.16. Let γ ∈ S and let σγ be the

corresponding component of σ:

σγ : B0 → R+.

We want to show that this function is continuous in (m0i , s

0i , t

0i )i. As we remarked in

II.6 (Expose 5), we can find an immersion γ′0, quasitransverse to F0, which is the limit

of embeddings and whose homotopy class is γ. Let us say that

γ′0 = α01 ∗ β0

1 ∗ α02 . . . ,

where α0i is transverse to F0 and where β0

i is contained in the leaves (and singular

points). We remark right away that the µ-length of γ′0 varies continuously; it follows

that σγ is upper semicontinuous.

DR

AFT

13 J

un 2

008


By the properties 3) and 4) of σ, we construct for (mi, si, ti) close to (m0i , s

0i , t

0i )

another immersed curve:

γ′ = α1 ∗ β1 ∗ α2 ∗ · · ·homotopic to γ′0, with the following properties:

1) αi and βi are glued quasitransversally to F , where (F , µ) = σ((mi, si, ti)i);

2) αi is transverse to F and µ(αi) is close to µ0(α0i );

3) µ(βi) is small.

With endpoints fixed, βi is isotopic to βi, which is quasitransverse to F ; we have

µ(βi) ≤ µ(βi). Using property 3) of σ, we easily see that γ′ = α1 ∗ β1 ∗α2 ∗ · · · , which

is piecewise quasitransverse to F , is really globally quasitransverse to F . We therefore

have

I(F , µ; [γ]) = µ(α1) + µ(β1) + · · · ,a sum which, term by term, is close to

I(F0, µ0; [γ]) = µ0(α01) + µ(β0

1) + · · · =∑

i

µ0(α0i ).

6.6 Return to curves as functionals

6.6.1

We have the following commutative diagram (Section 5.3.2, Expose 5):

R∗+ × S

enlargement−→i∗

NF MF I∗RS

+

The arrow R∗+ × S →MF naturally factors through NF . In fact, if we represent

an element of S by a curve γ having a minimal intersection with each Kj , then a

partial enlargement of γ gives a foliation in normal form, as one sees by looking at

each pair of pants.

Using the function θ : I∗(MF)→ B of Proposition 6.5.1, we thus obtain

Φ : R∗+ × S (resp. S′)→ B.

DR

AFT

13 J

un 2

008


which to β ∈ S′ associates {mj(β), sj(β), tj(β) : j = 1, . . . , 3g − 3}. We recall that, in

Expose 4, for β ∈ S′, we defined Φ(β) = {mj(β), sj(β), tj(β)}. Unfortunately, Φ(β)

does not coincide with Φ(β); it is true that mj(β) = mj(β) = i(β,Kj), but the other

coordinates differ because the arc jaune is not chosen in the same way in the theory of

curves and in the theory of foliations. Moreover Φ(β) does not always have integer

coordinates.

To discuss this difference between the arcs jaune in the two theories, one must

again examine the models on the standard pair of pants P 2. We observe that the arc

jaune of a multi-arc, associated to ∂1P2, always coincides with that of the foliation

obtained by enlargement, except if

(∗) m1 > m2 +m3.

In revenge, the arc jaune of the theory of curves is appropriate for the foliations. Evi-

dently, the length of the arc A, which is associated to it, is only given by the formula

in Section 3.3 if (∗) is not satisfied. Otherwise we take

length A = m2.

yellow arc of foliations

curves

P22

P12

yellow arc of foliationsyellow arc of

Figure 6.21:

use psfrag

Reflecting this change through the formulas of Appendix C, we obtain a new

classification of foliations, via a homeomorphism

θC : I∗(MF)→ B,

which, this time, makes the following diagram commutative:

DR

AFT

13 J

un 2

008


S′ i∗

Φ

I⋆(MF)

θC

BTherefore, i∗(S′) is a “lattice” in I∗(MF); as we know that i∗(S′) is contained in

i∗(R+ × S), we see that i∗(R∗+ × S) is dense in I∗(MF). We have therefore demon-

strated at the same time Theorem 4.10 (Expose 4) and Proposition 6.17.

6.6.2 Proposition.

Proposition 6.17. In P(RS+), the set πi∗(S) = S is dense in πI∗(MF). Therefore, I∗(MF)∪

{0} = i∗(R+ × S).

6.7 Minimality of the action of π0(Diff M) on PMF6.7.1

Let M be a compact connected orientable surface without boundary, of genus ≥ 1.

We always denote by π the projection RS+ − {0} → P(RS

+), and by PMF the image

of MF under π. The natural action of π0(Diff M) on MF gives, by passage to the

quotient, a natural action of π0(Diff M) on PMF .

6.7.2

The goal of this section is to show the following theorem.

Theorem 6.18. The action of π0(Diff M) on PMF is minimal.

We recall that the action of a group on a topological space is called minimal if the

orbit of each point is dense.

6.7.3

If α is a simple curve in M , we denote by tα : M →M a twist of one turn about α.

Proposition 6.19. Let α be a simple curve and F a measured foliation. For all curves β and

for all integers n ≥ 0, we have the inequality:

|I(tnαF , [β])− nI(F , [α])i([β], [α])| ≤ I(F , [β]).

DR

AFT

13 J

un 2

008


Proof. If F is a foliation defined by a curve, the proposition is a particular case of

Proposition A.1 of Appendix A. Considering that the inequality is homogeneous in

F , the proposition is again true for F in i∗(R∗+ × S).

DR

AFT

13 J

un 2

008


As i∗(R∗+ × S) is dense inMF , the inequality is true for every foliation F .

Corollary 6.20. Let F be a measured foliation and α a curve, such that I(F , [α]) 6= 0. We

have

limn→∞

π(tnαF) = π([α]).

Proof.

As a consequence of the preceding proposition, we have:

limn→∞

1

ni(α,F)tnα F = [α] inMF .

6.7.4

We prove the following particular case of Theorem 6.18.

Lemma 6.21. If γ is a curve that does not separate M , the orbit of π([γ]) under π0(Diff M)

is dense in PMF .

Proof. We begin by remarking that the orbit of γ under π0(Diff M) consists of the

(isotopy classes of) curves that do not separate M . Since S is dense in PMF , it suf-

fices to show that the closure of the orbit of γ contains also the curves that separate

M . Let γ be such a curve; we can find a curve γ′ that does not separate M and such

that i(γ′, γ) 6= 0. By Corollary 6.20, we have: limn→∞

tnγγ′ = γ in PMF . Thus γ is in the

closure of the orbit of γ′ and also in that of γ, since these two orbits are the same.

6.7.5 Proof of Theorem 6.18

Proof. Let F be a measured foliation. We can find a curve γ that does not separate

M and such that I(F , γ) 6= 0. By Corollary 6.20, the closure of the orbit of F in PMFcontains γ, and thus also the orbit of γ. It follows from Lemma 6.21 that the orbit of

F is dense in PMF .

DR

AFT

13 J

un 2

008


6.8 Existence of complementary measured foliations

By definition a “complement” of a measured foliation (F , µ) is a measured foliation

(F ′, µ′) transverse to (F , µ) (see Expose 1, Section 1.5).

Proposition 6.22. If (F , µ) is a measured foliation, there exists (F ′′, µ′′) ∼m (F , µ) such

that (F ′′, µ′′) admits a complement.

DR

AFT

13 J

un 2

008


Proof.

By Section 6.5 above, comes (F ′′, µ′′), equivalent to (F , µ), and a pair of pants

decomposition such that:

∀j, i(F ′′,Kj) 6= 0.

By enlarging the multicurve provided by the Kj , we obtain the desired F ′.

Remark. This result is equivalent to the theorem of Hubbard–Masur [HM79] and

Kerckhoff [Ker80], which states thatMF(M2) is realizable by the holomorphic quadratic

differentials on M2. We refer to [DV 6], [HM79] for details on the relation between

quadratic differentials and measured foliations.

DR

AFT

13 J

un 2

008

Expose 7

Teichmuller Space

by A. Douady

Notes by F. Laudenbach

Given a compact surfaceM with negative Euler characteristic χ(M), we consider the

space H of metrics of curvature −1 on M , for which the boundary of M is geodesic;

it is nonempty, and is endowed with the C∞ topology for contravariant tensor fields.

The group Diff0(M) of diffeomorphisms of M isotopic to the identity, equipped with

the C∞ topology, acts on H on the left by the general formula that comes from the

naturality of the field of contravariant tensors m ∈ H, φ ∈ Diff0(M) → φ∗m ∈ H.

The quotient space T = H/Diff0(M) is the Teichmuller space of M ; when M is ori- Teichmuller space

entable, this definition coincides with the classical definition as the space of complex

structures up to isotopy, by the Uniformization Theorem [Spr66]. It is known that

this space is homeomorphic to a “cell” [FK98]; Earle and Eells have shown that H is

the total space of a principal bundle over the Teichmuller space [EE69].

The program here is to establish a parameterization of Teichmuller space that

depends only on the lengths of the simple closed geodesics.

Recall that S is the set of isotopy classes of simple closed curves that are not

homotopic to a point in M . If m is a hyperbolic metric, and α ∈ S, then ℓ(m,α) is the

length of the unique geodesic in the isotopy class α; we thus have a map

ℓ∗ : T → RS+

given by the formula 〈ℓ∗(m), α〉 = ℓ(m,α).

133

DR

AFT

13 J

un 2

008


Proposition 7.1. For a fixed α ∈ S, the map that associates to m ∈ H the m-geodesic in the

class α is continuous in the C∞ topology.

Corollary 7.2. The map ℓ∗ is continuous.

Proof. [Proof of Proposition 7.1] One way to do this is to use the convexity of thefix proof argu-

ment “displacement function” (a theorem of Bishop-O’Neill [BO69]; see the paper of Bour-

guignon [Bou85]). We give a different proof.

We denote by Γ the set of pairs (m, γ) wherem is a hyperbolic metric and γ : S1 →M is a constant speed parameterization of the m-geodesic of α. We give Γ the topol-

ogy induced from the C∞ topology on the product space

H× C∞(S1,M).

We consider the projection p : Γ → H onto the first factor. We wish to show that p is

proper.

We let TM denote the tangent bundle of M , and we consider the subset of H ×TM given by

C = {(m, v) | ∀ t, expm(t+ 1)v = expm tv

and the closed curve t ∈ [0, 1]→ expm tv is in the class α}.

In the product topology on H × TM , C is closed. If S1 is obtained by identifying

the endpoints of [0, 1], one has an obvious map C → Γ which is surjective; by the

theory of differential equations, it is continuous. The properness of p follows from

the properness of the projection q : C → H, as we shall prove.

We know that m ∈ H 7→ ℓ(m,α) is an upper semicontinuous function. Hence if

m belongs to a compact set K , the set {ℓ(m,α) | m ∈ K} is bounded. Let (m, v) ∈q−1(K); the quantity

√m(v, v) = ℓ(m,α) is then bounded. Let m0 ∈ K ; there exists

λ > 0 such that, for all w ∈ TM , and all m ∈ K , one has

m0(w,w) ≤ λm(w,w).

Thus, if (m, v) ∈ q−1(K), m0(v, v) is bounded. Finally, q−1(K) is compact since it is

closed in a product of compact sets.

The groupO(2) of rotations acts naturally on Γ: for r ∈ O(2), (m, γ)∗r = (m, γ◦r).The quotient is the space of m-geodesics of α, m ∈ H. In negative curvature, p in-

duces a bijection Γ/O(2) → H, which is continuous and proper by the above. Since

the spaces considered are metrizable, the inverse

DR

AFT

13 J

un 2

008


is also continuous.

From now on, we suppose, to simplify the exposition, thatM is closed; we denote

by g its genus. We fix a decomposition K ofM into pairs of pantsRi, i = 1, . . . , 2g−2,

bounded by curves Kj , j = 1, . . . , 3g − 3. Each pair of pants is given with a param-

eterization onto some model, and every curve Kj is given with an orientation. We

have a continuous map

L : T → (R∗+)3g−3

defined by L(m) = (ℓ(m,Ki); i = 1, . . . , 3g − 3), where m is a hyperbolic metric

making the Ki geodesic (a so-called metric adapted to the decomposition.) adapted

Remark. From now on,H denotes the space of metrics adapted to K. One sees easily

that T is in bijection with the quotient of H by Diff(M,K) ∩Diff0(M). To see that the

topology is the same, we use Proposition 7.1 and the fact that the action of Diff(M) num

on the space of simple curves admits local sections [Pal60].

The “twists” along the curves Ki defines a continuous action θ of R3g−3 on T .

More precisely, let Ki × [0, 1] be a collar of Ki = Ki × {0}, given once and for all; all

of the collars are assumed to be pairwise disjoint. Being given an adapted hyperbolic

metric m and a number α, there exists a diffeomorphism φi(m,α) of the collar Ki ×[0, 1] with the following properties:

1. φi(m,α) is the identity on a neighborhood of Ki × {1};

2. φi(m,α) is an isometry of m in a neighborhood of Ki × {0};

3. The lift of φi(m,α) to the universal covering R× [0, 1], that is the identity on R×{1}, is a translation of distance αℓ(m,Ki) on R× {0}, in the direction indicated

by the sign of α (the universal cover is given the lifted metric).

The twisted metric θi(m,α) is defined by θi(m,α) = φ∗i (m,α)m for points of the

collar Ki × [0, 1] and by θi(m,α) elsewhere.

For (α1, . . . , α3g−3) ∈ R3g−3, let θ(m,α1, . . . , α3g−3) be the metric defined by

θ1(m,α1) in K1× [0, 1], . . . , by θ3g−3(m,α3g−3) in K3g−3× [0, 1], and by m elsewhere.

As the metric is adapted, its isotopy class is well-defined.

Remarks. (1) By the classification of metrics on pairs of pants (Expose 3),

DR

AFT

13 J

un 2

008


the orbits of the action θ coincide exactly with the fibers of L. Corollary 7.4 below

implies that this action is free.

(2) The Dehn twist ρ along Ki, which is a global diffeomorphism of the surface,Dehn twist

with support in a collar of Ki, is an isometry (up to isotopy) of the metric θi(m, 1)

onto m. One therefore has, for all curves K ′,

ℓ(θi(m, 1), [K ′]) = ℓ(m, ρ([K ′])).

LetR andR′ be the two pairs of pants adjacent toKi; suppose thatR contains the

collar Ki × [0, 1]. Let K ′i be a simple curve in R ∪R′, intersecting Ki in two essential

points (K ′i is not isotopic to a curve disjoint from Ki) – compare with Section 6.4.

We denote by K ′′i the curve in R ∪ R′ obtained from K ′

i by a Dehn twist along Ki:

ρ(K ′i) = K ′′

i .

Proposition 7.3. The length ℓ(θi(m,α), [K ′i]) is a strictly convex function of α, which takes

a minimum.

Corollary 7.4. (1) Being given the metric m0, there exists an isotopy class γi in R ∪ R′,

such that the function

α 7→ ℓ(θi(m0, α), γi)

is strictly increasing for α > 0.

(2) The length ℓ(θi(m,α), [K ′i]) tends uniformly to +∞ as α tends to +∞ or to −∞

and m remains in a compact set.

Proof. [Proof of Corollary 7.4] (1) We suppose that ℓ(θi(m,α), [K ′i]) is increasing fromproof argument

α = k, where k is an integer. We then take γi = ρk([K ′i]) and we apply Remark (2)

above.

(2) This is a general property of families of functions of a real variable that are

strictly convex and take a minimum, and that depend continuously on a parame-

ter (compact open topology). Let fλ(x) be such a family, and let x = m(λ) be the

point that realizes the minimum. Then m(λ) is a continuous function. Indeed, ǫ be-

ing given, if λ is sufficiently close to λ0, we have

fλ(m(λ0)) < inf[fλ(mλ0 − ǫ), fλ(mλ0 + ǫ)];

thus m(λ) belongs to the open interval (m(λ0)− ǫ,m(λ0) + ǫ).

DR

AFT

13 J

un 2

008


Now, let x0 > m(λ0) and let K lie between fλ0(m(λ0)) and fλ0(x0). Then if λ is

sufficiently close to λ0, one has fλ(x0) > K and fλ is strictly increasing on [x0,+∞];

thus fλ([x0,+∞)) ⊂ (K,+∞).

Lemma 7.5. Let γ be a geodesic in the hyperbolic plane and let τ be an isometry leaving γ

invariant. Let x be a point of γ and y a point not on γ; then

d(x, τx) < d(y, τy),

where d denotes hyperbolic distance.

Proof.

x

y

τxτy

τα

Figure 7.1:

We can take for x the foot of the perpendicular α from y onto γ. Then γ is the

unique common perpendicular to α and τα. This gives the inequality.

Lemma 7.6. Let γ1 and γ2 be two geodesics that do not intersect in the hyperbolic plane.

Then the function d(x, y), x ∈ γ1, y ∈ γ2, is strictly convex.

Proof. Let x, x′ (resp. y, y′) be two points of γ1 (resp. γ2); without loss of generality

we suppose that x 6= x′. Let i be the midpoint of the arc xx′, j that of yy′, and δ the

geodesic segment ij. Denote by σi (resp. σj) the refelction through the point i (resp.

j); σjσi is an isometry that leaves δ invariant. Let z = σjσi(x), z′ = σjσi(x

′), and

k = σjσi(i). Then σj takes x to z′ and y to y′. Therefore

d(x, y) = d(y′, z′).

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008


z’x

y

x′

y′

j

kz′

γ1

γ2

δ

Figure 7.2:

Also, by the triangle inequality, we have

d(x′, z′) ≤ d(x′, y′) + d(y′, z′).

By Lemma 7.5, we have

2d(i, j) = d(i, k) < d(x′, z′).

(Note that since γ1 does not intersect γ2, the point x′ is not on δ).

Finally, we obtain the inequality of convexity:

2d(i, j) < d(x, y) + d(x′, y′).

Proof. [Proof of Proposition 7.3] The surface being equipped with the metric m and fix begin proof

its metric universal cover being identified with hyperbolic space H2, there exists an

element τ of π1(M, ∗) that acts as an isometry of H2, leaving invariant a geodesic δ,

which lifts the geodesic K ′i. Let x be a point of δ, projecting to a point of K ′

i ∩ Ki;

denote by K1 the lift of Ki through x and K3 that through τx. The segment (x, τx)

intersects exactly one other lift K2 of Ki in a point y. In Figure 7.3, we show the

orientations of these three lifts.

DR

AFT

13 J

un 2

008


K1

K2

K3

y

x

τx

δ

Figure 7.3:

If we twist the metric by an “angle α” in the collars indicated in the figure, the lift of

the θi(m,α)-geodesic of [K ′i] intersects K1 in a point x′ and K2 in y′; this is a geodesic

from x′ to y′, with the metric of the hyperbolic plane; but from y′ to τx′, its length is

the hyperbolic distance d(y′ + α, τx′ + α); in this formula + denotes the translation

along the geodesics K2 and K3. Finally, one has

ℓ(θi(m,α), [K ′i]) = inf

x∈ fK1,y∈ fK2

(d(x, y) + d(y + α, τx + α)).

We are going to show that f(x, y, α) = d(x, y) + d(y + α, τx + α) is a proper and

strictly convex function. For this, we use the fact that d(x, y) is proper, because the

geodesics on which the points are moved have a common perpendicular (at a finite

distance), and that d is strictly convex by Lemma 7.6.We show the properness of f . Let (xn, yn, αn) → ∞; if (xn, yn) → ∞, then

d(xn, yn) → +∞, hence f(xn, yn, αn) → +∞. If (xn, yn) stays in a compact set, then

αn → ∞ and (yn + αn, τxn + αn) tends to ∞, hence d(yn + αn, τxn + αn) tends to

+∞.One verifies immediately that f is strictly convex.For α fixed, the function f(x, y, α) has a minimum g(α) by the properness of f .

The convexity of f implies that g is also convex; since g(α) is a value attained by

f(x, y, α), we see that g is strictly convex.The function f has an absolute minimum (f is proper and bounded below); it is

the minimum of g.

Proposition 7.7. The map L : T → (R∗+)3g−3 is a principle fibration of the group R3g−3

acting by θ.

DR

AFT

13 J

un 2

008


Corollary 7.8. The Teichmuller space of a closed surface of genus g is homeomorphic to

R6g−6.

Proof. The important point is to show that there exist local sections for L. We know

from Theorem 3.5 in Expose 3 that, for the model pair of pants P 2, the map

H(P 2)→ (R∗+)3

that, to a metric adapted to the boundary, associates the three lengths of the bound-

ary, admits local sections at the level of the metrics.

We know that to glue together two hyperbolic metrics along a geodesic,

DR

AFT

13 J

un 2

008


it is enough to specify an isometry of the geodesic along which we glue. Now, if one

has a metric on P 2 and if one considers a curve C of the boundary, one has a unique

simple geodesic arc that meets C in its two endpoints (arc jaune1). By Proposition 7.1,

its origin (which one distinguishes from the other endpoint by an orientation chosen

once and for all) varies continuously with the metric.

The desired local section is now obtained as follows. Above the 3g−3 lengths one

chooses a metric with the following property: ifKj is adjacent toRi1 andRi2 , the two

origins on Kj of the arcs jaune of the two pairs of pants coincide. By imposing this

condition, we obtain a continuous local section.

LetD be a ball of (R∗+)3g−3 over which L admits a section σ. Define a map T : D×

R3g−3 → T by:

T (x, α1, . . . , α3g−3) = θ(σ(x), α1, . . . , α3g−3).

It remains to show that T is a homeomorphism onto its image. Since T is second

countable, it is enough to show that T is injective and proper.

If two metrics differ from one another by a twist, they are distinguished by the

length of a geodesic (Corollary 7.4); this proves injectivity.

To simplify, denote (α1, . . . , α3g−3) by α. Let (xn, αn) be a sequence tending to

infinity in D × R3g−3. The second part of the same corollary gives that the image

under T of this sequence cannot be a compact set in Teichmuller space. Hence T is

proper.

Theorem 7.9. The map ℓ∗ : T → RS+ is a proper map that is a homeomorphism onto its

image.

Actually, we are going to prove a stronger proposition, stated in terms of the

system of curves Ki,K′i,K

′′i described before Proposition 7.3.num

Proposition 7.10. The map Λ: T → R9g−9+ that tom ∈ T associates the triple (ℓ(m, [Ki]),

ℓ(m, [K ′i]), ℓ(m, [K

′′i ])) is injective and proper (hence a homeomorphism onto its image).

Proof.

We choose a section s of the fibration L; that is, we write everym ∈ T in the form:

m = θ(s(x), α)

1Compare with the terminology of measured foliations (Expose 6).

DR

AFT

13 J

un 2

008


where α = (α1, . . . , α3g−3) ∈ R3g−3 is a “multi-angle” and where x ∈ (R∗+)3g−3 is the

tuple of lengths of the curves Ki.

The variable x being fixed, the function ℓ(m, [K ′i]) is a strictly convex and proper

function gi(αi) of the ith component of α; moreover, ℓ(m, [K ′′i ]) = gi(αi + 1).

Lemma 7.11. If g : R → R is a strictly convex proper function then t 7→ (g(t), g(t + 1))

defines a proper immersion of R into R2.

Thus, the (6g−6)-tuple (ℓ(m, [K ′i]), ℓ(m, [K

′′i ])) is an injective proper function of multi-

angle α. From this, it follows that Λ is injective.

To show that Λ is proper, we consider a sequence (xn, αn) tending to infinity. If

xn tends to ∞, it is clear that Λ(xn, αn) tends to ∞; otherwise the xn remain in a

compact set and, by Corollary 7.4, the length of one of the curves K ′i tends to infinity.

We complete the proof of the theorem with the following proposition, following

a proof indicated by S. Kerckhoff. Recall that π denotes the projection RS − {0} →P (RS).

Proposition 7.12. The composite map π ◦ ℓ∗ : T → P (RS+) is an injection.

Proof. We use the upper half-plane model for H2: {x + iy | y > 0}, with the metric

ds = dx2+dy2

y2 . The group of isometries is PSL(2,R) = SL(2,R)/{±Id}, where the

action of A =(a bc d

)is given by z 7→ az+b

cz+d .

IfA is a hyperbolic element (i.e. leaves invariant a geodesic), we define the displacementdisplacement

ℓ(A) = infz∈H2

d(z,A · z).

The minimum is attained on the invariant geodesic.

Lemma 7.13. If A ∈ SL(2,R) is hyperbolic, we have:

Tr(A) = 2 cosh(ℓ(A)

2).

Proof. By conjugating within SL(2,R), we reduce to the case where the invariant

geodesic is postive y axis. One then has:

DR

AFT

13 J

un 2

008


A =

(ρ 0

0 ρ−1

), ρ > 0.

As a consequence, A · i = ρ2i. Thus, we have

ℓ(A) = d(i, ρ2i) =

∫ ρ2

1

dt

t= 2 log(ρ)

and

Tr(A) = ρ+ ρ−1 = 2 cosh(ℓ(A)

2).

Lemma 7.14. Let A,B ∈ SL(2,R). We have:

Tr(A) · Tr(B) = Tr(AB) + Tr(A−1B)

The proof is a direct calculation.

Consider on the surface M two simple oriented curves γ1 and γ2 that intersect

transversely at the base-point. The homotopy classes of based loops γ1∗γ2 and γ−11 ∗γ2

can both be represented by simple curves γ3 and γ4. If M is given a metric m of

curvature −1, these elements of the fundamental group correspond to hyperbolic

isometries of H2 for which the displacement is ℓi = ℓ(m, [γi]). The preceding lemmas

thus give the formulas:

2 cosh(ℓ12

) cosh(ℓ22

) = cosh(ℓ32

) + cosh(ℓ42

)

or

cosh(ℓ1 + ℓ2

2) + cosh(

ℓ1 − ℓ22

) = cosh(ℓ32

) + cosh(ℓ42

) (∗)

(H) Suppose that there is another metric of curvature −1 for which the lengths of all

closed geodesics are multiplied by k 6= 1. For this metric, the equality ∗ becomes

cosh(kℓ1 + ℓ2

2) + cosh(k

ℓ1 − ℓ22

) = cosh(kℓ32

) + cosh(kℓ42

). (∗∗)

Lemma 7.15. Let α, β, γ, δ be four numbers ≥ 0 and let k > 0, k 6= 1. The relations

coshα+ coshβ = coshγ + cosh δ and

coshkα+ cosh kβ = coshkγ + coshkδ

imply that {α, β} = {γ, δ}.

DR

AFT

13 J

un 2

008


Proof. One may restrict to k > 1. The reader may check that the function cosh(kArg cosh x)

is a strictly convex function of x. Now, if c is a common value of the first equality and

if one sets x = coshα, y = coshγ, the second relation is:

cosh(kArg cosh x) + cosh(kArg cosh(c− x))

= cosh(kArg cosh y) + cosh(kArg cosh(c− y)).

We may suppose that y ≤ x ≤ c− x ≤ c− y. If y < x, by strict convexity, the left side two of the k’s

are in italics and

two are not

will be strictly less than the right.

Consequently, (∗) and (∗∗) give

{ℓ1 + ℓ2, ℓ1 − ℓ2} = {ℓ3, ℓ4}.

Up to change of notation, we can say that

ℓ3 = ℓ1 + ℓ2.

Since the angle between γ1 and γ2 is nonzero, it is not possible for ℓ1 + ℓ2 to be a

shorter distance; hence, the above equality cannot be true, and the hypothesis (H) is

absurd.

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008

Expose 8

How Thurston compactifies

Teichmuller space

by A. Fathi and F. Laudenbachproblems with

labels in pics?

fig 3 needs to be

foliated?in symbold for

tri ineq, be con-

sistent on order

In the space of functionals RS+, we have embedded Teichmuller space T and the space

of Whitehead classes of measured foliations MF (Exposes 6 and 7). In this expose,

we identify these spaces with their images in the functional space; for any functional

f ∈ RS+, we denote by i(f, α) the value of the functional on α ∈ S.

We recall that π : RS+ − {0} → PRS

+ denotes projection onto the space of rays

and that PMF is the image ofMF . Moreover,MF = π−1(PMF). We construct a

topology on the union of T and PMF . We prove that it is the topology of a manifold

with boundary; since the interior is homeomorphic to an open ball, and the boundary

is homeomorphic to a sphere, the manifold with boundary is homeomorphic to a

closed ball.

The key is in the inequalities of the “Fundamental Lemma” (below) whose proof

rests on length estimates from hyperbolic geometry that are gathered in the appendix

of this expose. ref

147

DR

AFT

13 J

un 2

008


8.1 Preliminaries

8.1.1

Proposition 8.1. In RS+, the spaces T andMF are disjoint.

Proof.

If f belongs to T , then, since the surface is compact, the set of numbers i(f, α),

α ∈ S, is bounded below by a strictly positive constant. We are going to prove that,

for f ∈MF , the closure of the set of numbers i(f, α) contains zero.

Let (F , µ) be a measured foliation representing f ; let γ be a small arc transverse

to F , with µ(γ) ≤ ε, where ε is given. By Poincare Recurrence (Theorem 5.1) almost

every leaf departing from a point of γ returns to γ; we thus obtain a simple closed

curve γ′, formed by an arc of γ, and an arc carried by a leaf of F ; if α is the isotopy

class of γ′, we have:

i(f, α) ≤ µ(γ′) ≤ µ(γ) ≤ ε.

8.1.2 Construction of a projection q : T →MFThe projection we construct will give the charts for the manifold with boundary

structure. It depends on the choice of a family K = {K1, . . . ,Kk} of mutually dis-

joint simple curves cutting the surface into (embedded) pairs of pants R1, . . . , Rk′ ; if

the surface is closed and of genus g, then k = 3g − 3 and k′ = 2g − 2.

Let m ∈ T ; we represent m by a metric m, of curvature −1, for which the curves

Kj are geodesics. The foliation that will represent q(m) will be transverse to eachKj ;

for each j, we specify

i(q(m),Kj) = i(m,Kj).

Let R be one of the pairs of pants; we say that ∂R = K1 ∪ K2 ∪ K3, and we set

2mj = i(m,Kj). Let gjj′ be the simple m-geodesic of R orthogonal to Kj and to Kj′ .

First case. (m1,m2,m3) ∈ (∇ ≤), triangle inequality.

Let T12 be the (closed) geodesic tube of points in R at a distance from g12 at most

(m1 +m2−m3)/2; it is foliated by equidistant lines; the distance between two leaves

gives the transverse measure. We consider in the same way the foliated tubes T23 and

T13. The tubes have, pairwise, only two common points, which are on the boundary;

for example:

T12 ∩ T13 = T12 ∩ T13 ∩K1.

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008


g12

g23

g13

K1

K2

K3

Figure 8.1:

This is due to the fact thatK1 is the unique common perpendicular to g12 and g13;

on the other hand, by adding the two thicknesses, we see that K1 is totally covered;

the same for K2 and K3. We obtain this way a “partial” measured foliation of R

(Figure 8.1.2). We can associate to it a true measured foliation by collapsing each non-

foliated triangle to a tripod. Actually, for what follows, we are interested in keeping

the partial foliation, in which the measure is directly given by the metric.

Second case. m1 > m2 +m3.

Up to change of numbering, there are no other cases. Here T12 is the tube of ra-

dius m2 and T13 is the tube of radius m3. The set of points of K1 that are neither in

the interior of T12 or of T13, forms two arcs A and A′, symmetric to one another by

the isometric involution of R that admits g12 ∪ g13 ∪ g23 as the locus of fixed points

(Lemma 3.7, Expose 3). Let T11 be the union of lines of equal distance to the geodesic

g11, emanating fromA (Warning! g11 cannot be in T11). We see that T11∩K1 = A∪A′.

These three tubes give a “partial foliation” that looks like those in Figure 8.1.2.num should be

8.2

DR

AFT

13 J

un 2

008


A

A'

T11

T12

T23

Figure 8.2:

We remark that, in the two cases, the leaves are perpendicular to the curves of

the boundary. When we reglue the pairs of pants, we obtain a “partial” measured

foliation Fm, which represents q(m); the leaves are only C1 at the junction of two

pairs of pants, but this is not important.

8.1.3

Proposition 8.2. The map q is a homeomorphism of T onto the open set U(K) of MFconsisting of the functionals taking nonzero values on each component of K.

Proof.

1o We construct the inverse map q−1 in the following way. An element of U(K) is

represented by a measured foliation (F , µ), transverse to the curves of K.

In the pair of pants R (notation of Section 8.1.2), we construct a metric m of cur-

vature−1 with the following properties:

(i) m|Kj = µ|Kj , for j = 1, 2, 3;

(ii) denoting 2mj = µ(Kj), if (m1,m2,m3) ∈ (∇ ≤), the smooth leaf that goes

from K1 to K2 (resp. to K3) and whose µ-distance to the singularities is m1+m2−m3

2

(resp. m1+m3−m2

2 ) is declared to be a geodesic of m orthogonal to the boundary;

DR

AFT

13 J

un 2

008


(iii) if m1 > m2 +m3, the smooth leaf that goes from K1 to K2 (resp. to K3) and

whose µ-distance to a singularity is m2 (resp. m3), is declared to be a geodesic of m

orthogonal to the boundary.

By the classification of hyperbolic metrics of pairs of pants (Expose 3), if two

metrics satisfy the conditions above, then they are conjugate by a diffeomorphism

isotopic to the identity, by an isotopy that is constant on the boundary. Therefore, when

we reglue all the pairs of pants, we obtain a hyperbolic metric, well-defined up to

isotopy. By the classification of measured foliations of pairs of pants (Expose 6), we

see that the map constructed in this way is the inverse of q.

2o For continuity, we proceed as follows.

We utilize the parameterization {mj, sj , tj} ofMF (see Expose 6). The projection

{mj, sj , tj} → {mj} restricted to U(K) is a principal bundle, for which the structure

group is the group of twists along K. Indeed, one has an obvious section σ({mj}) =

{mj, 0,mj}; moreover, if we act by a twist αj alongKj on this section, the pair (sj , tj)

parameterizing the twisted foliation is given by semi-linear formulas (exercise). This

establishes, for each mj , a homeomorphism of R onto the set of the (sj , tj) such that

(mj , sj , tj) belongs to ∂(∇ ≤); since U(K) is a manifold, these arguments suffice to

provide the structure of the principal bundle.

We also recall that T is fibered over the space of lengths of the components of

K (Proposition 7.7, Expose 7). And, by construction, the map q is equivariant with

respect to these two principal bundle structures and it extends the identity map of

their common base.

The continuity of q is equivalent to that of q−1, since the source and the target

are manifolds. For the continuity of q−1, by the above, it suffices to verify this on the

section σ. However, over the closed set (∇ ≤) ∪ {m1 ≥ m2 +m3}, the section σ lifts

to a section σ with values in the space of foliations on the pair of pants R, where the

middle leaves (specified in (ii) and (iii)) are fixed. Starting from this, we can construct

m continuously in R, by applying Theorem 3.5 of Expose 3. We do the same for all

the pairs of pants.

8.2 The Fundamental Lemma

8.2.1

Lemma 8.3. Let ε > 0 and let V (K, ε) be the open subset of T defined by the metrics for

which each component of K is a geodesic of length > ε. For

DR

AFT

13 J

un 2

008


each α ∈ S, there exists a constant C such that, for all m ∈ V (K, ε), we have:

i(q(m), α) ≤ i(m,α) ≤ i(q(m), α) + C

Proof.

First we show i(q(m), α) ≤ i(m,α).

If m is a metric representing m, the transverse measure of the foliation Fm, con-

structed as a representative of q(m), is given by the metricm on the geodesics orthog-

onal to the leaves. Therefore, the m-length of an arc is bigger than the Fm-measure.

Further, by the definition of the functional, the Fm-measure of a closed curve of the

class α bounds i(q(m), α) from above, which proves the first inequality.

Next we show: i(m,α) ≤ i(q(m), α) + C.

It suffices to prove it on the dense subset of V (K, ε), consisting of the m for which

the foliation Fm has simple (tripod) singularities without connections between these

singularities; such classes of metrics are called generic.

By Proposition 5.7 of Expose 5, if m is generic, α can be represented by a simple

curve α′ transverse to the foliation Fm; its measure Fm(α′) has the value i(q(m), α).

We can, moreover, choose α′ so that, for all j, we have

card(Kj ∩ α′) = i([Kj], α).

In fact, if this is not already the case, there is a disk (with angular boundary)

whose boundary is formed by an arc of α′ and an arc of Kj . Each of these being

transverse to the foliation, the disk is foliated as in Figure 8.2.1 and the assertion is num should be

8.3clear.

Kj

α'

Figure 8.3:

Now, two curves that are isotopic and in minimal position with K are isotopic by

an isotopy leavingK invariant (Proposition 3.13). Therefore α′ is cut byK into n arcs,

where n only depends on α:

DR

AFT

13 J

un 2

008


α′ = α′1 ∪ α′

2 ∪ · · · ∪ α′n,

where α′j is an essential arc of one of the pairs of pants of the decomposition, trans-

verse to the foliation. The inequality is a consequence of Lemma 8.4 below.

8.2.2

Lemma 8.4. Let ε > 0. There exists a constant C′ with the following property. For each

hyperbolic metric m on the pair of pants P 2 where each component of the boundary is a

geodesic of length ≥ ε, and for each simple arc β of P 2 going from boundary to boundary

transverse to the foliation Fm, there exists an arc γ, homotopic to β with endpoints fixed,

such that the m-length of γ is less than or equal to Fm(β) + C′.

Proof.

We consider separately each type of foliation (Figures 8.1.2 and 8.1.2) and we takenum

the bigger of the constants. We first do the argument for the foliation satisfying the

triangle inequality.

We replace β by an immersed arc β′, having the same endpoints, by applying the

two processes shown in Figure 8.2.2.num should be

8.4fix labels in fig-

ure

DR

AFT

13 J

un 2

008


δP2

δP2

β

leaves

β'

β'β

Figure 8.4:

We remark that β′ is transverse to Fm, with

Fm(β′) = Fm(β),

and that β′ is close to a simple arc. By construction, β′ is formed from an arc of the

boundary and from “diagonals” in the foliated rectangles (extending the usual defi-

nition of a diagonal of a rectangle to include the arcs containing the endpoints of β′,

which are not found, in general, at the vertex of a rectangle). We deduce from the

topology that β′ contains at most three diagonals (each one covered one time). For

example, if δ1 is the first diagonal met in following β′ (see Figure 8.2.2), then δ2 is nec-

essarily the second and δ3 the third. Upon leaving δ3, we travel along the boundary

in a way that cannot travel along any diagonal more than one time. num should be

8.5We replace each diagonal by an arc of a leaf and an arc of the boundary; in this

way we construct an arc β′′ with the same Fm-measure and containing at most three

leaves. Finally we form γ by replacing the leaves by the geodesics with the same end-

points. The length of γ is the sum of the lengths of the geodesics and the lengths of

the arcs along the boundary; the second term has value Fm(β′′) = Fm(β); the contri-

bution of the first term is bounded, by the Proposition of Section 7 in the appendix.

DR

AFT

13 J

un 2

008


δ1

δ2

δ3

Figure 8.5:

number the

prop and the

appendix

If Fm is the foliation of Figure 8.1.2, then β′ contains at most three diagonals (Fig-

ure 8.2.2); to bound from above the length of a geodesic joining the endpoints of a

should be figs 8.2 and8.6

leaf of the tube T11, one must use the Corollary in Section 7 of the appendix.

num the cor and

the app

DR

AFT

13 J

un 2

008


Figure 8.6:

8.2.3

Corollary 8.5. Let xn be a sequence in V (K, ε) tending to infinity in T . Then π(xn) con-

verges if and only if π ◦ q(xn) converges, and in this case the two sequences have the same

limit.

Proof.

Say that π(xn) converges. That is to say that there exists a sequence of scalars

λn > 0 such that the sequence λnxn converges. Since the topology of T is defined by

a finite number of curves γ1, . . . , γk, we have:∑

j

i(xn, γj) → ∞ and∑

j λni(xn, γj)

converges. Therefore, λn → 0.

By Lemma 8.3, for all α ∈ S,

|i(λnxn, α)− i(λnq(xn), α)| → 0.

so π ◦ q(xn) converges to the same limit as π(xn). The converse is analogous.

8.3 The manifold T8.3.1 Topology

On the disjoint union T ∪ PMF , we take as a basis the open sets of T (open sets of

type 1), and the sets of the form (T ∩ π−1(U))∪ (PMF ∩ U), where U is an open set

of the projective space (open sets of type 2). As π−1(U) ∩ T is an open set of T , the

DR

AFT

13 J

un 2

008


intersection of an open set of type 1 and an open set of type 2

DR

AFT

13 J

un 2

008


is an open set of type 1. We then easily verify the axioms of a topology. This topolog-

ical space is denoted T .

The space T is equipped with a continuous map to the projective space which is

an injection; in fact, π injects T (Proposition 7.12 of Expose 7) and π(T ) avoids PMFby Proposition 8.1. In particular, T is a separable space. The topology of T is second num

countable.

8.3.2 Map of the neighborhood of a foliation

Let f ∈ PMF . By Lemma 6.15, there exists a decomposition of the surface into pairs

of pants along a systemK of curvesK1, . . . ,Kk such that i(f, [Kj ]) 6= 0 for all j, where

f denotes some lift of f to RS+ (nonzero-ness is a projective property). Let {K ′

j,K′′j }

be a system of curves that parameterize T with the {Kj} (Proposition 7.10).

Let ε > 0 be arbitrary. We consider the open set V (K, ε) of T (see Lemma 8.3)

and the open set W of PMF of the “projective” functionals that are nonzero on the

components of K; we have π−1(W ) = U(K) (see I.3) and π ◦ q(V (K, ε)) = W . We

define

φ : W ∪ V (K, ε) −→W × [0, 1]

by φ(x) = (x, 0), if x ∈W ,

φ(x) = (π ◦ q(x), e−Σ{i(q(x),Kj)+i(q(x),K′

j)+i(q(x),K′′

j )}) if x ∈ V (K, ε).

Lemma 8.6. 1o W ∪ V (K, ε) is an open set of T .

2o φ is a homeomorphism onto an open subset of W × [0, 1].

Proof.

1o Let x ∈ W . Suppose that the designated set is not a neighborhood of x in T .

Then there exists a sequence xn in T , xn /∈ V (K, ε), such that π(xn) tends to x. Up to

change of indices and extraction of a subsequence, one can say that i(xn,K1) ≤ ε.The sequence xn does not have a subsequence converging to a point of T be-

cause of Proposition 8.1. Therefore xn tends to infinity. But, moreover, there exists a

sequence of scalars λn > 0 such that λnxn converges to a measured foliation f in the

fiber of x. We deduce that λn → 0. But then i(f,K1) = 0, which contradicts x ∈W .

2o The map φ is continuous in x ∈ W . We just saw that if xn ∈ T converges to x in

T , then xn tends to infinity in T . As xn belongs to V (K, ε), by Corollary 8.5, the first

component is also continuous.

DR

AFT

13 J

un 2

008


For the same reason,∑

j

(i(q(x),Kj)+i(q(x),K′j)+i(q(x),K

′′j )) tends to∞. Therefore

the second component of φ(xn) tends to 0.

The map φ is injective. A priori, the only failing of injectivity can only come from

two elements x and y of T . If q(x) = λq(y), the equality of the second components

implies λ = 1. But q is injective.

x

λq(x)

q(x)

TMF

Figure 8.7:

We remark that if φ(x) = (z, t), then φ(q−1(λq(x))) = (z, tλ).

There exists a continuous section σ : W → MF of π. Up to multiplication by a

scalar, one can take it to have values in q(V (K, ε)). The manifold φ ◦ q−1 ◦σ(W ) is the

graph, inW×[0, 1), of a strictly positive function defined onW . The neighborhood of

W × {0} bounded by this graph is surely in the image of φ by the preceding remark.

The inverse of φ is continuous on this neighborhood. We verify only that if (zn, tn)→(z, 0), then π ◦ φ−1(zn, tn) converges to z in the projective space. As tn tends to 0,

q ◦ φ−1(zn, tn) tends to infinity. By Lemma 8.3, φ−1(zn, tn) tends to infinity in T . We

know that zn = π ◦ q ◦φ−1(zn, tn) converges to z; therefore π ◦φ−1(zn, tn) goes to the

same limit by Corollary 8.5.

8.3.3

We already know that T is a manifold; we just saw that T is a manifold with bound-

ary, bounded by PMF . In particular, T is locally compact and, as the topology is

second countable, it is paracompact. Thus, the boundary admits a collar neighbor-

hood (theorem of M. Brown, see [Rus73], Chapter 1, Theorem 17.4, page 40). As

PMF is homeomorphic to a sphere, the interior boundary of the collar neighbor-

hood is an embedded sphere in the interior of T , hence in a Euclidean space; then,

by the Schoenflies Theorem, generalized by Mazur and Brown ([Rus73], Chapter 1,

Theorem 18.2, page 48), this sphere bounds a ball. Finally, T is homeomorphic to

DR

AFT

13 J

un 2

008


a ball. In particular, it is a compact set; by Propositions 7.12 and 8.1, the projection

π : RS+−{0} → PRS

+ induces a continuous injection of T into PRS+, which is therefore

a homeomorphism

DR

AFT

13 J

un 2

008


onto its image. We have finally proven the following theorem.

Theorem 8.7. The space T = PMF∪T , given the topology induced by PRS+, is a compact

manifold with boundary, homeomorphic to a ball and bounded by PMF .

If the surface is closed and of genus g > 1, T is homeomorphic to D6g−6.

The group of isotopy classes of diffeomorphisms of the surface acts continuously on T , by

the transposed action of the direct image action on S.

Remark. The described action thus is the opposite of the one usually used on mea-

sured foliations, given by taking the direct images of the measures.

DR

AFT

13 J

un 2

008

Expose 9

Classification of surface

diffeomorphisms

by V. Poenaru

9.1 Preliminaries

Let M be a closed, orientable surface of genus g > 1. Its compactified Teichmuller

space T = T (M) is homeomorphic to D6g−6. The natural action of π0(Diff(M)) on

T (M) and PMF(M) combine to give a continuous action on

T (M) = T (M) ∪ PMF(M).

Let ϕ ∈ Diff(M) and let [ϕ] be its isotopy class. By the Brouwer Fixed Point Theorem,

there is an x ∈ T (M) such that [ϕ] · x = x.

If x belongs to T (M), then x determines a hyperbolic metric on M , up to isotopy,

and ϕ is isotopic to an isometry in this metric. By Theorem 3.20 in Expose 3, ϕ is iso-

topic to a diffeomorphism of finite order.

163

DR

AFT

13 J

un 2

008


If x belongs to the boundary of T (M), i.e., x ∈ PMF(M); the equality [ϕ] · x = x

tells us that there exists a measured foliation whose measure class in the projective

space P (RS+) is preserved by ϕ. In other words, there exists a measured foliation

(F , µ)1 and a scalar λ ∈ R+ such that

ϕ(F , µ) ∼m ϕ(F , λµ) = λ(F , µ). (∗)

Notes: (1) Here ∼m is the relation of ‘measure equivalence’ between measured

foliations. Recall that

(F1, µ1) ∼m (F2, µ2)

means that the two measured foliations define the same functional in RS+ (Schwartz

equivalence). By the results of Expose 6, this relation is the same as Whitehead equiv-

alence, defined in Section 5.2.1 of Expose 5.

(2) ϕ(F , µ) denotes the image foliation of F under ϕ, equipped with the (direct

image) measure: the measure of a transverse arc α is the µ-measure of ϕ−1(α).

Now, we define a partial measured foliation of M as a measured foliation (F ′, µ′)

that is supported on a compact submanifold N of dimension 2, and that satisfies the

following:

(i) Every connected component of ∂N is a cycle of leaves.

(ii) If Γ is a component of ∂N that bounds a disk in M \ int(N), then the number

of separatrices that leave the set Sing(F ′ ∩ Γ), and enter N , is at least 2.

If we start with a measured foliation (F ′, µ) of M , we may “unglue” F along

all of the leaves that join the singularities, and “blow-up” the singularities that are

not connected to other singularities. We obtain, then, a partial measured foliation

U(F , µ), called the unglue of (F , µ), whose singularities are all on the boundary.unglue

One easily verifies the following facts:

(a) i∗(F , µ) = i∗(U(F , µ)) ∈ RS+

(b) if i∗(F1, µ1) = i∗(F2, µ2), that is to say, if (F1, µ1) ∼m (F2, µ2), then U(F1, µ1)

and U(F2, µ2) are isotopic.

(c) We denote by βU(F , µ) the union of of the boundary components of the sup-

port of U(F , µ) that do not bound a disk in M . As an element of S′, βU(F , µ) only

depends

1Notation: F denotes the foliation and µ denotes a transverse invariant measure on F .

DR

AFT

13 J

un 2

008


on the measure class of (F , µ).

Returning to (∗), the Brouwer Fixed Point Theorem gives three possibilities:

i) βU(F , µ) 6= ∅,ii) βU(F , µ) = ∅ and λ = 1,

iii) βU(F , µ) = ∅ and λ 6= 1.

In the rest of this expose, we will analyse the three cases. We show that (i) is

the “reducible” case, that (ii) is again a case of “finite order”, whereas case (iii) is

“pseudo-Anosov” (see Expose 1). The classification theorem is stated at the end of

Section 9.5. In this expose, the surfaces are always orientable, but the diffeomor-

phisms do not necessarily preserve orientation, which complicates certain arguments,

in particular Lemma 9.9. num

9.2 The reducible case: βU(µ) 6= ∅

The relation (∗) implies that U(ϕ(F , µ)) and U(F , λµ) are isotopic. Hence, in S′, we

have the equality

βU(F , µ) = βU(F , λµ).

Further,

βU(ϕ(F , µ)) = ϕ(βU(F , µ)),

βU(F , λµ) = βU(F , µ).

Hence, the element βU(F , µ) of S′ is invariant under [ϕ], with the various components

possibly permuted.

Under these conditions, ϕ is isotopic to a diffeomorphism ϕ′ that leaves invari-

ant the submanifold βU(F , µ). By cutting M along this family of curves, we obtain

a manifold (with boundary) W , possibly not connected, on which ϕ induces a diff-

eomorphism ψ. We start over with an analogous study of ψ by applying Thurston’s

theory for surfaces with boundary, which is sketched in Expose 11. Observe that W

is simpler than M in the sense that every component of W has either smaller genus

than M , or the same genus but smaller Euler characteristic in absolute value. Hence,

in a finite number of stages we may give the structure of ϕ up to isotopy.

DR

AFT

13 J

un 2

008


9.3 Arational Measured Foliations

By definition, a measured foliation (F , µ) is arational if βU(F , µ) is empty.arational

Lemma 9.1. (1) If (F , µ) is an arational measured foliation, the compact invariant set X ,

consisting of all singularities and the leaves joining two singularities, only has connected

components that are contractible.are we taking

this as the def of

invar set?

(2) F does not have any smooth closed leaves.

Proof. The manifold M − SuppU(F , µ) collapses onto X , which gives (1). Suppose

that Γ is a smooth leaf of (F , µ); applying the Stability Lemma of Expose 5 to one of

the sides of Γ, one may find a maximal cylinder Φ: Γ× [0, 1]→M such that

1) Φ(Γ× {0}) = Γ;

2) Φ(Γ× [0, 1)) is an embedding starting from the chosen side of Γ.

The genus being > 1, if the cylinder is maximal, then Φ(Γ×{1}) ⊂ X . In view of

(1), the invariant set Φ: (Γ×{1}) is contractible and we may show without difficulty

that Φ: (Γ × [0, 1]) is a disk D2 with spine Φ(Γ × {1}). As there does not exist a

measured foliation on D2 such that ∂D2 is a leaf, the existence of Γ is absurd. Hence,

every half-leaf of F that does not go to a singularity is infinite.

Remark. On the torus T 2, by the definition, every foliation is arational, whereas a

foliation that satisfies the conditions of Lemma 9.1 is conjugate to a linear foliation

with irrational slope.

Corollary 9.2. Under the same conditions as in the preceding lemma, there exists (F ′, µ′),

equivalent to (F , µ), that does not have any connections between singularities. This foliation

is unique up to isotopy in its measure class.

Proof. We obtain (F ′, µ′) by collapsing every component of the F -invariant set X

described above. The result of collapsing remains unchanged, up to isotopy, if we

perform a Whitehead smoothing on X ; uniqueness follows.

DR

AFT

13 J

un 2

008


Convention. In what follows, we will consistently represent a class of arational foli-

ations by the canonical model described above.

Lemma 9.3. If (F , µ) is the canonical model of a class of arational measured foliations and

if ϕ is a diffeomorphism such that ϕ(F , µ) ∼m λ(F , µ) for some λ ∈ R∗+, then ϕ is isotopic

to ϕ′ such that

ϕ′(F , µ) = (F , λµ);

that is to say ϕ′ takes leaves to leaves and, for every arc α transverse to F we have

µ(ϕ′−1(α)) = λµ(α).

N.B. If λ > 1, this says that ϕ contracts the transverse distance (by a factor of 1/λ),

whereas if λ < 1, this says that ϕ dilates the transverse distance (by a factor of 1/λ).

Proof. The foliations ϕ(F , µ) and (F , λµ) are two canonical models of the same type;

hence they are isotopic. Changing ϕ by this isotopy, one obtains the required ϕ′.

Definition. Let (F , µ) be any measured foliation. An (F , µ)-rectangle (or briefly, an (F, µ)-rectangle

F -rectangle), is the image of an immersion ϕ : [0, 1] × [0, 1] → M with the following

properties.

(a) ϕ∣∣(0,1)×(0,1)

is a C∞ embedding.

(b) ϕ({t} × [0, 1]) is contained in a finite union of leaves and singularities; if t ∈(0, 1) then the image is contained in a single leaf.

(c) ϕ([0, 1]× {0}) and ϕ([0, 1]× {1}) are transverse to the leaves.

For anF -rectangleR, we consider the decomposition ∂R = ∂FR∪∂τR where we

define

∂FR = ϕ({0, 1} × [0, 1]) and ∂τR = ϕ([0, 1]× {0, 1}).We will denote by ∂0

FR and ∂1FR the images, respectively, of {0} × [0, 1] and

{1} × [0, 1]; an analogous notation will be used for ∂τR. Further, we will find it con-

venient to write intR = ϕ((0, 1) × (0, 1)), which in general is not the interior of the

image; it is easy to see that intR and ∂R are disjoint.

DR

AFT

13 J

un 2

008


Definition. A good system of transversals for F is a finite system τ = {τi | i ∈ I} ofgood system of transversals

simple arcs with the following properties:

(a) every arc is transverse to F and may only meet a singularity at one of its

endpoints;

(b) two arcs do not meet, except possibly at a single endpoint; if this is a singu-

larity, the two arcs fall into two distinct sectors.

Remark. We do not require that every arc contains a singularity.

Lemma 9.4. Given a measured foliation F and a good system of transversals τ , there exists

a system of rectangles R1, . . . , RN , with the following properties.

(1) intRi ∩ intRj = ∅ for i 6= j.

(2) ∂ǫτRi is contained in a single arc of τ , ǫ ∈ {0, 1}.(3) Every ∂ǫFRi contains a point of Sing(F) ∪ ∂τ ; in other words, every rectangle Ri is

maximal with respect to condition (2).

(4) The two sides of each arc of τ are covered by the rectangles.

The system (R1, . . . , RN ) is unique.

Remark. It is very instructive to take a small transversal to an irrational foliation of

T 2 and to construct the corresponding rectangles.

Proof. Cut the manifold as indicated in Figure 9.1.num

DR

AFT

13 J

un 2

008


Figure 9.1:

We obtain a manifold with boundary M with a foliation F ′; the boundary τ ′ of

M is the “double” of τ . Consider the finite set Z of τ ′, defined by any one of the double

following conditions:

(1) x ∈ Sing F ′;

(2) x is one of the points giving an endpoint of τ ; or

(3) the leaves departing x run into a singularity of F , or a point that gives an

endpoint of τ .

By Poincare Recurrence (Theorem 5.1), all leaves that depart from a point of τ ′−Zreturn to τ ′ − Z .

For every component αi of τ ′ − Z , by the Stability Lemma (Lemma 5.5), we may

find a rectangle Ri such that ∂0τRi = αi; ∂

1τRi gets attached to another component of

τ ′ − Z . When we view these in M , the rectangles are the desired rectangles. Unique-

ness is left as an exercise.

Lemma 9.5. If, in the hypotheses of Lemma 9.4, F is an arational foliation, then

R1 ∪ · · · ∪RN = M.

DR

AFT

13 J

un 2

008


Proof. The union of the Ri is a closed F -invariant set. If the boundary is not empty,

there is a closed F -invariant set consisting of cycles of leaves. If F is arational, such

a cycle cannot exist, hence the boundary is empty and M =⋃Ri.

Lemma 9.6. If F is an arational foliation, every half-leaf L of F that does not lead to a

singularity is dense.

DR

AFT

13 J

un 2

008


Proof. We know that L is “infinite” (Lemma 9.1). Let τ be a small arc transverse to

F and R1, . . . , RN be the system of rectangles from Lemma 9.4. By the above lemma,⋃Ri = M and, since L is infinite, it contains plaques in

⋃intRi, so L meets τ . Since

τ is arbitrary, L is dense.

9.4 Case II: (F , µ) is arational and λ = 1

Lemma 9.7. If ϕ is a diffeomorphism and (F , µ) is an arational foliation such that

ϕ(F , µ) = (F , µ),

then ϕ is isotopic to a diffeomorphism of finite order that preserves (F , µ).

Proof. In the neighborhood of every singularity, we choose transverse arcs, one in

each sector, all of the same length with respect to the measure µ, as indicated in

Figure 9.2.

Figure 9.2:

Since λ = 1, we may choose the system of arcs τ so that, possibly after an isotopy

of ϕ through diffeomorphisms that preserve F , we have ϕ(τ) = τ .

Let R1, . . . , RN be the system of rectangles associated to τ (Lemma 9.4). Since

ϕ(τ) = τ and ϕ(F) = F , we have that every ϕ(Ri) is again an F -rectangle satisfy-

ing condition (2) of Lemma 9.4. It is easy to see that there exists a permutation π of

(1, . . . , N) such that ϕ(Ri) = Rπ(i). In particular, ϕ acts on the graph Γ =⋃i ∂Ri as

well. Hence ϕ permutes the edges of Γ among themselves. Working with the cycles

of this permutation we may isotope ϕ to ϕ′, by diffeomorphisms that preserve F ,

such that ϕ′∣∣Γ

is periodic and ϕ′(Ri) = Rπ(i).

Working on the cycles of π, we may make a second isotopy to obtain a periodic

diffeomorphism, through diffeomorphisms that preserve F .

DR

AFT

13 J

un 2

008


Remark. Such a diffeomorphism always has a fixed point in T (M). Indeed, if ϕ is of

finite order, ϕ′ is an isometry in a certain metric m (whose curvature we cannot con-

trol); ϕ is hence an automorphism of the underlying conformal structure. Further-

more, by the Uniformization Theorem cited in Expose 7, underlying this structure

there is a unique hyperbolic structure which, as a consequence, is invariant under ϕ.

9.5 Case III; (F , µ) is arational and λ 6= 1

We now suppose that we are in the situation where ϕ(F , µ) = (F , λµ), with λ 6= 1,

where F is a canonical model for a class of arational foliations. By changing ϕ to ϕ−1

if necessary, we may assume that λ > 1.

Lemma 9.8. The multiplicative factor λ (respectively 1/λ) is an algebraic integer of degree

bounded by a quantity which is a function only of the genus of the surface.

Proof. There is a branched covering M overM in which (F , µ) lifts to a closed 1-form

ω (the “orientation cover”). If γ is a loop ofM−Sing(F), along which F is orientable,

then ϕ(γ) has the same property; it follows that ϕ lifts to a diffeomorphism ψ of the

open covering M → M − Sing(F). This extends to a diffeomorphism ψ of M . We

have (ϕ−1)∗(ω) = λω.

Hence λ is an eigenvalue of an automorphism of H1(M,Z). Now, the rank of the

cohomology group is bounded by a quantity that depends only on the genus of M .

Lemma 9.9. Under the hypothesis given above, up to changing ϕ by an isotopy leaving Finvariant, we may find a good system of transversals τ with the following properties.

(1) In every sector of a singularity, there is an arc of τ . (figure 9.4).

(2) ϕ(τ) ⊂ τ , that is to say, ϕ takes every arc of τ into an arc of τ .

(3) If x ∈ ∂τ − Sing(F), x belongs to the separatrices of a singularity; we denote by Fxthe arc of the leaf joining x to Sing(F).

(4) Every separatrix contains an Fx.

(5) (⋃Fx) ⊂ ϕ(

⋃Fx).

DR

AFT

13 J

un 2

008


Proof. Since λ > 1, ϕ contracts the transversals (see the definition of the direct image

of a measure). Up to modifying ϕ by an isotopy that preserves F , it is easy to find a

good system of transversals τ ′′ that satisfies (1) and (2) and that is formed of one arc

in each sector. Let α′′ be an arc of τ ′′ and L a separatrix emanating from a singularity

s. Considering the density of the half-leaves, there is, starting from s, a first point of

intersection of Lwith α′′. By considering all the separatrices, we obtain on α′′ a finite

number of such points; we subdivide α′′ and we truncate it at the furthest point. Let

τ ′ be the good system of transversals obtained by this operation on each of the arcs

of τ ′′. It satisfies (1), (3), (4).

The system τ ′ also satisfies (2). Let α′ ∈ τ ′, with endpoints x and y. We have

α′ ⊂ α′′, α′′ ∈ τ ′′, and ϕ(α′′) ⊂ β′′ for a certain β′′ ∈ τ ′′. We suppose, for the moment,

thatϕ(α′) is already contained in∪{β′|β′ ∈ τ ′}. If ϕ(α′) is not contained in a single arc

of τ ′, there exists a separatrix L where the first point of intersection with β′′ is a point

z between ϕ(x) and ϕ(y). But ϕ−1(L), before intersecting α′′ in ϕ−1(z), intersects α′′

in t 6= ϕ−1(z). Thus L intersects β′′ in ϕ(t), which is before z on L; contradiction.

Now an analogous reasoning proves that ϕ(α′) ⊂ ∪{β′|β′ ∈ τ ′} for all α′ ∈ τ ′, which

completes the proof that τ ′ satisfies (2).

Let n be the first integer ≥ 0, for which ϕn+1 leaves invariant each separatrix. Let

τ be the subdivision of τ ′ defined by τ ′∨ϕ(τ ′)∨· · ·∨ϕn(τ ′); an arc α of τ is contained

in an arc of τ ′ and is bounded by two consecutive points of the form ϕj(x), ϕj′

(x′),

with x, x′ ∈ τ ′ and 0 ≤ j, j′ ≤ n. Properties (1), (3), (4) are evident.

For (2), we suppose that ϕ(α), which, since τ ′ satisfies (2), is contained in a certain

β′ of τ ′, is subdivided; that is to say that between ϕj+1(x) and ϕj′+1(x′), there will

be a ϕj′′

(x′′), with x′′ ∈ ∂τ ′, j′′ ≤ n. We claim that j′′ ≥ 1; this is true because

ϕ(α), which is contained in β′, is not subdivided by a point of ∂τ ′. Thus α contains

ϕj′′−1(x′′); contradiction.

We now prove (5). Let x ∈ ∂τ ; if ϕ−1(x) ∈ ∂τ , property (5) is evident. If ϕ−1(x) /∈∂τ , then x ∈ ∂τ . The leaf L of Fx also contains ϕn+1(Fx); by the same construction

as τ ′, x is the first point of intersection of L with the arc of τ ′′ that passes through x.

Thus ϕn+1(Fx) contains Fx and ϕ−1(Fx) ⊂ Fϕn(x).

Remark. Let x ∈ ∂τ − SingF and let L be the leaf containing Fx. Starting from the

singularity s of L, we consider the first point y that belongs to τ and not to Fx. We

denote by F ′x the segment from s to y on L.

DR

AFT

13 J

un 2

008


Let F (resp. F ′) be the union of the Fx (resp.F ′x). Seeing that ϕ(τ) ⊂ τ and ϕ(F ) ⊃ F ,

we verify without difficulty that ϕ(F ′) ⊃ F ′.

Definition. Let (F , µ) be an arational measured foliation and ϕ a diffeomorphism

such that ϕ(F , µ) = λ(F , µ) with λ > 1. A Markov pre-partition for (F , ϕ) is byMarkov pre-partition

definition a collection of F -rectangles R1, . . . , Rm such that

1) intRi ∩ intRj = ∅,2) ∪iRi = M ,

3) ϕ(∪i∂τRi) ⊂ ∪i∂τRi ;

4) ϕ−1(∪i∂FRi) ⊂ ∪i∂FRi.

Lemma 9.10. A Markov pre-partition also satisfies:

5) for each i = 1, . . . ,m and ǫ = 0, 1, ϕ(∂ǫτRi) is covered on the side of ϕ(Ri) by a single

rectangle: ϕ(∂ǫτRi) ⊂ ∂η(ǫ,i)τ Rj(ǫ,i) ;

6) in the same way,ϕ−1(∂ǫFRi) is covered on the side with ϕ−1(Ri) by a single rectangle.

This is to say that the image under ϕ of a rectangle Ri is something like in Fig-

ure 9.3.

Proof.

If (5) does not hold, there exists an x ∈ int(Ri) such that ϕ(x) ∈ ∂FRj , which

contradicts (4). Similarly, (6) follows from (3).

i 2

Riδτ1φ( )δτ

0 Riφ( )

Rip

Ri1leaf of F

1 2 p(N.B. : i , i , . . . , i are not necessarily distinct)

R

Figure 9.3:

DR

AFT

13 J

un 2

008


Lemma 9.11. Let (F , µ) be the canonical model of a class of arational foliations and ϕ a

diffeomorphism such that ϕ(F , µ) = (F , λµ) with λ > 1. After possible performing an

isotopy of ϕ preserving F , there exists a Markov pre-partition for (F , ϕ).

Proof. Let τ be as in Lemma 9.9 and let R′1, . . . , R

′l be the system of rectangles at-

tached to τ . We construct the system R1, . . . , Rm by taking the closures of the com-

ponents of ∪i intR′i − F ′, where F ′ is described in the remark following Lemma 9.9.

The conditions 1) and 2) of a Markov pre-partition are clearly satisfied. For con-

dition 3), we see that τ = ∪i∂τR′i = ∪i∂τRi and we know that ϕ(τ) ⊂ τ . Moreover,

by construction, each ∂ǫFR′i is an arc α of a leaf joining a point x ∈ ∂τ to a point y ∈ τ ,

and not intersecting τ in its interior. If x /∈ SingF , α is thus contained in F ′x; if x is

a singularity, α is contained in Fy ; thus ∂ǫFR′i ⊂ F ′. The subdivision guarantees that

F ′ is covered by the union of the ∂FRi. We have remarked that ϕ−1(F ′) ⊂ F ′; thus

condition 4) is satisfied.

In the rest of this expose, we will work with a Markov pre-partition R1, . . . Rmadapted to the measured foliation (F , µ) and to the diffeomorphism ϕ. We denote

by xi the µ-length of the rectangle Ri and by aij the number of times that ϕ(intRi)

crosses intRj (i.e. the number of components of the intersection). Since ϕ−1 dilates

the transverse distances by a factor of λ and since aij is also equal to the number of

times that ϕ−1(intRj) crosses intRi, we find

λxj =∑

i

xiaij .

In other words, the column vector xi is an eigenvector, with eigenvalue λ, for the

transpose matrix of A = (aij).

Lemma 9.12. There exist numbers ξ > 0 and y1, . . . , ym > 0 such that

yi = ξ∑

j

aijyj .

In other words, A admits an eigenvalue ξ−1 > 0, with an eigenvector whose coordinates are

all strictly positive.

Proof. Since aij ≥ 0 and, for each j, there exists an i such that aij > 0, A acts projec-

tively on the fundamental simplex. The Brouwer Fixed Point Theorem then implies

that A has an eigenvalue > 0 with an eigenvector y1 ≥ 0, . . . , yn ≥ 0, with∑yi > 0.

It suffices to show that, for all i, yi is

DR

AFT

13 J

un 2

008


nonzero.

Let us say, to fix ideas, that y1 = y2 = · · · = yl = 0 and that yl+1 > 0, . . . , yn > 0.

It follows that, for i ≤ l, we have

aij > 0 =⇒ j ≥ l.

In other words, the set J = ∪i=1Ri is invariant under ϕ and is not dense. To show

that this is a contradiction, we can make the following remarks.

1o First of all, if N > 0 is an integer, then R1, . . . , Rm is a Markov pre-partition

for ϕN . Thus, without loss of generality, we can reduce to the case where ϕ fixes each

sector of a singular points (in particular fixes each singularity).

2o As ϕ(J) ⊂ J and as each segment of τ is contracted by ϕ towards its singular

point, there exists among the boxes R1, . . . , Rℓ, a box, let us say R1, which is in the

configuration of Figure 9.4.

L

1

s1 SingFε

τ1

τ2

p

R

Figure 9.4:

3o Since τ2 is contracted by ϕ towards its singularity, the points ϕn(p) form an

infinite set. Further, they all belong to the same leaf L which is ϕ-invariant. Thus

the sequence converges towards infinity in the topology of the leaf. If F denotes the

segment of s1 at p on L, we have

L =

∞⋃

n=1

ϕn(F ).

4o The leaf L being dense in M , the preceding equality implies that ∪∞n=1ϕn(R1)

is dense in M , while, in addition, this union is contained in J which

DR

AFT

13 J

un 2

008


is not dense; contradiction.

9.5.1 Construction of a measured foliation F ′

We are going to construct a measured foliation F ′ having the same singularities as Fand transverse to F outside of these points, with the following properties:

A) Each segment of τ is contained in a leaf of F ′ and each rectangleRi is foliated

by F ′, as in Figure 9.5.

i

yi

Ri

= leaf of

= leaf of F

F’x

Figure 9.5:

B) The F ′-width of Ri is the yi of Lemma 9.12.

C) LetA0, A1, . . . , Ak be the sequence of points of τ found on ∂0FRi. The segments

[A1, A2], . . . , [Ak−2, Ak−1] are all of the type ∂ǫFRj ; thus their F ′-width is prescribed

by condition B). Let u be the F ′-width of [A0, A1]; we determine u in the following

way: if q is a large enough integer, ϕq(A0) and ϕq(A1) do not belong to ∂τ − SingF ;

then ϕq([A0, A1]) is a sum of segments (plaques), contained in a finite union of leaves

and singularities, each traversing an Rj from side to side. We claim then:

u = ξq × (sum of the F ′-widths of these segments).

We do the same for [Ak−1, Ak], and for the intervals of ∂1FRi. The lemma below says

that all these choices are consistent.

Lemma 9.13. The F ′-width of [A0, Ak] is the sum of the F ′-widths of the [Aj , Aj+1], j =

0, . . . , k − 1.

Proof. To fix ideas, let us say that ∂0FRi is [A0, A1]∪ [A1, A2]∪ [A2, A3] where [A1, A2]

is a plaque of

DR

AFT

13 J

un 2

008


R1 and where [A2, A3] is a plaque of R2; also, say that ϕ([A0, A1]) is the sum of a

plaque of R3 and of a plaque of R4. It suffices to prove that

yi = ξy3 + ξy4 + y1 + y2.

We set a(q)ij = cardπ0(ϕ

q(intRi) ∩ intRj)

aqij = cardπ0(ϕq(∂0

FRi) ∩Rj).

We remark that a(q)ij is the coefficient (i, j) of the matrix Aq ; further, the integer δqij =

aqij − a(q)ij is included between 0 and 4. Indeed, ϕq(∂0

FRi) ∩ Rj can contain two arcs

of ∂FRj such as the two images of the endpoints of ∂0FRj , in addition to the intersec-

tions of the interiors.

Finally, if q is big enough so that ϕq(∂τ) ∩ ∂τ = ∅, we have the equality given by

the geometry:

aqij = aq−13j + aq−1

4j + aqij + aq2j .

Thus:

∑

j

[a(q)ij − a

(q−1)3j − a(q−1)

4j − a(q)ij − a

(q)2j ]yj =

∑

j

[δq−13j + δq−1

4j + δq1j + δq2j − δqij ]yj .

The first terms is equal to [yi − ξy3 − ξy4 − y1 − y2]/ξq; as for the second term, it

only takes a finite number of values as q varies. This forces the numerator above to

be zero.

We provide each rectangle Ri with a system of coordinates X i, Y i such that

Ri = {0 ≤ X i ≤ xi, 0 ≤ Y i ≤ yi},

and that, for each segment [Aj , Aj+1] of ∂FRi, the difference Y i(Aj+1)−Y i(Aj) is the

width prescribed by condition C). We can thus interpret the rectangles as being an

atlas that defines the foliation (F ′, µ′), where the plaques are Y = constant and where

the transverse measure is µ′ = |dY |.

9.5.2 Construction of a “diffeomorphism”

We are going to construct a “diffeomorphism” ϕ′, isotopic to ϕ, such that ϕ′(F , µ) =

(F , λµ) and ϕ′(F ′, µ′) = (F ′, ξµ′). Actually, ϕ′ is going to be a diffeomorphism on

the complement of the singularities, but will not be C1 on the singularities (see the

definition of a “pseudo-Anosov diffeomorphism” below).

DR

AFT

13 J

un 2

008


We define ϕ′ by the following conditions:

α) ϕ′(Ri) = ϕ(Ri) ;

β) ϕ′(X i = c) = ϕ(X i = c) ;

γ) Let V be a component of Ri ∩ ϕ−1(Rj); then ϕ′(V ∩ (Y i = const.)) ⊂ (Y j =

const.);

δ) For p, q ∈ V , we have

ξ|Y j(ϕ′(p)) − Y j(ϕ′(q))| = |Y i(p)− Y i(q)|.Lemma 9.14. We have ξ = 1

λ .

Proof. Once (F , µ) and (F , µ′) are given, transverse to each other, we have a measure

M on M given locally by the product of µ and µ′. Clearly

ϕ⋆M = λξM.

As M is compact,M is of finite total measure and the above equality is only possible

if λξ = 1.

Remarks.1) The measureM is thus ϕ′-invariant. We can show that (M, ϕ′) is a Bernoulli

process. In particular, (M, ϕ′) is ergodic (see Section 10.6).2) We note the contrast between the fact that there does not exist a compact space

X equipped with a measure M and a homeomorphism ψ such that ψ⋆M = λM,

λ 6= 1, and the fact that there exists a compact manifold, equipped with a measured

foliation (F , µ) and a diffeomorphism ϕ : M → M such that ϕ(F , µ) = (F , λµ),

λ 6= 1.On the level of “the noncommutative space” that is the set of leaves ofF , equipped

with µ, this is precisely a “homeomorphism” as above. Such a “paradoxical” situa-

tion only happens for a “discrete” spectrum of values λ satisfying certain “arith-

metic” conditions. I am confused

by this para-

graph9.5.3 Pseudo-Anosov “diffeomorphisms”

By definition, a “diffeomorphism” ϕ : M → M is pseudo-Anosov, if there exist two

mutually transverse invariant measured foliations, (Fs, µs) and (Fu, µu), and anλ > 1, such that:

ϕ(Fs, µs) = (Fs, 1

λµs)

ϕ(Fu, µu) = (Fu, λµu)

DR

AFT

13 J

un 2

008


We call Fs (resp. Fu), the stable (resp. unstable) foliation; the diffeomorphism isstable

unstable contracting on the leaves of the stable foliation, where the lengths are measured by

µu.

From the point of view of “smoothness”, a pseudo-Anosov “diffeomorphism”

is a true diffeomorphism on M − SingF ; but it is never C1 at the singularities. Of

course, at the singularities, there are, topologically speaking, canonical local models

of the pseudo-Anosov, coming from quadratic differentials.

We have just proven the following lemma.

Lemma 9.15. If the diffeomorphism ϕ satisfies the conditions of Case III in Section 9.1, that

is, if there exists an arational measured foliation (F , µ) and λ 6= 1 such that ϕ(F , µ) ∼m(F , λµ), then ϕ is isotopic to a pseudo-Anosov “diffeomorphism”.

Remark. The hypothesis that F is arational is essential. It follows from the existence

of pseudo-anosov diffeomorphisms on a manifold with boundary, that on a closed

surface M (of genus > 1) there exists a measured foliation (F , µ) having a cycle of

leaves and a diffeomorphism ϕ satisfying ϕ(F , µ) ∼m (F , λµ) with λ 6= 1. As we will

see, this ϕ is not isotopic to a pseudo-Anosov diffeomorphism of M .

Theorem 9.16. Let ϕ be a diffeomorphism of a surface of genus > 1. Up to isotopy, ϕ is in

one of the following situations:

(1) Isometry for a hyperbolic structure

(2) “Reducible” (that is to say that ϕ fixes a system of simple curves that are mutually

disjoint and not homotopic to a point),

(3) Pseudo-Anosov.

Situations (1) and (3) (resp. (2) and (3)) are mutually exclusive.

Once we have extended the theory to nonempty boundary (Expose 11), we will

be able to write that ϕ is isotopic to ϕ′, for which there exists a decomposition M =

M1∪· · ·∪Mn into surfaces with boundary with disjoint interiors, such that ϕ′(Mi) =

Mi and that ϕ′|Mi is isotopic (as a diffeomorphism of Mi, the boundary being free)

to a hyperbolic isometry or to a pseudo-Anosov diffeomorphism. Of course, this de-

composition can not take into account any Dehn twists that ϕ′ does on the curves

along which the Mi are glued.

DR

AFT

13 J

un 2

008


Proof. The classification theorem is completely proven; it remains to prove the ex-

clusions.

The equality ϕ(F , µ) = (F , λµ) with λ 6= 1 prohibits the isotopy class of ϕ from

being periodic; thus, we obtain the incompatibility of (1) and (3).

We suppose that ϕ fixes an element of S′; up to replacing ϕ by one of its powers,

we can suppose that ϕ is pseudo-Anosov and preserves the isotopy class of a curve

γ. We thus have

I(Fs, µs; [γ]) = λI(Fs, µs; [ϕ(γ)]) = λI(Fs, µs; [γ]).

We deduce that I(Fs, µs; [γ]) = 0. By Expose 5, there is a foliation equivalent to Fs,having a cycle (nontrivial) of leaves. Since Fs is arational, this is a contradiction.

Remark. The incompatibility relations are in fact consequences of the dynamics of a

pseudo-Anosov of the completed Teichmuller space: there are only two fixed points,

represented by the stable and unstable foliations, which are respectively attracting

and repelling (see Expose 12).

9.6 Some properties of pseudo-Anosov diffeomorphisms

Lemma 9.17. The stable and unstable foliations of a pseudo-Anosov do not have connections

between singularities; they are thus canonical models for the classes of arational foliations.

Proof. Considering the case where the diffeomorphism fixes each singularity, such a

connection must be contracted or dilated, which is impossible.

Proposition 9.18. If U is a nonempty open set, invariant by a pseudo-Anosov diffeomorph-

ism, then U is dense.

Proof. It suffices to consider the case where the diffeomorphism ϕ fixes the singular-

ities of the stable and unstable foliations. Let F be a separatrix of the stable foliation,

emanating from a singularity s. Since F is dense in M , there exists a segment J of

F contained in U . Let a and b be the endpoints of J ; the sequences ϕn(a) and ϕn(b)

converge to s. Let T be a plaque of Fu contained in U and intersecting J

DR

AFT

13 J

un 2

008


in one point; ϕn(T ) is going to be lengthened from the point of view of the transverse

measure of the stable foliation and is going to approach the two separatrices adjacent

to F (Figure 9.6). Precisely, F ′ ∪F ′′ is contained in the closure of ∪n≥0ϕn(T ). Thus U

contains a separatrix of Fu entirely. A separatrix is dense, therefore U is dense.

b

s

=F u

φn(T)

F’’

F’

F

TJ

a

=F

Figure 9.6:

Corollary 9.19. A pseudo-Anosov diffeomorphism is topologically transitive (there exists a

dense orbit).

Proof. Let {Ui} be a countable basis of open sets. The intersection ∩i(∪n∈Zϕn(Ui))

is nonempty (Baire Category Theorem). All points in the intersection have a dense

orbit.

Proposition 9.20. The periodic points of a pseudo-Anosov diffeomorphism are dense.

This is a generalization of the analogous classical fact for Anosov diffeomor-

phisms.

Proof.

The singular points are periodic. Let x0 ∈M be a regular point and U a rectangle

adapted to the foliationsFs andFu that is neighborhood of x0. Let V be another rect-

angle neighborhood of x0, strongly included inU (Figure 9.7). As the diffeomorphism

ϕ leaves invariant a measure that assigns a nonzero measure to all nonempty open

sets (the measure is given locally by the product of µs and µu), Poincare Recurrence

[Sin76, p. 7] applies: for any n0, there exists n ≥ n0, such that ϕn(V ) ∩ V 6= ∅.

DR

AFT

13 J

un 2

008


U

1

x0

1(x )φn

(V)nφ

nφ (J)

u

s

J

V

x

Figure 9.7:

Let x1 be a point of V such that ϕn(x1) ∈ V . Let J be the Fs-plaque of U passing

through x1; we have

µu(ϕn(J)) = λ−nµu(J)

where λ is the dilatation factor of ϕ. We see that if n0 is chosen to be large enough (U

and V being given), we will be able to ensure that ϕn(J) is contained in U .

Identifying ϕn(J) to an interval of J , the identification being given by following

the Fu-plaques, we see that ϕn has a “fixed point in J”, which is to say that there

exists a point x2 of J where the Fu-leaf is invariant by ϕn.

Let L be the Fu-plaque of x2; if n0 is chosen large enough, we are sure that ϕn(L)

contains L, since ϕn(L) and L already have ϕn(x2) in common (this new condition

on n0 does not depend on the µs-widths of U and V ). Thus, there is a fixed point for

ϕn|L.

Proposition 9.21. Let g be a Riemannian metric on M and α ∈ S; we denote by lg(α) the

length of a minimizing geodesic of the class α. Let ϕ be a pseudo-Anosov diffeomorphism of

M , of dilatation λ > 1; the isotopy class of ϕ(α) is well-defined. We have

limn→∞

n

√lg(ϕn(α)) = λ.

Proof. If (Fs, µs) and (Fu, µu) are the stable and unstable foliations of ϕ, we can

define the “metric” µ =√

(µs)2 + (µu)2; this comes from a “singular norm” on the

tangent bundle, where the zeros are the singularities of the invariant foliations. We

note in passing that the metric µ is flat in the complement of the singularities and that

the curvature is constituted of Dirac masses at the singularities. Let

DR

AFT

13 J

un 2

008


c be a curve in the class α. We have:

I(Fs, µs;α) ≤ lµ(α) ≤∫

v

dµs +

∫

c

dµu

and

I(Fs, µs;ϕn(α)) ≤ lµ(ϕn(α)) ≤∫

ϕn(c)

dµs +

∫

ϕn(c)

dµu;

by the properties of ϕ, it follows that:

λnI(Fs, µs;α) ≤ lµ(ϕn(α)) ≤ λn∫

c

dµs + λ−n∫

c

dµu.

As we saw in the proof of the classification theorem, I(Fs, µs;α) 6= 0.

Therefore lim n√lµ(ϕn(α)) = λ. The proof now follows from Lemma 9.22.

Lemma 9.22. There exist constants K and k > 0 such that, for any class of loops α,

k ≤ lg(α)

lµ(α)≤ K

Proof. [A. Douady] Let a1, . . . , aq be the singularities of µ; letD(a, ρ) be the ball with

center a and of radius ρ for the metric µ. We choose ρ small enough so that the balls

D(ai, 2ρ) are disjoint. In the complement of the balls of radius ρ/2, the two metrics

give norms on the tangent bundle; thus there exist constantsK ′ and k′ > 0, such that,

for all rectifiable arcs β, we have

k′ ≤ Lg(β)

Lµ(β)≤ K ′, (9.1)

where Lg (resp. Lµ) denotes the geometric length.

Moreover, there exist constants K ′′ and k′′ > 0 such that, for all x, y ∈ ∂D(ai, ρ),

we have

k′′ ≤ dg(x, y)

dµ(x, y)≤ K ′′. (9.2)

Now, if x and y are close enough, the inequality (9.1) applies; on the other hand, if we

prohibit (x, y) from a neighborhood of the diagonal, the quotient above is defined,

continuous and > 0 on a compact set. In the two cases, the inequality (9.2) is clear.

We take k = inf(k′, k′′, 1) and K = sup(K ′,K ′′, 1). These constants depend on

the choice of radius ρ. We take these small enough so that, for all x, y ∈ ∂D(ai, ρ), the

shortest g-geodesic joining x to y is the identity in π1(M,D(ai, ρ)).

DR

AFT

13 J

un 2

008


Let c1 be a minimizing µ-geodesic of the class α and c′1 the loop obtained by

replacing each diagonal of c1 in the D(ai, ρ) by the g-geodesic joining the entry point

to the exit point (a diagonal is a connected component of c−11 (D)). We thus have:

klµ(α) ≤ Lg(c′1) ≤ Klµ(α).

From this we deduce that lg(α) ≤ Klµ(α). To obtain the other inequality, we start

from a minimizing g-geodesic c2 and we replace its diagonals in the balls by µ-

geodesic arcs.

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008

Expose 10

Some dynamics of

pseudo-Anosov

diffeomorphisms

by A. Fathi and M. Shub

We prove in this “expose” that a pseudo-Anosov diffeomorphism realizes the mini-

mum of topological entropy in its isotopy class. In Section 10.1 we define topological

entropy and give its elementary properties. In Section 10.2 we define the growth of

an endomorphism of a group and show that the topological entropy of a map is

greater than the growth of the endomorphism it induces on the fundamental group.

In Section 10.3, we define subshifts of finite type and give some of their properties. In

Section 10.4, we prove that the topological entropy of a pseudo-Anosov diffeomorph-

ism is the growth rate of the automorphism induced on the fundamental group, it

is also logλ, where λ > 1 is the stretching factor of f on the unstable foliation. In

Section 10.5, we prove the existence of a Markov partition for a pseudo-Anosov diff-

eomorphism, this fact is used in Section 10.4. In Section 10.6, we show that a pseudo-

Anosov map is Bernoulli.

187

DR

AFT

13 J

un 2

008


10.1 Topological Entropy

Topological entropy was defined to be a generalization of measure theoretic en-

tropy [AKM65]. In some sense, entropy is a number (possibly infinite) which de-

scribes “how much” dynamics a map has. Here the emphasis, of course, must be

on asymptotic behaviour. For example, if f : X → X is a map and Nn(f) is the

cardinality of the fixed point set of fn, then lim sup 1n logNn(f) is one measure of

“how much” dynamics f has; but, if we consider f × Rθ : X × T 1 → X × T 1 to be

(f×Rθ)(x, α) = (f(x), θ+α) where T 1 = R/Z and θ is irrational thenNn(f×Rθ) = 0,

and yet f ×Rθ should have at least as “much” dynamics as f . Topological entropy is

a topological invariant which overcomes this difficulty.

We describe a lot of material frequently without crediting authors.

Definition. Let f : X → X be a continuous map of a compact topological space

X . Let A = {Ai}i∈I and B = {Bj}j∈J be open covers of X . The open cover {Ai ∩Bj}i∈I,j∈J will be denoted by A ∨ B. If A is a cover, Nn(f,A) denotes the minimum

cardinality of a subcover ofA∨f−1A∨· · ·∨f−n+1A, and h(f,A) = lim sup 1n logNn(f,A).

The topological entropy of f is h(f) = supAh(f,A) where the supremum is taken overtopological entropy

all open covers of X .

Proposition 10.1. LetX and Y be compact spaces. Let f : X → X, g : Y → Y and h : X →Y be continuous. Suppose that h is surjective and hf = gh:

X

h

fX

h

Yg

Y

then h(f) ≥ h(g).In particular, if h is a homeomorphism, then h(f) = h(g). So topological entropy is a

topological invariant.

Proof. Pull back the open covers of Y to open covers of X .

For metric spaces, compact or not, Bowen has proposed the following definition.

Suppose f : X → X is a continuous map of a metric space X and suppose

DR

AFT

13 J

un 2

008


K ⊂ X is compact. Let ǫ be > 0. We say that a set E ⊂ K is (n, ǫ)-separated if, (n, ǫ)-separated

given x, y ∈ E with x 6= y, there is 0 ≤ i < n such that d(f i(x), f i(y)) ≥ ǫ. We

let sK(n, ǫ) be the maximal cardinality of an (n, ǫ)-separated set contained in K .

We say that the set E is (n, ǫ)-spanning for K if, given y ∈ K , there is an x ∈ E (n, ǫ)-spanning

such that d(f i(x), f i(y)) < ǫ for each i with 0 ≤ i < n. We let rK(n, ǫ) be the

minimal cardinality of an (n, ǫ)-spanning set contained in K . It is easy to see that

rK(n, ǫ) ≤ sK(n, ǫ) ≤ rK(n, ǫ/2). We let sK(ǫ) = lim sup 1n log sk(n, ǫ) and rK(ǫ) =

lim sup 1n log rk(n, ǫ). Obviously sK(ǫ) and rK(ǫ) are decreasing functions of ǫ, and

rK(ǫ) ≤ sK(ǫ) ≤ rK(ǫ/2). Hence, we may define hK(f) = limǫ→0

sK(ǫ) = limǫ→0

rK(ǫ).

Finally, we put hX(f) = sup{hK(f) | K compact ⊂ X}.

Proposition 10.2. [Bow70, Din71]. If X is a compact metric space and f : X → X is

continuous, then hX(f) = h(f).

Proof. The proof is rather straightforward. By the Lebesgue covering lemma, every

open cover has a refinement which consists of ǫ-balls.

The number hX(f) depends on the metric on X and makes best sense for uni-

formly continuous maps.

Suppose that X and Y are metric spaces, we say that p : X → Y is a metric metric covering map

covering map if it is surjective and satisfies the following condition: there exists ǫ > 0

such that, for any 0 < δ < ǫ, any y ∈ Y and any x ∈ p−1(y), the map p : Bδ(x) →Bδ(y) is a bijective isometry (here Bδ(·) is the δ-ball).

The main example we have in mind is the universal covering p : M → M of a

compact differentiable manifold M .

DR

AFT

13 J

un 2

008


Proposition 10.3. Suppose p : X → Y is a metric covering and f : X → X, g : Y → Y are

uniformly continuous. If pf = gp, then hX(f) = hY (g).

Proof. It should be an easy estimate. The clue is that for ℓ > 0 and for any sequence anwe have lim sup 1

n log(ℓan) = lim sup 1n log an. If K ⊂ X and K ′ ⊂ Y are compact and

p(K) = K ′, then there is a number ℓ > 0 such that [cardinality(p−1(y))] ≤ ℓ for all y ∈K ′. In fact, we may choose ℓ such that, if δ > 0 is small enough, then p−1(Bδ(y)) ∩Kcan be covered by at most ℓ 2δ-balls centered at points in p−1(Bδ(y)) ∩K .

By the uniform continuity of f , we can find a δ0 (< ǫ) such that x, x′ ∈ X and

d(x, x′) < δ0 implies d(f(x), f(x′)) < ǫ, where ǫ > 0 is the one given in the definition

of a metric covering. If 2δ < δ0, it is easy to see that if E′ ⊂ K ′ is an (n, δ)-spanning

set for g, then there exists an (n, 2δ)-spanning set E ⊂ K for f , such that cardE ≤ℓ cardE′. So, we have rK(n, 2δ) ≤ ℓrK′(n, δ), hence rK(f, 2δ) ≤ rK′(g, δ) and hK(f) ≤hK′(g).

On the other hand, if E ⊂ K is (n, η)-spanning (with 0 < η < ǫ) then p(E) ⊂ K ′

is (n, η)-spanning. So rK′(n, η) ≤ rK(n, η), hence hK′(g) ≤ hK(f). Consequently

hK(f) = hK′(g). Since we sup over all compact sets and since p is surjective, we

obtain hX(f) = hY (g).

We add one additional fact.

Proposition 10.4. If X is compact and f : X → X is a homeomorphism, then h(fn) =

|n|h(f).

For a proof, see [AKM65] or [Bow70].

DR

AFT

13 J

un 2

008


10.2 The Fundamental Group

Given a finitely generated group G and a finite set of generators G = {g1, . . . , gr} of

G, we define the length of an element g of G by LG(g) = minimum length of a word length

in the gi’s and the g−1i ’s representing the element g.

It is easy to see that if G′ = {g′1, . . . , g′s} is another set of generators, then:

LG(g) ≤ (maxLG(g′i))LG′(g).

If A : G→ G is an endomorphism, let:

γA := supg∈G

lim sup1

nlogLG(Ang) = sup

gi∈Glim sup

1

nlogLG(Angi).

So γA is finite and by the inequality given above, γA does not depend on the set of

generators.

Proposition 10.5. If A : G → G is an endomorphism and g ∈ G, define gAg−1 : G → G

by [gAg−1](x) = gA(x)g−1. We have γA = γgAg−1 .

Caution: (gAg−1)n 6= gAng−1.

First, we need a lemma.

Lemma 10.6. Let (an)n≥1 and (bn)n≥1 be two sequences with an and bn ≥ 0 and k be > 0.

We have:

i) lim sup 1n log(an + bn) = max(lim sup 1

n log an, lim sup 1n log bn)

ii) lim sup 1n log kan = lim sup 1

n log an

iii) lim sup 1n log an ≤ lim sup 1

n log(a1 + · · ·+ an) ≤ max(0, lim sup 1n log an)

Proof. Put a = lim sup 1n log a and b = lim sup 1

n log bn.

(i) The inequality max(a, b) ≤ lim sup 1n log(an + bn) is clear.

DR

AFT

13 J

un 2

008


If c > max(a, b), then we can find n0 ≥ 1 such that n ≥ n0 implies an ≤ enc and

bn ≤ enc. We obtain for n ≥ n0:

1

nlog(an + bn) ≤

1

nlog(2enc).

Hence lim sup 1n log(an + bn) ≤ lim sup 1

n log(2enc) = c.

(ii) is clear.

(iii) The inequality a ≤ lim sup 1n log(a1 + · · ·+ an) is clear.

Suppose c > max(0, a). We can find then n0 ≥ 1 such that an ≤ enc for n ≥ n0.

We have for n ≥ n0:

a1 + · · ·+ an ≤n0−1∑

i=1

ai +e(n+1−n0)c − 1

ec − 1en0c.

It follows clearly that lim supn

1n1(a1 + · · ·+ an) ≤ c.

Proof. (Proposition 10.5). If x ∈ G, we have:

(gAg−1)n(x) = gA(g) · · ·An−1(g)An(x)An−1(g−1) · · ·A(g−1)g−1.

Suppose first that An0(g) = e for some n0, then it is clear that by Lemma 10.6 (i):

lim sup1

nlogLG [(gAg−1)n(x)] ≤ lim sup

1

nlogLG(An(x)).

If An(g) 6= e for each n ≥ 1, we have LG(An(g)) ≥ 1, for each n ≥ 1; hence

lim sup 1n logLG(An(g)) ≥ 0. By Lemma 10.6 (i) & (iii), we obtain:

lim sup1

nlogLG [(gAg−1)n(x)] ≤ max(lim sup

1

nlogLG((An(g)), lim sup

1

nlogLG(An(x))).

This gives us γgAg−1 ≤ γA, and by symmetry, we have γgAg−1 = γA.

DR

AFT

13 J

un 2

008


For a compact connected differentiable manifold, we interpret π1(M) as the group

of covering transformations of the universal covering space M of M . If f : M → M

is continuous, then there is a lifting f : M → M . If f1 and f2 are both liftings of

f , then f1 = θf2 for some covering transformation θ. A given lifting f1 determines

an endomorphism f1# of π1(M) by the formula f1α = f1#(α)f1 for any covering

transformation α. If f1 and f2 are two liftings of f , then f1 = θf2 for some covering

transformation θ and f1α = θf2α = θf2#(α)f2 = θf2#(α)θ−1f1, so f1# = θf2#θ−1

and γf1# = γf2# . Thus, we may define γf# = γf# for any lifting f : M → M of f . If f

has a fixed point m0 ∈M , then there is also a map f# : π1(M,m0)→ π1(M,m0). The

group π1(M,m0) is isomorphic to the group of covering transformations of M and f

may be lifted to f such that f# : π1(M)→ π1(M) is identified with f# : π1(M,m0)→π1(M,m0) by this isomorphism. Thus γf# makes coherent sense in the case that f

has a fixed point as well.

We suppose now that M has a Riemannian metric and we put on M a Rieman-

nian metric by lifting the metric on M via the covering map p : M → M . The map p

is then a metric covering and the covering transformations are isometries. We have

the following lemma due to Milnor [Mil68].

Lemma 10.7. Fix x0 ∈ M . There exist two constants c1, c2 > 0 such that for each g ∈π1(M), we have:

c1LG(g) ≤ d(x0, gx0) ≤ c2LG(g).

Proof. [Mil68]. Let δ = diam(M), and define N ⊂ M by N = {x ∈ M | d(x, x0) ≤ δ}.We have p(N) = M . Remark that {gN}g∈π1(M) is a locally finite covering of M by

compact sets. Choose as a finite set of generators G = {g ∈ π1(M) | gN ∩N 6= ∅} and

notice that g ∈ G ⇐⇒ g−1 ∈ G. Suppose LG(g) = n, then we can write g = g1 · · · · ·gn,

with giN ∩N 6= ∅. It is easy to see then that d(x0, gx0) ≤ 2δn. Hence, we obtain:

DR

AFT

13 J

un 2

008


d(x0, gx0) ≤ 2δLG(g).

Now, put ν = min{d(N, gN) | N ∩ gN = ∅}, by compactness ν > 0. Let k be the

minimal integer such that d(x0, gx0) < kν. Along the minimizing geodesic from x0

to gx0, take k + 1 points y0 = x0, y1, . . . , yk−1, yk = gx0 such that d(yi, yi+1) < ν for

i = 0, . . . , k− 1. Then, for 1 ≤ i ≤ k− 1, choose y′i ∈ N and gi ∈ G such that yi = giy′i

and put g0 = e and gk = g. We have d(giy′i, gi+1y

′i+1) < ν, hence g−1

i gi+1 ∈ G. From

g = (g−10 g1) · · · (g−1

k−1gk), we obtain LG(g) < k.

Since k is minimal, we have:

LG(g) ≤ 1

νd(x0, gx0) + 1 ≤

(1

ν+

1

µ

)d(x0, gx0)

where µ = min{d(x0, gx0) | g 6= e, g ∈ π1(M)}.

Consider now f : M → M and let f : M → M be a lifting of f . Applying the

lemma above, we obtain, for each x0 ∈ M :

γf# = maxg∈π1(M)

lim sup1

nlog d(x0, f

n#(g)x0).

We next prove the following lemma:

Lemma 10.8. Given x, y ∈ M , we have:

lim sup1

nlog d(fn(x), fn(y)) ≤ h(f).

Proof. Choose an arc α from x to y. If y1, . . . , yℓ ∈ α is (n+1, ǫ)-spanning for α and f ,

then fn(α) ⊂ ∪ℓi=1B(fn(yi), ǫ). Since fn(α) is connected, this implies diam fn(α) <

2ǫℓ. Hence:

d(fn(x), fn(y)) ≤ 2ǫℓ.

By taking ℓ to be minimal, we obtain:

d(fn(x), fn(y)) ≤ 2ǫrα(n+ 1, ǫ).

DR

AFT

13 J

un 2

008


From this, we get:

lim supn→∞

1

nlog d(fn(x), fn(y)) ≤ lim sup

n→∞

1

nlog[2ǫrα(n+ 1, ǫ)] = rα(ǫ)

≤ hα(f) ≤ h(f) = h(f).

We are now ready to prove:

Theorem 10.9. If f : M →M is a continuous map, then:

h(f) ≥ γf# .

Proof. Since γf# = maxg∈π1(M)

[lim sup 1n log d(x0, f

n#(g)x0], we have to prove that for

each g ∈ π1(M):

lim sup1

nlog d(x0, f

n#(g)x0) ≤ h(f).

We have:

d(x0, fn#(g)x0) ≤ d(x0, f

n(x0)) + d(fn(x0), fn#(g)fn(x0)) + d(fn#(g)fn(x0), f

n#(g)x0).

Since f#(g)fn = fng, and the covering transformations are isometries, we obtain:

d(x0, fn#(g)x0) ≤ 2d(x0, f

n(x0)) + d(fn(x0), fn(gx0)).

Remark also that:

d(x0, f(x0)) ≤ d(x0, f(x0)) + d(f (x0), f2(x0)) + · · ·+ d(fn−1(x0), f

n(x0)).

By applying Lemma 10.8 and Lemma 10.6 (together with the fact h(f) > 0), we ob-

tain:

lim sup1

nlog d(x0, f#(g)x0) ≤ h(f).

The proof of the following lemma is straightforward.

Lemma 10.10. If G1 and G2 are finitely generated groups, if A : G1 → G1, B : G2 → G2

DR

AFT

13 J

un 2

008


and p : G1 → G2 are homomorphisms with p surjective and pA = Bp:

G1p

A

G2

B

0

G1p

G2 0

then, γA ≥ γB .

Applying this lemma to the fundamental group of M mod the commutator sub-

group, we have

π1(M)p

f#

H1(M)

f1∗

0

π1(M)p

H1(M) 0

so we obtain Manning’s theorem [Man74].

Theorem 10.11. If f : M → M is continuous, then h(f) ≥ γf1∗ = max logλ , where λ

ranges over the eigenvalues of f1∗.

Remark 1. For α ∈ π1(M,m0), we denote by [α] the class of loops freely homotopic to

α. If M has a Riemannian metric, let ℓ([α]) be the minimum length of a (smooth) loop

in this class. If f : M → M is continuous, f [α] is clearly well defined as a free homo-

topy class of loops. Let Gf ([α]) = lim supn

1n log[ℓ(fn[α])] and let Gf = sup

αGf ([α]).

It is not difficult to see that Gf ≤ γf# . In fact, we have ℓ(fn[α]) ≤ d(x0, fn#(α)x0),

since the minimizing geodesic from x0 to fn#(α)x0 has an image in M which repre-

sents fn[α].

Remark 2. It occurred to various people that Manning’s theorem is a theorem about

π1.

DR

AFT

13 J

un 2

008


Among these are Bowen, Gromov and Shub. Manning’s proof can be adapted. The

proof above is more like Gromov [Gro] or Bowen [Bow70], but we take responsibility

for any error. At first, we assumed that f had a periodic point or we worked with

Gf . After reading Bowen’s proof [Bow78], we eliminated the necessity for a periodic

point.

Remark 3. If x ∈ M and ρ is a path joining x to f(x), we call ρ# the homomorphism

π1(M, f(x))→ π1(M,x). Since f# : π1(M,x)→ π1(M, f(x)), the composition

ρ#f# : [γ] 7→ [ρ−1γρ]

is a homomorphism of π1(M,x) into itself. This homomorphism can be identified

with f# for a lifting f of f . Thus our result is the same as Bowen’s [Bow78].

10.3 Subshifts of finite type

Let A = (aij) be a k × k matrix such that aij = 0 or 1, for 1 ≤ i, j ≤ k, that is A is

a 0 − 1 matrix. Such a matrix A determines a subshift of finite type as follows. Let

Sk = {1, . . . , k} and let Σ(k) =

i=∞∏

i=−∞Sik, where Sik = Sk for each i ∈ Z. We put on

Sk the discrete topology and on Σ(k) the product topology. The subset ΣA ⊂ Σ(k)

is the closed subset consisting of those bi-infinite sequences b = (bn)n∈Z such that

abibi+1 = 1 for all i ∈ Z.

Pictorially, we imagine k boxes 1 2 · · · k and a point which at discrete “time

n” can be in any one of the boxes. The bi-infinite sequences represent all possible

histories of points. If we add the restriction that a point may move from box i to box

j, if and only if aij = 1, then the set of all possible histories is precisely ΣA.

The shift σA : ΣA → ΣA is defined by σA[(bn)n∈Z] = (b′n)n∈Z where b′n = bn+1 shift

for each n ∈ Z. Clearly, σA is continuous. Let Ci ⊂ Σ(k) be defined by Ci = {x ∈Σ(k) | x0 = i}. Let Di = Ci ∩ ΣA, then D = {D1, . . . , Dk} is an open

DR

AFT

13 J

un 2

008


cover of ΣA by pairwise disjoint elements. For any k × k matrix B = (bij), we de-

fine the norm ||B|| of B by ||B|| =

k∑

i,j=1

|bij |. It is easy to see that Nn(σA,D) =

card(D ∨ · · · ∨ σ−n+1A D) ≤ ||An−1|| because the integer a

(n)ij is equal to the num-

ber of sequences (i0, . . . , in) with iℓ ∈ {1, . . . , k}, i0 = i, in = j and aiℓiℓ+1= 1.

So lim sup 1n log(Nn(σA,D)) ≤ lim sup 1

n log ||An−1|| = lim sup log ||An||1/n. This lat-

ter number is recognizable as log(spectral radius A) or log λ, where λ is the largest

modulus of an eigenvalue of A. In fact, we have:

Proposition 10.12. For any subshift of finite type σA : ΣA → ΣA, we have h(σA) = logλ,

where λ is the spectral radius of A.

Proof. We begin by noticing that each open cover U of ΣA is refined by a cover of the

formℓ∨

i=−ℓσ−iA D. This implies, with the notations of Section 10.1:

Nn+1(σA,U) ≤ card(

j=n+ℓ∨

j=−ℓσ−jA D) = card(

j=n+2ℓ∨

j=0

σ−jA D) = Nn+2ℓ+1(σA,D).

Hence, we obtain: h(σA,U) ≤ h(σA,D).

This shows that h(σA) = h(σA,D).

We now compute h(σA,D). We distinguish two cases.

First case. Each state i = 1, . . . , k occurs. This means that Di 6= ∅ for eachDi ∈ D.

It is not difficult to show by induction that we have in fact

Nn+1(σA,D) = card(D ∨ · · · ∨ σ−nA D) = ‖An‖.

This proves the proposition in this case, as we saw above.

Second case. Some states do not occur. One can see that a state i occurs, if, and

only if, for each n ≥ 0, we have:

k∑

j=1

a(n)ij > 0 and

k∑

j=1

a(n)ji > 0

where An = (a(n)ij ).

DR

AFT

13 J

un 2

008


Notice that ifk∑

j=1

a(n0)ij = 0 then

k∑

j=1

a(n)ij = 0 for all n ≥ n0. This is because each

aℓm is ≥ 0.

Now, we partition {1, . . . , k} into three subsets X,Y, Z, where:

X = {i | ∀n ≥ 0

k∑

j=1

a(n)ij > 0 ,

k∑

j=1

a(n)ji > 0}

Y = {i | ∃n > 0

k∑

j=1

a(n)ij = 0} = {i | for n large

k∑

j=1

a(n)ij = 0}

Z = {1, . . . , k} − (X ∪ Y ).

We have:

Z ⊂ {i | for n largek∑

j=1

a(n)ji = 0}.

By performing a permutation of {1, . . . , k}, we can suppose that we have the follow-

ing situation:

{1, . . . , t︸︷︷︸X

, t+ 1, . . . , s︸︷︷︸Y

, s+ 1, . . . , k︸︷︷︸Z

}

If B is a k × k matrix, we write:

B =

BXX BXY BXZBYX BY Y BY ZBZX BZY BZZ

where BKL corresponds to the subblock of B having row indices in K and column

indices in L.

It is easy to show that:

Nn+1(σA,D) = card(D ∨ · · · ∨ σ−nA D) = ‖AnX,X‖.

On the other hand, by the definition of Y and Z , for n large, An has the form:

An =

(An)X,X (An)X,Y 0

0 0 0

(An)Z,X (An)Z,Y 0

DR

AFT

13 J

un 2

008


This implies that for n large, An and (An)X,X have the same non-zero eigenval-

ues, in particular:

log(spectral radius AnX,X) = n logλ.

Remark also that we get, for n large and k ≥ 1:

(Akn)X,X = [(An)X,X ]k.

This gives us, for n large:

lim supk→∞

1

kn+ 1logNkn+1(σA,D) = lim sup

k→∞

1

kn+ 1

∥∥[(An)X,X ]k∥∥ = logλ

This implies that:

logλ ≤ h(σA,D) = lim supn→∞

1

nlogNn(σA,D).

As we showed the reverse inequality, we have:

logλ = h(σA,D) = h(σA).

10.4 The entropy of pseudo-Anosov diffeomorphisms

Now we suppose that we have a compact, connected 2-manifold M without bound-

ary with genus ≥ 2, and a pseudo-Anosov diffeomorphism f : M →M . Hence there

exists a pair (FU , µU ) and (FS , µS) of transverse measured foliations with (the same)

singularities such that f(FS , µS) = (FS , 1λµ

S) and f(FU , µU ) = (FU , λµU ) where

λ > 1. This means, in particular, that f preserves the two foliations FS and FU ; it

contracts the leaves of FS by 1λ and it expands the leaves of FU by λ.

Let us recall that for any non trivial simple closed curve α we have logλ = Gf (a)

(see Proposition 9.21), hence we get logλ ≤ Gf . [For the definition of Gf , look at thefix ref

end of Section 10.2.]

Proposition 10.13. If f : M →M is pseudo-Anosov, then h(f) = γf# . So in particular, f

DR

AFT

13 J

un 2

008


has the minimal entropy of anything in its homotopy class. Moreover h(f) = logλ where λ

is the expanding factor of f .

Proof. Since Gf ≥ logλ, it suffices to show that h(f) ≤ logλ for a pseudo-Anosov

diffeomorphism f . To do this, we find a subshift of finite type σA : ΣA → ΣA and a

surjective continuous map ΣA →M such that:

ΣAσA

θ

ΣA

θ

Mf

M

commutes, and log(spectral radius A) = h(σA) = log λ for this same λ. Thus we will

have logλ ≤ Gf ≤ γf# ≤ h(f) ≤ h(σA) or logλ ≤ h(f) ≤ logλ.

In the following, we construct A and θ via Markov partitions.

First some definitions.

Definition. (compare Expose 9). A subset R of M is called a (FS ,FU )-rectangle, or (FS , FU )-rectangle

birectangle, if there exists an immersion φ : [0, 1]× [0, 1]→M whose image is R and birectangle

such that:

• ϕ|(0, 1)× (0, 1) is an imbedding;

• ∀ t ∈ [0, 1], ϕ({t}× [0, 1]) is included in a finite union of leaves and singularities

of FS , and in fact in one leaf if t ∈ (0, 1)

• ∀ t ∈ [0, 1], ϕ([0, 1]×{t}) is included in a finite union of leaves and singularities

of FU , and in fact in one leaf if t ∈= (0, 1).

We adopt the following notations:

intR = φ((0, 1)× (0, 1)

)

∂0FSR = φ({0} × [0, 1])

∂1FSR = φ({1} × [0, 1])

∂FSR = ∂0FSR ∪ ∂1

FSR

and in the same way, we define ∂0FUR, ∂1

FUR, ∂FUR.

DR

AFT

13 J

un 2

008


Remark that intR is disjoint from ∂FSR ∪ ∂FUR, because φ|(0, 1) × (0, 1) is an

imbedding.

We call a set of the form φ({t} × [0, 1]) ( resp. φ([0, 1] × {t}) ) a FS-fiber, (resp. a

FU -fiber) of R. We will call a birectangle good if φ is an embedding.birectangle, good

If R is good birectangle, a point x of R is contained in only one FS-fiber which

we will denote by FS(x,R). In the same way, we define FU (x,R).

good

rectangle

birectangle (not good)

Remark 1) If R is a FU -rectangle (see Expose 9) and ∂0τR and ∂1

τR are contained in a

union of FS-leaves and singularities, it is easy to see that R is in fact a birectangle.

2) We used the word birectangle instead of rectangle, even though rectangle is

the standard word in Markov partitions, because this word was already used in Ex-

pose 9.

3) If R1 and R2 are birectangles and R1 ∩R2 6= ∅ then it is a finite union of birect-

angles and possibly of some arcs contained in (∂FSR1 ∪ ∂FUR1)∩ (∂FSR2 ∪ ∂FUR2).

DR

AFT

13 J

un 2

008


Moreover the birectangles are the closures of the connected components of intR1 ∩intR2.

If R is a birectangle, we define the width of R by: width

W(R) = max{µU (FS-fiber), µS(FU -fiber)}.

Lemma 10.14. There exists ǫ > 0 such that, if R is a birectangle withW(R) ≤ ǫ, then it is

a good rectangle.

Proof. [Sketch] If a birectangle is contained in a coordinate chart of the foliations,

then it is automatically a good birectangle. The existence of ǫ follows from compact-

ness.

Lemma 10.15. There exists ǫ > 0 such that if α (resp. β ) is an arc contained in a finite

union of leaves and singularities of FS (resp. FU ) with µU (α) < ǫ (resp. µS(β) < ǫ), then

the intersection of α and β is at most one point.

Definition. A Markov partition for the pseudo-Anosov diffeomorphism f : M →M Markov partition

is a collection of birectangles R = {R1, . . . , Rk, } such that:

1.k⋃

i=1

Ri = M ;

2. Ri is a good rectangle;

3. intRi ∩ intRj = ∅ for i 6= j;

4. If x is in int(Ri) and f(x) is in int(Rj), then

f(FS(x,Ri)) ⊂ FS(f(x), Rj), and f−1(FU (f(x), Rj)) ⊂ FU (x,Ri);

5. If x is in int(Ri) and f(x) is in int(Rj), then

f(FU (x,Ri)) ∩ (Rj) = FU (f(x), Rj) and f−1(FS(x,Rj)) ∩Ri = FS(x,Ri);

This means that f(Ri) goes across Rj just one time.

DR

AFT

13 J

un 2

008


Rj

f(Ri) {s

We will show in next section how to construct a Markov partition for a pseudo-

Anosov diffeomorphism.

Given a Markov partition R = (R1, . . . , Rk}, we construct the subshift of finite

type ΣA and the map h : ΣA → M as follows. Let A be the k × k matrix defined by

aij = 1 if f(intRi)∩ intRj 6= ∅, and aij = 0 otherwise. If b ∈ ΣA then⋂

i∈Z

f−i(Rbi) is

non empty and consists in fact of a single point. This will follow from the following

lemma.

Lemma 10.16. i) Suppose aij = 1, then f(Ri)∩Rj is a non empty (good) birectangle which

is a union of FU -fibers of Rj .

ii) Suppose moreover that C is a birectangle contained in Ri which is a union of FU -

fibers of Ri, then f(C) ∩ Rj is a non empty birectangle which is a union of FU -fibers of

Rj .

iii) Given b ∈ ΣA, for each n ∈ N,n⋂

i=−nf−i(Rbi) is a non empty birectangle. Moreover,

we haveW(n⋂

i=−nf−i(Rbi)) ≤ λ−n max{W(R1), . . . ,W(Rk)}.

Proof. Since aij = 1, we can find x ∈ int(Ri) ∩ f−1(intRj). We have f(FS(x,Ri)) ⊂FS(f(x), Rj) ⊂ Rj . Since each FU -fiber of Ri intersects FS(x,Ri), we obtain that the

image of eachFU -fiber ofRi intersectsRj . Moreover, by condition 5) f [Ri−∂FURi]∩Rj is an union of FU -fibers of Rj , hence f(Ri) ∩ Rj = f(Ri − ∂FURi) ∩Rj is also a

union of FU -fibers of Rj . This proves i). The proof of ii) is the same.

To prove iii), remark first that it follows by induction on n using ii) that

DR

AFT

13 J

un 2

008


each set of the form fnRbi ∩ fn−1(Rbi+1) ∩ · · · ∩ Rbi+n is a non empty birectangle

which is a union of FU -fibers of Rbi+n . In particular,n⋂

i=−nf−i(Rbi) is a non empty

birectangle in Rb0 . The estimate of the width is clear.

By the lemma, if b ∈ ΣA the set⋂i∈Z f−i(Rbi) is the intersection of a decreas-

ing sequence of non empty compact sets, namely the sets⋂ni=−n f

−i(Rbi) for n ∈ N.

Hence⋂i∈Z f−i(Rbi) is non-void. It is reduced to one point becauseW(

⋂ni=−n f

−i(Rbi))

tends to zero as n goes to infinity.

The map θ : ΣA →M given by θ(b) =⋂i∈Z f

−i(Rbi) is well defined, it is easy to

see that it is continuous and that the following diagram commutes:

ΣAσA

θ

ΣA

θ

Mf

M

We show now that θ is surjective. First remark that, for each i = 1, . . . , k, the

closure of int(Ri) is Ri. Hence V =⋃ki=1 int(Ri) is a dense open set. By the Baire

category theorem U =⋂i∈Z f

−i(V ) is dense in M . If x ∈ U , then for each n ∈ Z,

the point fn(x) is in a unique int(Rbn) and b = {bn}n∈Z is an element of ΣA. It is

clear that θ(b) = x. Thus θ(ΣA) ⊃ U . As ΣA is compact and h continuous, we have

θ(ΣA) = M .

Up to now, we have obtained that:

logλ ≤ Gf ≤ γf# ≤ h(f) ≤ h(σA) = log(spectral radius of A)

All that remains is to show that:

(spectral radius of A) = λ.

To see this, we do the following thing. Put yi = µU (FS-fiber of Ri), it is clear that this

quantity is independent of the FS-fiber of Ri and also yi > 0.

DR

AFT

13 J

un 2

008


We have trivially the following equality:

yj =k∑

i=1

yiλaij ,

which gives:

λyj =

k∑

i=1

yiaij

[in particular λ is an eigenvalue of A]. Hence. we obtain:

λyj ≥( k∑

i=1

aij

)miniyi.

This gives us:

λ(∑

j

yj

)≥ ||A||min

iyi

where || || is the norm introduced in Section 10.3.

In the same way, we obtain for each n ≥ 2:

λn(∑

j

yj

)≥ ||An||min

iyi.

Hence:

λ ≥ ||An||1/n(min(y1, . . . , yk)∑

j

yj

)1/n

.

Since min(y1, . . . , yk) > 0,

limn→∞

(min(y1, . . . , yk)∑

j

yj

)1/n

= 1.

We thus obtain:

λ ≥ limn→∞

||An||1/n = spectral radius of A.

Since λ is an eigenvalue of A, we obtain:

λ = spectral radius of A.

DR

AFT

13 J

un 2

008


10.5 Construction of Markov partitions for pseudo-Anosov

diffeomorphisms

In this section, we still consider f : M → M a pseudo-Anosov diffeomorphism and

we keep the notations of the last section. We sketch the proof of the following propo-

sition.

Proposition 10.17. A pseudo-Anosov diffeomorphism has a Markov partition.

Proof. Using the methods given in Section 9.5, it is easy, starting with a family of

transversals to FU contained in FS-leaves and singularities, to construct a family Rof FU -rectangles R1, . . . , Rℓ, such that:

(i)ℓ⋃i=1

Ri = M ;

(ii) int(Ri) ∩ int(Rj) = ∅ for i 6= j;

(iii) f−1(

ℓ⋃

i=1

∂FURi) ⊂ℓ⋃

i=1

∂FURi, f(⋃ℓi=1 ∂FSRi) ⊂

ℓ⋃

i=1

∂FSRi.

By the remark following the definition of birectangles, the Ri’s are birectangles

since the system of transversals is contained in FS-leaves and singularities.

We define for each n a family of birectangles {Rn} in the following way: the

birectangles of {Rn} will be the closures of the connected components of the non

empty open sets contained in

n∨

i=−nf iR◦ =

{n⋂

i=−nf i(intRai) | Rai ∈ R

}.

It is easy to see that Rn still satisfies the properties (i), (ii) and (iii) given above.

Moreover, if R ∈ Rn, we haveW(R) ≤ λ−n max{W(Ri) | Ri ∈ R}. In particular, by

Lemma 10.14 of last section, for n sufficiently large, each birectangleR inRn is a good

one.

DR

AFT

13 J

un 2

008


We assert that for n sufficiently large Rn is a Markov partition. All that remains

is to verify properties (4) and (5) of a Markov partition. It is an easy exercise to show

that property (4) is a consequence of property (iii) given above (see Lemma 9.1). Byfix numbering

Lemma 10.15, if n is sufficiently large and R,R′ ∈ Rn, then if x ∈ R, f(FU (x,R))

intersects in at most one point each FS-fiber of R′. Property (5) follows easily from

the combination of this fact and of property (4).

Example of Markov partition on T 2. Let A : T 2 → T 2 be the linear map defined by

A =

(2 1

1 1

).

Here T 2 = R2/Z2; and A acts on R2 preserving Z2, thus A defines a map of T 2. The

translates of the eigenspaces ofA foliate T 2. The mapA on T 2 is Anosov. The foliation

of T 2 corresponding to the eigenvalue 3+√

52 is expanded, the foliation corresponding

to 3−√

52 is contracted.

We draw a fundamental domain with eigenspaces approximately drawn in.

Figure 10.1:

The endpoints of the short stable manifold are on the unstable manifolds after

equivalences have been made. Filling in to maximal rectangles gives us the following

picture.

The hatched line is the extension of the unstable manifold. Identified pieces are

numbered similarly. One rectangle is given by 1,2,3,6 and the other by 4,5. This

DR

AFT

13 J

un 2

008


6

2

36

11

5

3

5

4

Figure 10.2:

partition in two rectangles gives a Markov partition by taking intersections with di-

rect and inverse images.

The construction of the Markov partition of a pseudo-Anosov diffeomorphism

f : M → M , which preserves orientation and fixes the prongs of FS and FU , is the

same as in the example above. We sketch here the argument, hoping that it will aid

the reader to understand the general case.

Since the unstable prongs are dense, we may pick small stable prongs whose

endpoints lie on unstable prongs.

Roughly, the picture is:

small stable prong

We may extend these curves to maximal birectangles leaving the drawn curves

as boundaries. By density of the leaves, every leaf crosses a small stable prong, so

the rectangles obtained this way cover M2. The extension process requires that the

unstable prongs be extended perhaps but the extension remains connected. Thus we

have a partition by birectangles with boundaries the unions of connected segments

lying on stable or unstable prongs. Consequently an unstable leaf entering the inte-

DR

AFT

13 J

un 2

008


rior of a birectangle under f can’t end in the interior, because the stable boundary

has been taken to the stable boundary, etc. . .

f

f

but not

DR

AFT

13 J

un 2

008


The only thing left is to make the partition sufficiently small. To do this, it is sufficient

to take the birectangles obtained by intersections f−nR ∨ · · · ∨ R ∨ · · · fn(R) for n

sufficiently large.

10.6 Pseudo-Anosov diffeomorphisms are Bernoulli

A pseudo-Anosov diffeomorphism f : M → M has a natural invariant probability

measure µ which is given locally by the product of µS restricted to plaques of FUwith µU restricted to plaques of FS . The goal of this section is to sketch the proof of

the following theorem.

Theorem 10.18. The dynamical system (M, f, µ) is isomorphic (in the measure theoretical

sense) to a Bernoulli shift.

Recall that a Bernoulli shift is a shift (Σ(ℓ), σ) together a measure ν which is the

infinite product of some probability measure on {1, . . . , ℓ}. Obviously, ν is invariant

under σ, see [Orn74], [Sin76].

We will have to use the notion and properties of measure theoretic entropy, see [Sin76].

We will also need the following two theorems on subshifts of finite type.

Let A be a k× k matrix and (Σ, σA) be the subshift of finite type obtained from it.

Theorem 10.19 (Parry [Par64]). Suppose that A has all its entries > 0 for some n. Then,

there is a probability measure νA invariant under σA such that the measure

DR

AFT

13 J

un 2

008


theoretic entropy hνA(σA) is equal to the topological entropy h(σA). Moreover, νA is the only

invariant probability measure having this property, and (ΣA, σA, νA) is a mixing Markov

process.

Theorem 10.20 (Friedman-Ornstein [Orn74]). A mixing Markov process is isomorphic to

a Bernoulli shift. In particular, the (ΣA, σA, νA) above is Bernoulli.

Now we begin to prove that (M, f, µ) is Bernoulli. For this, we will use the sub-

shift (ΣA, σA) and the map θ : (ΣA, σA)→ (M, f) obtained from the Markov partition

R = {R1, . . . , Rk}.

Lemma 10.21. There exists n ≥ 1 such that An has entries > 0.

Proof. GivenRi, we can find a periodic point xi ∈ R◦i , call ni its period. Consider the

unstable fiber FU (xi, Ri); we have, for ℓ ≥ 0, f ℓni(FU (xi, Ri)) ⊃ FU (xi, Ri). More-

over the µS-length of f ℓni(FU (xi, Ri)) goes to infinity, since it is λℓniµS(FU (xi, Ri)).

This implies that

f ℓni(FU (xi, Ri)) ∩R◦j 6= ∅ ∀ j = 1, . . . , k,

for ℓ large because the leaves of FU are dense. Now, if n = ℓ ·k∏

i=1

ni with ℓ large

enough, we get fn(R◦i ) ∩ R◦

j 6= ∅ for each pair (i, j). Hence, we obtain that a(n)ij > 0

for each (i, j), where An = (a(n)ij ).

This lemma shows that (ΣA, σA, νA) is Bernoulli by the results quoted above. All

we have to do now is to prove that (M, f, µ) is isomorphic to (ΣA, σA, νA).

Lemma 10.22. The measure theoretic entropy hµ(f) is logλ.

Proof. Since topological entropy is the supremum of measure theoretical entropies,

see [Bow70, Goo71], we have hµ(f) ≤ logλ. Consider now the partitionR◦ = {intRi};its

DR

AFT

13 J

un 2

008


µ-entropy hµ(f,R◦) with respect to f is given by:

hµ(f,R◦) = limn− 1

n

∑a(n)ij λ

−nyixj log(λ−nyixj)

where yi = µU (FS-fiber of Ri) and xj = µS(FU -fiber of Rj). As we saw at the end of

Section 10.4,a(n)ij

λn ≤ ||An||λn is bounded by (

P

yi

min yi). This implies:

limn− 1

n

∑a(n)ij λ

−nyixj log yixj = 0.

We have also: do we want the

interior symbol

directly above

the R?

∑a(n)ij λ

−nyixj =∑

yjxj =∑

µ(R◦j ) = µ(M) = 1.

By putting these facts together, we obtain: hµ(f,R◦) = logλ. Hence, hµ(f) = logλ,

because log λ = hµ(f,R◦) ≤ hµ(f) ≤ h(f) = logλ.

Proof. [Proof of the theorem] Put ∂R =k⋃

i=1

∂Ri, we have µ(∂R) = 0. This implies that

the set Z = M −⋃

i∈Z

f i(∂R) has µ-measure equal to one. We know by Section 10.4

that θ induces a (bicontinuous) bijection of θ−1(Z) onto Z , we can then define a prob-

ability measure ν on ΣA by ν(B) = µ(θ([θ−1(Z) ∩ B]) for each borel set B ⊂ ΣA. It

is easy to see that ν is σA invariant; moreover, θ gives rise to a measure theoretic iso-

morphism between (ΣA, σA, ν) and (M, f, µ). In particular hν(σA) = hµ(f) = logλ.

Since logλ is also the topological entropy of σA we obtain from Parry’s theorem that

ν = νA and that (ΣA, σA, ν) is a mixing Markov process. By the Friedman-Ornstein

theorem, (ΣA, σA, ν) is Bernoulli, hence (M, f, µ) is also Bernoulli.

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008

Expose 11

Thurston’s theory for surfaces

with boundary

replace phi with

varphi in other

chaptersby F. Laudenbach

Let M be a compact connected surface with nonempty boundary, whose Euler char-

acteristic is negative; for simplicity, we will limit ourselves to the case where M is

orientable. Let g be the genus of M and b the number of boundary components. The

Euler characteristic of M is given by

χ(M) = 2− 2g − b;

thus χ(M) < 0 is equivalent to b > 2− 2g.

Such a surface may be cut by 3g − 3 + b curves into 2g − 2 + b pairs of pants. The

excluded surfaces are S2, T 2, D2 and S1 × [0, 1]. The pair of pants is the only surface

with χ < 0 and b ≤ 3 − 3g. In what follows, we will restrict ourselves to the case

b > 3− 3g.

11.1 The space of curves and measured foliations

Here, S denotes the set of isotopy classes (= homotopy classes) of simple curves in M

that are not homotopic to a point or to a boundary component. Also, we consider the set

MF of Whitehead classes of measured foliations,

215

DR

AFT

13 J

un 2

008


which are subject to the following condition on the boundary:

each boundary curve is a cycle of leaves containing at least one singularity.

The relation of Whitehead equivalence is generated by the following relations:

– isotopy, free on the boundary;

– contraction (to a point) of a leaf in the interior joining two singularities, at most

one of which is on the boundary;

– contraction (to a point) of a leaf in the boundary joining two singularities;

– inverse operations of the two preceding ones.

As in the case of surfaces without boundary, the geometric intersection gives risegeometric intersection

to maps

i∗ : S → RS+ and

I∗ : MF → RS+.

Let π be the projection onto the projective space

π : RS+ − {0} → P (RS

+).

We denote by PMF the image of π ◦ I∗.

Theorem 11.1.

1o The maps i∗ and I∗ are injective.

2o PMF is homeomorphic to the sphere S6g−7+2b.

3o The image π ◦ i∗(S) is dense in PMF .

Proof.

The proof is very close to that of the case without boundary (see Exposes 4 and

6); we only give an explanation for the dimension.

Consider inM a system of 3g−3+b simple curvesKi, that partitionM into pairs

of pants such that each Ki belongs to two distinct pairs of pants; such a system does

not exist if M is a one-holed torus (3g− 3+ b = 1); we will revisit this case at the end.

Once we fix a ’normal form’ with respect to this decomposition, a foliation (M, µ) is

characterized (up to equivalence) by triples (mi, si, ti), i = 1, . . . , 3g−3+b, belonging

to the boundary ∂(≤ ∇) of the triangle inequality. We have

mi = I(F , µ; [Ki]).

Because the curves of the boundary have measure zero, knowledge of the mi deter-

mines the foliations in each pair of pants, up to equivalence. Thus, by the theory of

the “arc jaunes”, the pair (si, ti) describes how to glue the foliations in the two

DR

AFT

13 J

un 2

008


pairs of pants adjacent to Ki.

Finally, the set of equivalence classes of measured foliations in normal form with

respect to the given decomposition of M , is in bijection with a punctured positive

cone, on the base S6g−7+2b.

To obtain the theorem, it remains to show that si and ti are determined by I∗(F , µ)

and that the image I∗(MF) is a topological manifold. These two points are proven

as in the case without boundary; to be precise, si and ti are calculated with the aid of

the measures of classes [K ′i], [K ′′

i ] associated with the decomposition (see Expose 6);

it suffices then to remark that these classes are truly elements of S.

In the case 3g − 3 + b = 1 (M is a one-holed torus), we take for K1 and K ′1 two

“generators” of the torus and for K ′′1 the curve obtained from K ′

1 by a positive twist

along K1 (see Figure 11.1). We leave to the reader the exercise of establishing the

formulas that give s1 and t1 as functions of the measures of [K1], [K ′1], and [K ′′

1 ].

K1’’

K1K ′

1

hole

11.2 Teichmuller space and its compactification

We consider the topological space H of Riemannian metrics of curvature −1, for

which each boundary curve is a geodesic of length 1. The group Diff0(M) of diffeomor-

phisms ofM isotopic to the identity acts naturally onH; we define the Teichmuller space1Teichmuller space

of M to be the topological quotient space

T = H/Diff0(M).

1Classically [Har77], one does not fix the length of the curves of the boundary.

DR

AFT

13 J

un 2

008


We parameterize T by fixing a pair of pants decomposition as in the proof of

Theorem 11.1. A Teichmuller structure (i.e., a point of T ) is completely determined

by the lengths mi of the geodesics isotopic to the curves Ki and by the “angles” (real

numbers) θi given by the gluing. We will show, via this parametrization, that T is

homeomorphic to (R∗+ × R)3g−3+b.

Further, for each α ∈ S, we may speak of the length with respect to the Te-

ichmuller structure in consideration. We thus have a map

ℓ∗ : T → RS+.

As in the case without boundary, the “angle” αi is determined by the lengths of the

geodesics of [K ′i] and of [K ′′

i ]. We therefore have the following theorem.

Theorem 11.2. The Teichmuller space being defined as above, the map ℓ∗ is a proper function

which is a homeomorphism onto its image. In particular, ℓ∗(T ) is homeomorphic to R6g−6+2b.

From here on, we will identifyMF and T with their respective images in RS+.

Lemma 11.3. In RS+, the spacesMF and T are disjoint.

Proof. It suffices, for example, to find for each foliation (F , µ) a sequence of αn ∈ Ssuch that I((F , µ);αn) → 0. Let q : M → M be the ramified covering of transverse

orientations of F . Let F = q∗F . Let z ∈ int M \ sing F be a limit point for a leaf L; we

may form a simple curve Cn, with an arc of L and an arc transverse to the measured

foliation of measure ≤ 1n , contained in a “flow box” neighborhood of z. As F is

transversely orientable, Cn can be approximated by a true transversal to F ; we may

suppose in addition that any double points of q(Cn) are isolated. By a modification

around each double point, we construct a curve C′n that is a simple curve in M ,

transverse to F and of measure≤ 1n . We set αn = [C′

n].

It remains to show that C′n is not isotopic to a curve of the boundary. If it is, we

have an annulus equipped with a measured foliation, where one boundary curve

is transverse to the foliation and the other is a cycle; this is forbidden by Poincare

Recurrence (Theorem 5.1).

Theorem 11.4. The projection π injects T into P (RS+) by a homeomorphism onto its image,

which is disjoint from PMF . Endowed with the induced topology, π(T ) ∪ PMF is

DR

AFT

13 J

un 2

008


a manifold with boundary T , and is homeomorphic to a ball of dimension 6g − 6 + 2b. The

group π0(Diff(M)) acts continuously on T .

For the proof, we follow the same procedure as in the case without boundary,

(Expose 8), and not the one suggested by the order of the sentences in the statement

of the theorem.

11.3 Preparation for the classification of diffeomorphisms

11.3.1

It is good to emphasize that we are dealing with a classification of diffeomorphisms

up to isotopy, where the isotopy on the boundary is free.

Let ϕ ∈ Diff(M) and let [ϕ] be its isotopy class. By the Brouwer Fixed Point Theo-

rem, there exists a point x ∈ T such that

[ϕ] · x = x.

If x ∈ T , then ϕ is isotopic to an hyperbolic isometry of x; in this case, [ϕ] is of

finite order (Expose 9).

If x ∈ PMF , there is a foliation (F , µ) and a λ > 0, such that

ϕ(F , µ) ∼ (F , λµ),

where the equivalence is in the sense of Whitehead; from this point on, everything

depends on (F , µ) and λ.

Let Σ be a complex consisting of the singularities and the leaves joining two sin-

gularities (possibly joining some singularity to itself); Σ contains ∂M since each com-

ponent of the boundary contains a singularity. We denote by U(F) the complement

of a regular neighborhood of Σ in M . We see that, up to isotopy, U(F) only depends

on the Whitehead class of F .

We define βU(F) as the union of the boundary components of U(F) that repre-

sent the elements of S. We distinguish the following cases:

1. βU(F) 6= ∅ (reducible case),

2. βU(F) = ∅ and λ = 1 (periodic case),

3. βU(F) = ∅ and λ 6= 1 (pseudo-Anosov case).

DR

AFT

13 J

un 2

008


11.3.2 Reducible diffeomorphism

Definition. We say that ϕ is reducible if there exist mutually disjoint simple curvesreducible

γ1, . . . , γn, each representing a distinct element of S, such that ϕ(γ1 ∪ · · · ∪ γn) =

γ1 ∪ · · · ∪ γn.

Lemma 11.5. If ϕ(F , µ) ∼ (F , λµ) and if βU(F) is not empty, then ϕ is isotopic to a

reducible diffeomorphism.

Proof. Uo to changing ϕ by an isotopy, we may suppose that ϕ(U(F)) = U(F). Let

γ1 be a component of βU(F), γ2 = ϕ(γ1), etc.; we stop at γn = ϕn−1(γ1) if it is the

first iterate such that ϕ(γn) is isotopic to γ1. Since ϕ(γ1) and γ1 bound an annulus, it

is not difficult to produce an isotopy of ϕ for which γ1 ∪ · · · ∪ γn is invariant.

If we cut M along γ1, . . . , γn, we obtain a “simpler” surface M on which ϕ in-

duces a diffeomorphism. A small difficulty for what follows arises because M is not

in general connected; we will revisit this in Section 11.4. Observe that each compo-

nent of M is either a pair of pants, or satisfies b > 3g − 3; indeed, two curves among

the γi are not isotopic, and no γi is isotopic to a curve of the boundary. Note that the

number of possible successive reductions has an upper bound only depending on

M ; when all of the pieces are pairs of pants, no further reduction is possible.

11.3.3 Arational foliations

If βU(F) is empty, we say that F is arational. There is then a distinguished repre-arational

sentative in the class of F ; the singularities of the interior are without connections

(neither between themselves nor with those of the singularities on the boundary)

and the singularities on the boundary are simple (a single separatrix enters into the

interior). This representative is unique up to isotopy. In what follows, we suppose

that F is this canonical representative. The equivalence ϕ(F , µ) ∼ (F , λµ) therefore

gives rise to an equality:

ϕ(F , µ) = (F , λµ)

under the condition that we may have to modify ϕ by a suitable isotopy.

To each system τ of arcs transverse to F is associated a system of F -rectangles,

where the union is a subset N of M , which has as a frontier a union of cycles of

leaves. Since βU(F) is empty, the frontier of N is in

DR

AFT

13 J

un 2

008


∂M , thus N = M . From this, we deduce that each half-leaf that does not meet a

singularity is everywhere dense.

11.3.4 Case ϕ(F , µ) = (F , µ).

As in the case of closed surfaces, we consider a “good” system of transverse arcs τ

(see Expose 9). Up to changing ϕ by an isotopy that preserves F , we reduce to the

case where ϕ(τ) = τ and thus where ϕ preserves the system of rectangles; from this

we deduce that ϕ is isotopic to a periodic diffeomorphism.

11.3.5 Case ϕ(F , µ) = (F , λµ), λ > 1.

With the aim of constructing a second invariant foliation, one has to modify the con-

struction of a “good” system of transverse arcs given in Lemma 9.9. In each sector of

an interior singularity of M , we take a small arc transverse to F ; however, one does

not put any in the sectors adjacent to the boundary. In addition, for each smooth leaf

of the boundary, we choose a point that we make an endpoint of a small transversal

that enters the interior (Figure 11.3.5).

τ

ττ

τ

∂M

rectangle

If τ is such a system of arcs, then, by a suitable isotopy of ϕ along the leaves of F ,

we obtain ϕ(τ) ⊂ τ . From this, the techniques of Lemma 9.9 and Lemma 9.11 apply

to construct a Markov pre-partition {Ri}.Let aij be the number of components of ϕ(intRi)∩intRj . Let xi be the µ-measure

of ∂0τRi (or of ∂1

τRi). We have

λxj =∑

i

xiaij .

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008


In other words, ifA is the matrix (aij), the column vector (xi) is an eigenvector of

the transpose matrix At, with eigenvalue λ. By the same proof as in the case without

boundary, we prove that A also has an eigenvector (yi), whose coordinates are all

strictly positive, with an eigenvalue of 1/ξ > 0.

yi = ξ∑

aijyj .

Observe that the geometric proof (in the case without boundary) rests on the fact

that, for each i, ∪nϕn(∂0FRi) is dense. This is again true here because ∂0

FRi may not

be entirely contained in the boundary of M ; it necessarily contains an arc of a leaf of

the interior.

Construction of the foliation (F ′, µ′). As in the case without boundary, we start

by fixing the µ′-measure of the arcs ∂ǫFRi, ǫ ∈ {0, 1}. If such an arc is in ∂M, we

assign it measure 0, because one wants ∂M to also be a union of cycles of leaves for

F ′ (Figure 11.3.5).

∂0FRi

∂τRi

∂1FRi

Now, we draw in each rectangleRi a measured foliation (F ′, µ′) that is transverse

to F and that respects the assigned measures; this condition guarantees that we can

glue the pieces together. We observe that SingF ′ ∩ intM = SingF ∩ intM , while

the singularities of F on ∂M become regular points of F ′; we have SingF ′ ∩ ∂M =

τ ∩ ∂M .

Now that we have a measure µ′ on the leaves of F , we construct a pseudo-

Anosov “diffeomorphism” ϕ′ that respects the two foliations, dilating the leaves of

F by 1/ξ and contracting those of F ′ by 1/λ. We have thus proved that ξ = 1/λ. Note

that ϕ′ is the identity on the boundary.

DR

AFT

13 J

un 2

008


11.3.6 Pseudo-Anosov diffeomorphism

We say that ϕ is a pseudo-Anosov “diffeomorphism” if there exist two invariantpseudo-Anosov

measured foliations, (Fs, µs) and (Fu, µu), and a λ > 1 with the following properties:

1. ϕ(Fs, µs) = (Fs, 1λµ

s),

2. ϕ(Fu, µu) = (Fu, λµu),

3. Fs and Fu are transverse at each point of the interior,

4. Each component of ∂M is a cycle of leaves of Fs and of Fu and contains singu-

larities of these two foliations, and ϕ is the identity on the boundary.

N.B. ϕ is not C1 along the boundary.

The properties of pseudo-Anosov diffeomorphisms, indicated in Section 9.6, are

still true. Only Proposition 9.21 requires a modification: it only applies to isotopy

classes of curves not homotopic to a component of the boundary; moreover, the met-

ric√

(dµs)2 + (dµu)2 is singular along the whole boundary.

11.3.7 Example: a disk with 3 holes

Let A be an Anosov matrix acting on T 2. Let σ be the involution (x, y) 7→ (−x,−y);it has four fixed points; we may regard T 2 → T 2/σ as a ramified covering. As can be

seen by calculating the Euler characteristic, the base is a 2-sphere.

The transformation A leaves invariant two linear foliations of irrational slope

which therefore pass to the quotient, inducing on S2 two measured foliations (Fs, µs)and (Fu, µu), with singularities at the four ramifications points (on T 2, the transverse

measures are given by closed 1-forms with constant coefficients that define the respec-

tive foliations). Since the degree of ramification is 2, the singularities are of the type

indicated in Figure 11.3.7.

DR

AFT

13 J

un 2

008


FuFs

Since A commutes with σ, A induces on S2 a homeomorphism ϕ that leaves in-

variant Fu and Fs and that transforms the measures in the same way as A on T 2. To

obtain the disk with 3 holes, equipped with a pseudo-Anosov diffeomorphism, we

blow up the singularities are as shown in Figure 11.1.

Figure 11.1:

11.4 Thurston’s classification; Nielsen’s theorem

The arguments of the previous subsection lead, at least in the orientable case, to a

proof of the following theorem.

Theorem 11.6. For any diffeomorphism ϕ of a compact, connected surface satisfying b >

2− 2g, ϕ is isotopic to ϕ′ having one of the following three properties:

(1) ϕ′ is of finite order; it is thus an isometry of a hyperbolic structure;

(2) ϕ′ is pseudo-Anosov;

(3) ϕ′ is reducible.

DR

AFT

13 J

un 2

008


To pursue the analysis in the reducible case, it is necessary to indicate how to

work with a disconnected surface. We only need to consider the case where ϕ acts

transitively on π0(M). Therefore, let M = M1 ∪ · · · ∪Mn where the Mi are the con-

nected components of M ; ϕ(Mi) = Mi+1 for i = 1, . . . , n− 1 and ϕ(Mn) = M1.

We will say that ϕ is pseudo-Anosov if ϕn|M1 is pseudo-Anosov; then, by con-

jugation, we see that ϕn|Mi is pseudo-Anosov for all i. If one knows that ϕn|M1 is

isotopic to a pseudo-Anosov (resp. a diffeomorphism of finite order), then ϕ is iso-

topic to such a diffeomorphism (on each Mi, take the foliation associated to φn); it

suffices to read the isotopy of ϕn|M1 as an isotopy of

DR

AFT

13 J

un 2

008


ϕ : Mn →M1. In this way, one obtains the final result below, in which one can avoid

restrictions on the genus or the Euler characteristic, since the cases ofM = S2, T 2, D2,

pair of pants, Mobius band, and Klein bottle are known.

Theorem 11.7. Let ϕ be a diffeomorphism of a compact surfaceM . Then, there exist surfaces

(perhaps disconnected) M1, . . . ,Mk having the following properties:

1o M = M1 ∪ · · · ∪Mk;

2o For i 6= j, Mi ∩Mj is the union of closed curves of their common boundary compo-

nents; we denote by C1, . . . , Cr these “cutting” curves;

3o For i 6= j, Ci is not isotopic to Cj ;

4o ϕ is isotopic to a diffeomorphism ϕ′ that that for each i, ϕ′(Mi) = Mi;

5o ϕ′|Mi is a isotopic in Diff(Mi) to a periodic diffeomorphism or a pseudo-Anosov “diff-

eomorphism”.

We must note that, paradoxically, if ϕ is a Dehn twist (twist along a curve C), this

classification drops mention of ϕ entirely. That is, if one cuts M along C and allows

a free isotopy on the boundary, one arrives at the identity.

Theorem 11.8 (Nielsen [Nie44a]). Let ϕ be a diffeomorphism of a compact surface repre-

senting an element of order n in π0(Diff(M)). Thenϕ is isotopic to a periodic diffeomorphism,

of order n.

Proof. We limit ourselves to the nontrivial case b > 3 − 3g (notation from the intro-

duction). Let M1 ∪ · · · ∪Mk be a decomposition of the surface as in the preceding

theorem; for each cutting curve Ci, one considers a small tubular neighbourhood

Ni = Ci × [−1, 1]; we denote by M ′j the part of Mj that remains after removing the

open collars. After the first isotopy, we have ϕ(M ′j) = M ′

j for each j and ϕ(Ci × {t})is of the form C′

i × {t′}; furthermore, ϕ|M ′j

is periodic or pseudo-Anosov.

get rid of end of

proof symbolLemma 11.9. The diffeomorphism ϕn preserves each Ci with its orientation and normal

orientation.

Proof. If i 6= j, Ci is not isotopic to Cj . Also, Ci cannot be

DR

AFT

13 J

un 2

008


isotoped to its opposite except on the Klein bottle (excluded by hypothesis). Finally,

an exchange of sides induces a nontrivial morphism on H1(M,Z).

Lemma 11.10. The isotopy of ϕn to the identity can be chosen through diffeomorphisms that

preserve C1 ∪ · · · ∪ Cr.

Proof. An isotopy of ϕn with the identity induces a loop, based at C1, in the space of

simple curves on M . Since M 6= T 2, or the Klein bottle, such a loop is homotopic to a

point. By lifting this homotopy to Diff(M), we find an isotopy of ϕn to the identity inREFERENCE:

GRAMAIN Diff(M,C1). We proceed in the same fashion for the other curves.

Consequently, for each j, ϕn|M ′j

is isotopic to the identity in Diff(M′j). Thus ϕ|M ′

j

can not be pseudo-Anosov; hence ϕ|M ′j

is periodic and, since it is an isometry for a

particular hyperbolic metric2 (see Expose 9), ϕn|M ′j

is the identity. Thus ϕn|Ni is a

certain iterate θqi of a Dehn twist θ along the curve Ci.

Lemma 11.11. The integer qi is zero for each i.

Proof. There exists a class β ∈ S such that i(β, [Ci]) 6= 0 and i(β, [Cj ]) = 0 for j 6= i; if

qi is not zero, then by Appendix A we have i(ϕn(β), β) 6= 0, which is forbidden since

ϕn is isotopic to the identity.

Suppose that ϕ(Ci) = Ci; the lemma thus signifies that ϕ makes the two bound-

aries of the collar Ni turn in the same direction. More precisely, in suitable coordi-

nates, ϕ(x,±1) = (x + 1n ,±1), where x ∈ R/Z, and ϕ({0} × [−1, 1]) is isotopic to

{ 1n} × [−1,+1], rel boundary. From here, it is easy to make an isotopy of ϕ|Ni , trivial

along ∂Ni, to a periodic diffeomorphism of period n.

If ϕ(Ci) 6= Ci, we proceed in the same fashion with the orbit of Ci.

Remark. The proof of Nielsen rests on the fact that ϕ lifts in the universal cover to a

ϕ that extends to the boundary of the Poincare disk; ϕ|∂D2

2The hyperbolic metric obtained by this argument may not give boundary curves of length 1; hence we do

not say here that ϕ|M′j

admits a fixed point in T (M ′

j). Question: is there a Teichmuller metric invariant

for ϕ|M′j

?

DR

AFT

13 J

un 2

008


only depends on the homotopy class of ϕ.

There is still another proof of the theorem In fact, as Fenchel announced (see [Fen50]

or the book of Fenchel and Nielsen [FN]), we may deduce from the Smith Fixed Point

Theorem [Smi34] that, if G is a finite solvable subgroup of π0(Diff(M)), then G admits a

fixed point in (open) Teichmuller space; from this we may deduce that G lifts to a sub-

group of Diff(M).

The argument, briefly, is as follows. Let F → G → Zp be an extension where p

is a prime number; suppose that the result holds for F . Let TF be the set of fixed

points of F in T . Let M ′ = M/F , for a chosen action of F on M , and let X be

the set of ramification points. Let T (M ′, X) be the set of conformal structures of M ′

modulo the identity component of Diff(M′,X); we show that this is a cell [using the

theorem of Earle and Eells [EE69] that the action of Diff0 on the metrics of curvature

−1 gives the structure of a principal fibration and the fact that Diff0(M′)/Diff0(M′,X)

is contractible; for example, if X = 1 point, this last quotient is homeomorphic to the

universal cover M ].

Furthermore, we show that TF is homeomorphic to T (M ′, X); thus TF is also a

cell. Finally, as F is invariant in G, G acts on TF via the quotient Zp. By Smith, there

is a fixed point.3

11.5 The spectral theorem

For a Riemannian metric g and a simple curve c, denote the length of c by Lg(c), and

let [c] denote its isotopy class, we set:

ℓg([c]) = inf{Lg(c′) | c′ is isotopic to c}.

Theorem 11.12. For each diffeomorphism ϕ of a compact surface M , there exists a finite

sequence λ1, . . . , λk ≥ 1 such that, for each α ∈ S and for any Riemannian metric g,

the sequence n√ℓg(ϕn(α)) converges to a limit that is independent of g and that belongs to

{λ1, . . . , λk}. The numbers λ1, . . . , λk are algebraic integers whose degrees admit an upper

bound that only depends on the Euler characteristic of M .

3I wish to thank Alexis Marin who communicated to me the essential elements of this remark.

DR

AFT

13 J

un 2

008


Proof. By Theorem 11.7, we may suppose M = M1 ∪ · · · ∪Mk, ϕ(Mi) = Mi; ϕ|Mi

is isotopic in Diff(Mi) to a diffeomorphism ϕi that is pseudo-Anosov with dilatation

factor λi for i = 1, . . . ,m, and that is periodic for i = m+ 1, . . . , k. For i > m, we set

λi = 1. We will prove that this “spectrum” satisfies the statement of the theorem.

Since all Riemannian metrics are equivalent, we may restrict ourselves to the case

where g is a hyperbolic metric that admits the ∂Mi as geodesics. Then, the geodesic c

in the class α intersects ∂Mi minimally. We cut c into arcs c1, . . . , cr corresponding to

the different segments of c in the Mi; cs ⊂ Mi(s). By Section 3.3 of Expose 3, cs is an

essential arc in Mi(s) (nontrivial in π1(Mi(s), ∂Mi(s))); hence, ϕ(cs) is also an essential

arc in Mi(s). Furthermore, the geodesic of the class ϕn(α) is c(n)1 ∪ · · · ∪ c(n)

r , where

c(n)s is isotopic to ϕn(cs) by an ambient isotopy of Diff(Mi(s)).

Let us say that λi(1) ≥ λi(s) for s = 1, . . . r. We will prove that n√ℓg(ϕn(α)) →

λi(1). For fixed α, all of the classes ϕn(α) traverse the same Mi; thus, it suffices to

prove the statement for a subsequence (ϕt)n; this allows us to reduce to the case

where ϕ is the identity on the ∂Mi and where λi(s) = 1 implies ϕi(s) = Id, as we shall

assume in what follows.

Consider the geodesic arc d(n)s (resp. h

(n)s ) that is homotopic to ϕn(cs) with end-

points fixed (resp. free). Let β(n)s (resp. δ

(n)s ) be the shortest path joining the origin

(resp. the endpoints) of d(n)s to that of h

(n)s , such that d

(n)s is homotopic, with end-

points fixed, to β(n)s ∗ h(n)

s ∗ [δ(n)s ]−1. Let us provisionally accept the following result.

Lemma 11.13. The growth of Lg(β(n)s ) and of Lg(δ

(n)s ) is subexponential; that is to say,

lim sup 1n log(Lg(β

(n)s )) = lim sup 1

n log(Lg(δ(n)s )) = 0.

If λi(s) = 1, it is clear that Lg(h(n)s ) is bounded. If λi(s) > 1, then

n

√Lg(h

(n)s ) →

λi(s); indeed ϕi(s) is pseudo-Anosov with dilatation coefficient λi(s) and, in this case,

the result is given by Proposition 9.21 (with one slight difference that here the man-

ifold has boundary and that it acts on the free isotopy classes of paths going from

boundary to boundary, but the proof is the same). By the lemma, we haven

√Lg(d

(n)s )→

λi(s). In addition, we have the following inequalities:

∑

s

Lg(h(n)s ) ≤

∑

s

Lg(c(n)s ) = ℓg(ϕ

n(α)) ≤∑

s

Lg(d(n)s ).

Considering the growth of each term, we find that n√ℓg(ϕn(α)) tends to λi(1).

DR

AFT

13 J

un 2

008


Proof. [Proof of the lemma] We know, by a theorem of Lickorish, that ϕ|Mi(s)is iso- fix beginproof

topic rel boundary to a composition of Dehn twists along simple curves in Mi(s) (see

Expose 15). We consider therefore the situation of a surface with boundary N , en-

dowed with a hyperbolic metric g, and of a twist ψ along a geodesic α in N . Let c be

the minimizing geodesic for a nontrivial class of π1(N, ∂N); let h be the minimizing

geodesic in the class of ψ(c). In the homotopy of ψ(c) to h, each endpoint of the arc

shifts along the boundary: we have on ∂N geodesic arcs of ∂N , β and δ, such that

ψ(c) is homotopic to β ∗ h ∗ δ−1, with endpoints fixed. Say that each component of ∂N

has length equal to 1; then the lemma is a consequence of the following claim.

Claim: Lg(β), Lg(δ) ≤ 1.

Note that if α is isotopic to a component of the boundary, there is nothing to show

since h = c and the displacement of each endpoint of ψ(c) in the course of its isotopy

to c is exactly one turn.

In general, the intersection of h with c is minimal in the free homotopy class of

h. Let c′ be an arc parallel to c; if card(ψ(c′) ∩ c) = card(h ∩ c), then ψ(c′) and h are

isotopic by an isotopy that leaves c invariant (Proposition 3.13 in Expose 3); in this

case, the displacement of the endpoints during this isotopy is less than one turn.

By the appendix to Expose 4, card(ψ(c′) ∩ c) does not decrease as long as one num appendix

leaves the endpoints of ψ(c′) fixed. On the other hand, by shifting the origin of ψ(c′)

on top of that of c, we possibly reduce card(ψ(c′) ∩ c); we say that we drive a point

of intersection to the boundary. To shift the origin of ψ(c′) by more than one turn,

one must have an immersion of a triangle in N , as indicated in Figure 11.5, in the

domain of the immersion. From this, we deduce that α is isotopic to a component of

the boundary, which we have excluded at the beginning. Since the shift of the origin

of c to that of c′ is arbitrarily small, we finally have that the shift of the origin of ψ(c)

to h is less than one turn.

origin of c

ψ(c′)

∂N

DR

AFT

13 J

un 2

008


dN

origin

pc’

Figure 11.2:

DR

AFT

13 J

un 2

008

Expose 12

Uniqueness theorems for

pseudo-Anosov

diffeomorphisms

by A. Fathi and V. Poenaru

12.1 Statement of results

In what follows, M is a compact orientable surface without boundary of genus g > 1.

We are given a pseudo-Anosov diffeomorphism ϕ : M → M ; there exist therefore

two transverse measured foliations (Fs, µs) and (Fu, µu) and a number λ > 1, such

that ϕ(Fs, µs) = (Fs, 1λµ

s) and ϕ(Fu, µu) = (Fu, λµu) (i.e., ϕ contracts distances

between the leaves of Fu by a factor of 1λ ).

Theorem 12.1 (unique ergodicity). The stable and unstable foliations of a pseudo-Anosov

diffeomorphism are uniquely ergodic.

We recall what unique ergodicity means. First of all, an Fs-invariant “measure”

µ is given on each transversal T of (the nonsingular part of) Fs, by a

233

DR

AFT

13 J

un 2

008


Borel measure µT that is finite on each compact set, and such that these measures

are invariant under (the germs of) the holonomy of Fs. The foliation Fs is uniquelyuniquely ergodic

ergodic if there exists a single Fs-invariant measure up to multiplication by a scalar,

that is:

1. There exists a measure µ invariant under Fs.

2. If ν is another measure invariant under Fs, there exists a scalar λ ∈ R such that

νT = λµT for every transversal T .

Theorem 12.1 is a particular case of a result of Bowen and Marcus [BM77].

We recall that a pseudo-Anosov diffeomorphism gives a natural invariant posi-

tive measure, determined up to a constant > 0; it is given locally by the product of

µs and µu. Up to changing µs (or µu) and multiplying by a constant, we can suppose

that µs ⊗ µu is a probability measure, i.e., µs ⊗ µu(M) = 1.

Theorem 12.2. Let ϕ be a pseudo-Anosov diffeomorphism, and suppose that µs⊗µu(M) =

1. If α, β ∈ S, we have

limn→∞

i(ϕnα, β)

λn= I(Fs, µs;α)I(Fu, µu;β).

Corollary 12.3. If α ∈ S, and if [α], [Fs, µs], and [Fu, µu] are the images of α, (Fs, µs),and (Fu, µu) in PMF , we have

limn→∞

[ϕnα] = [Fu, µu] and

limn→∞

[ϕ−nα] = [Fs, µs].

In fact, the result of Thurston is stronger: if [F , µ] ∈ PMF and if [F , µ] 6= [Fs, µs],then lim

n→∞ϕn[F , µ] = [Fu, µu]. It is possible that our proof of Theorem 12.2 can also

recover this stronger result, by doing uniform estimates of convergence on compact

sets of PMF − {[Fs, µs]}.

Corollary 12.4. The only fixed points of the action ofϕ on the compactification of Teichmuller

space ¯T (M) are [Fu, µu] and [Fs, µs].

Theorem 12.5 (Uniqueness of pseudo-Anosovs). Two homotopic pseudo-Anosov diffeo-

morphisms are conjugate by a diffeomorphism isotopic to the identity.

DR

AFT

13 J

un 2

008


12.2 Reminder

12.2.1 The theorem of Perron–Frobenius

See [Gan98, chap. 13], or [Kar66, appendix].

Perron–Frobenius Theorem. Let A = (aij) be an n × n matrix with nonnegative

entries. We denote by a(k)ij the coefficients of Ak, the kth power of A. If there exists

ℓ ≥ 1 such that all the coefficients a(ℓ)ij ofAℓ are strictly positive, we have the following

properties.

1) The matrix A admits an eigenvalue λ > 0 that is strictly greater than the absolute

value of each other eigenvalue.

2) There exists an x = (x1, . . . , xn) ∈ Rn, with each xi positive, that is an eigenvector

with eigenvalue λ for A:

λxi =

n∑

j=1

aijxj , i = 1, . . . , n.

3) The eigenspace of A associated to the eigenvalue λ is of dimension 1.

4) If y = (y1, . . . , yn) ∈ Rn is an eigenvector with eigenvalue λ for At:

λyj =

n∑

i=1

yiaij , j = 1, . . . , n,

then, all the yj are positive. If y is normalized by

〈x, y〉 =

n∑

i=1

xiyi = 1,

we have

limk→∞

Ak

λk= 〈 , y〉x;

that is,

limk→∞

a(k)ij

λk= xiyj.

12.2.2 Markov partition (see Exposes 9 and 10)

We are going to consider a Markov partitionR = {R1, . . . , RN} for ϕ. We set:

xi = µs(Fu-fiber of Ri)

yi = µu(Fs-fiber of Ri).

DR

AFT

13 J

un 2

008


xi

yi

Fs

Fu

Figure 12.1:

We made the hypothesis that µs ⊗ µu(M) = 1; this is equivalent toN∑

i=1

xiyi = 1.

Let A = (aij) be the incidence matrix of R for ϕ, so aij = (number of times that

ϕ(IntRi) traverses IntRj). We have

(∗) λxi =

N∑

j=1

aijxj , and

(∗∗) λyi =

N∑

j=1

yjaji .

We saw in Expose 10 (end of Section 10.4) that λ is in fact the greatest eigenvalue

for A. Moreover, still in Expose 10 (Lemma 10.14), we showed that there exists an

integer ℓ > 0 such that the matrix Aℓ has only positive entries. We can thus apply the

theorem of Perron–Frobenius, which gives us the following.

Lemma 12.6. With the notations introduced above:

limk→∞

a(k)ij

λk= xiyj.

12.3 Proof of Theorem 12.1: unique ergodicity

Let ν be an invariant measure for Fu. Since Fu does not have any closed leaves in

M −Sing Fu, the measure ν does not have any atoms. For eachRi, we choose an Fs-fiber of Ri that passes through a point pi ∈ Ri; denote this fiber by Fi (Figure 12.3).

DR

AFT

13 J

un 2

008


Fi

Fs

Fu

Figure 12.2:

Lemma 12.7. Let ν be an invariant measure for Fu. There exists a constant C such that

ν(Fi) = Cµu(Fi), i = 1, . . . , N .

Proof. We can suppose ν ≥ 0. Let i ∈ {1, . . . , n} be fixed. For any k > 0, we have

Fi =

N⋃

j=1

[ϕk(IntRj) ∩ Fi]

∪ {a finite number of points}.

Now, since ν does not have atoms and since the IntRj are pairwise disjoint, we have

ν(Fi) =N∑

j=1

ν[ϕk(IntRj) ∩ Fi].

In addition, by the properties of Markov partitions, ϕk(IntRj)∩Fi is a disjoint union

of a certain number of intervals that, outside of their endpoints, all come from holon-

omy of ϕk(Fj); moreover, the number of these intervals is equal to the number of

times that ϕk(IntRj) traverses IntRi, that is to say a(k)ji , and so

(∗) ν(Fi) =

N∑

j=1

a(k)ji ν[ϕ

k(Fj)].

We obtain in particular that ν(Fi) ≥ a(k)ji ν[ϕ

k(Fj)]. Sincea(k)ji

λk tends to the nonzero

finite limit xjyi as k tends to infinity, it follows that λkν[ϕk(Fj)] stays bounded as k

tends to infinity, and in particular

limk→∞

(a(k)ji

λk− xjyi)λkν[ϕk(Fj)] = 0.

Combining this with the equality (∗), we obtain

ν(Fi) = limk→∞

N∑

j=1

xjyiλkν[ϕk(Fj)] = yi lim

k→∞

N∑

j=1

λkxjν[ϕk(Fj)]

.

DR

AFT

13 J

un 2

008


ϕk(Rj)

Figure 12.3:

This completes the proof of the lemma, since yi = µu(Fi), and limk→∞

N∑

j=1

λkxjν[ϕk(Fj)]

is a constant independent of i.

DR

AFT

13 J

un 2

008


Proof. [Proof of Theorem 12.1] fix beginproof

begin proof not

there. can we

remove the bpf

and still have

the box at the

end?

For each m ≥ 0, we consider the Markov partition {Γkm,i,j} given by the closure

of the connected components of ϕm(o

Ri)∩o

Rj . Lemma 12.7 gives that, for fixed m,

there exists a constant Cm such that

∀ k, i, j, ν(Fj ∩ Γkm,i,j) = Cmµu(Fj ∩ Γkm,i,j).

If we fix j in the above equalities, and if we sum over k and i, we obtain: ν(Fj) =

Cmµu(Fj); thus Cm is independent of m. We have thus shown the existence of a

constant C such that

(∗∗) ∀m, k, i, j, ν(Fj ∩ Γkm,i,j) = Cµu(Fj ∩ Γkm,i,j).

For fixed m, the Fj ∩ Γkm,i,j give a covering of Fj by intervals that only intersect at

their endpoints; moreover each Fj ∩ Γkm,i,j is included in one Fj ∩ Γk′

m,i′,j and the

diameter of Fj ∩Γkm,i,j tends to zero as m tends to infinity. From these properties and

the fact that ν and µu have no atomic masses, the equalities (∗∗) imply

ν|Fj = Cµu|Fj , j = 1, . . . , N.

It follows that ν = Cµu since each leaf intersects each Fj .

12.4 Proof of Theorem 12.2 and its corollaries

We begin with some generalities. We consider an orientable surfaceN without bound-

ary (compact or not) and F a measured foliation on N . We will say that an immersed

(closed) curve is quasitransverse to F if it is an immersion S1 j→ N with the follow- quasitransverse

ing properties:

(i) It is a limit of embeddings;

(ii) It only has a finite number of double points and moreover these double points

are points of singF ;

(iii) It is quasitransverse to F (cf. Section 5.1).

DR

AFT

13 J

un 2

008


Figure 12.4:

Proposition 12.8. Let α be a path that is quasitransverse to F , whose start point is equal to

its endpoint and that does not have any other double points. Suppose that it leaves and arrives

transversely to F . Then the closed curve defined by α is not null homotopic.

Proof. We call x0 the origin (= the endpoint) of α. We consider the case where x0 is a

regular point of F . The situation for a neighborhood of α is one of the two indicated

in Figure 12.5.

Figure 12.5:

In the first case, we have a quasitransverse curve which, by Proposition 5.3 in

Expose 5, cannot be null homotopic. In the second case, we construct a curve homo-

topic to α with a piece of α and a small piece of a leaf; this curve can never be null

homotopic by Proposition 5.3.

Considering the case x0 ∈ Sing F , we have three possible configurations (Fig-

ure 12.6).

The first case gives us an embedded quasitransverse curve. In the second and third

DR

AFT

13 J

un 2

008


Figure 12.6:

cases, we make the modifications shown in Figure 12.7.

Figure 12.7:

We obtain, again by Proposition 5.3, that α is not null homotopic.

DR

AFT

13 J

un 2

008


Corollary 12.9. Suppose moreover thatN is simply connected. Then every immersion R→N that is quasitransverse to F is simple.

Proof. If this immersion has double points, we can find a path quasitransverse to Fas in the hypothesis of the preceding lemma and that is null homotopic since N is

simply connected.

Proposition 12.10. Suppose that N ∼= S1 × R, and let α be an immersed curve that is

quasitransverse to F and homotopic to the core of the cylinder S1 × {0}. Then α is a simple

curve.

Proof. Let R2 = Np−→ N ∼= S1 × R be the universal covering of N . We think of α as

a map Rα→ N that is Z-periodic. Let α : R→ N be a lift of α. Since α is homotopic to

the core of the cylinder, we can see that p−1(α(0)) = {α(n)|n ∈ Z}, and we see that

p−1(α) = α(R). By Corollary 12.9, α(R) = p−1(α) is simple, from which it follows

that α is simple, since p−1(α)p−→ α is a covering.

In what follows, we consider a pseudo-Anosov diffeomorphism; we denote by

(Fs, µs) and (Fu, µu) its invariant foliations and by λ > 1 its dilatation coefficient.

Lemma 12.11. Let γ be an embedded curve in M that is not null homotopic. We can find an

immersed curve γ′ (resp. γ′′) quasitransverse to Fs (resp. Fu), that is homotopic to γ.

Proof. Recall thatFs does not have any connections between singularities. By Propo-

sition 5.7, we can find a foliation Fs1 equivalent to Fs and an embedded curve γ1

transverse to Fs1 and isotopic to γ. When we recover Fs by blowing down the con-

nections between the singularities, we transform the curve γ1 into the desired curve

γ′.

We endow M with the metric ds2 = (dµs)2 + (dµu)2, which is flat outside of

the singularities. Below, when we talk about angles being small, this will only make

sense away from the singularities. We remark that for this metric, the foliations Fsand Fu are orthogonal at all (regular) points.

Lemma 12.12. Let α be an immersed curve quasitransverse to Fs (resp. Fu). The angle of

ϕnα with Fu (resp. of ϕ−nα with Fs) tends to zero as n tends to infinity.

DR

AFT

13 J

un 2

008


The proof of this lemma is left to the reader.

Proposition 12.13. We consider in M , the universal cover of M , the two induced foliations

(Fs, µs), (Fu, µu) and the flat metric ds2 = (dµs)2 + (dµu)2. Let γ be a simple arc that

is quasitransverse to Fu and whose angle with Fs is < π4 , and let δ be a simple arc that is

quasitransverse to Fs and whose angle with Fu is < π4 . Then γ ∪ δ cannot be a simple closed

curve.

Proof. We suppose that γ ∪ δ is a simple closed curve; as M ∼= R2, it bounds a disk

∆. If δ passes through a singularity s0, then a local isotopy makes δ coincide with

arcs of the separatrices of Fu in a neighborhood of s0; we can perform this operation

while preserving the conditions on angles and keeping γ ∪ δ embedded. Now, by

the angle condition on δ, the field of tangent vectors to Fu along δ can be turned

without ambiguity until it becomes tangent to δ; the angle condition on γ allows us

to extend this field to a field that is quasitransverse to γ, and that coincides with Fuin a neighborhood of the singularities and outside of a neighborhood of δ. The new

foliation only has permissible singularities, which gives us a contradiction with the

Euler–Poincare formula.

Corollary 12.14. Let α and β be two immersed curves that are quasitransverse to Fs and

Fu, respectively, and such that the angle of α with Fu (resp. β with Fs) is < π4 . Two lifts α

and β in M intersect in at most one point.

Proof. By Corollary 12.9, the immersions α and β are embeddings. If card(α∩ β) ≥ 2,

we can find a disk ∆ with ∂∆ = γ ∪ δ where γ ⊂ β and δ ⊂ α, which is absurd by the

preceding proposition.

Let α and β be two simple curves inM that are not null homotopic. We denote by

α′ (resp. β′) an immersion quasitransverse toFs (resp.Fu) and homotopic to α (resp.

β). We denote by P (α′) (resp. P (β′)) the number of times that α′ (resp. β′) passes

through a singularity. We denote by Int(α′, β′) the number of points of intersection

of α′ and β′ counted with multiplicity in the following manner: let {p1, . . . , pk} =

α′ ∩ β′ (with pi 6= pj , i 6= j); we assign to pi the multiplicity mi = (number of times

β′ passes through pi) × (number of times α′ passes through pi); so, by definition,

Int(α′, β′) =k∑

i=1

mi.

Proposition 12.15. For all n ≥ 0 and all k ≥ 0 large enough, we have

DR

AFT

13 J

un 2

008


|i(ϕn(α), ϕ−k(β))− Int(ϕn(α′), ϕ−k(β′))| ≤ P (α′)P (β′).isn’t int always

bigger than i?

that would sim-

plify the state-

ment.

Proof. We remark that P (ϕn(α′)) = P (α′) and P (ϕ−k(β′)) = P (β′). By Lemma 12.12,

the angle of ϕn(α′) with Fu (resp. ϕ−k(β′) with Fs) is < π4 for n (resp. k) sufficiently

large. It thus suffices to show that |i(α, β) − Int(α′, β′)| ≤ P (α′)P (β′) if the angle of

α′ with Fu (resp. β′ with Fs) is < π4 .

We denote by Mp→ M the covering of M where p⋆(π1(M)) is the cyclic group

generated by α′. As M is orientable, we have M ∼= S1 × R. Let α′ be a closed lift of

α′ in M ; since α′ is quasitransverse to Fs = p−1(Fs) and it is homotopic to the core

of the cylinder, α′ is in fact a simple curve by Proposition 12.10. We consider β′ as a

Z-periodic map Rβ′

→M ; a lift of β′ in M is by definition a lift β′ : R→ M of the map

β′ : R→M .

We are going to show that a lift β′ of β′ intersects α′ in at most one point (a point

of α′∩β′ is counted with multiplicity if β′ passes through this point multiple times, so

a single point of intersection means: card(α′∩β′) = 1 and β′ only passes once through

α′ ∩ β′). To see this, suppose that β′(a) ∈ α′ and β′(b) ∈ α′ with a 6= b. As π1(M) is

generated by α′, it is easy to find a path γ : [a, b] → M such that γ([a, b]) ⊂ α′,

γ(a) = β′(a), γ(b) = β′(b) and such that β′|[a,b] is homotopic to γ|[a,b] with endpoints

fixed. If we go up to the universal cover M , we find a lift α′ of α′ and a lift β′ of β′ in

M such that card(α′ ∩ β′) ≥ 2; however this is impossible by Corollary 12.14 (here is

where we use the assumption on the angles).

It follows that Int(α′, β′) is equal to the number of lifts of β′ in M that intersect

α′. The lifts β′ that intersect α′ are partitioned into two categories. The first category

consists of lifts that lie on a single side of α′, and the second category consists of lifts

that join the 2 infinities of the cylinder M (Figure 12.8).

secondcategory

firstcategory

Figure 12.8:

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008


Since α′ and β′ are transverse outside of the singularities of Fs (or Fu), it is easy

to see that the point of contact for the first category is a singularity. We conclude

that the number of lifts of β′ that intersect α′ and that are in the first category is

≤ P (α′)P (β′). The reader will easily show that the number of lifts of β′ that intersect

α′ and that are in the second category is in fact exactly i(α, β).

We consider now a Markov partitionR = {R1, . . . , RN} for ϕ (see Section 12.2B).

We can, by a small perturbation of α′ (resp. β′), suppose that α′ (resp. β′) is transverse

to ∪Ni=1∂FuRi (resp. ∪Ni=1∂FsRi). As ϕ−1(∪Ni=1∂FuRi) ⊂ ∪Ni=1∂FuRi, the curve ϕℓ(α′)

is also transverse to ∪Ni=1∂FuRi for ℓ ≥ 0; in the same way ϕ−ℓ(β′) is transverse to

∪Ni=1∂FsRi.

For ℓ ≥ 0, we denote by αℓi the number of connected components of the pre-

image of Ri under a parametrization of ϕℓ(α′). The image of any such component

will be called a passage of ϕℓ(α′) in Ri. We say that a passage is good if it does notpassage

meet ∂FuRi; otherwise we say that it is bad. We denote by αℓi the number of good

passages of ϕℓ(α′) in Ri.

passages

good passage

bad∂FuRi

Figure 12.9:

We remark that αℓi − αℓi is bounded above by the number of times (with multi-

plicity) that ϕℓ(α′) intersects ∂FuRi. As ϕ−ℓ(∪Ni=1∂FuRi) ⊂ ∪i∂FuRi, if C1 denotes

the number of times that α′ intersects ∪Ni=1∂FuRi, we find that αℓi ≤ αℓi ≤ αℓi +C1. In

the same manner, we define βℓi and βℓi by replacing ϕℓ(α′) with ϕ−ℓ(β′) and Fu with

Fs. We also find a constant C2 such that βℓi ≤ βℓi ≤ βℓi + C2, ∀ i = 1, . . . , N , ∀ ℓ ≥ 0.

We set C = max(C1, C2).

DR

AFT

13 J

un 2

008


Since ϕ−n(∪Ni=1∂FuRi) ⊂ ∪Ni=1∂FuRi and ϕn(∪Ni=1∂FsRi) ⊂ ∪Ni=1∂FsRi for n ≥ 0, it

is easy to see that if P is a good passage of ϕℓ(α′) inRi, then ϕn(P )∩Rj is composed

of a(n)ij good passages of ϕℓ+n(α′), whereA = (aij) is the incidence matrix associated

to the Markov partition and An = (a(n)ij ). On the other hand, if P ′ is an arbitrary

passage of ϕℓ(α′) in Ri, then ϕn(P ′) ∩ Rj is composed of at most a(n)ij passages of

ϕℓ+n(α′) in Rj (here we use the fact that α′ is quasitransverse to Fs). We therefore

have the following inequalities:

N∑

i=1

αℓia(n)ij ≤ αℓ+nj ≤ αℓ+nj ≤

N∑

i=1

αℓia(n)ij ≤

N∑

i=1

(αℓi + C)a(n)ij .

We recall that xi = µs(Fu-fiber of Ri) and yi = µu(Fs-fiber of Ri).

Lemma 12.16.

a) limℓ→∞

N∑

i=1

xiαℓi

λℓ= I(Fs, µs;α)

b) limℓ→∞

N∑

j=1

yjβℓj

λℓ= I(Fu, µu;β).

Proof. I(Fs, µs;ϕℓ(α)) = λℓI(Fs, µs;α) is nothing other than the µs-length of ϕℓ(α′)

since ϕℓ(α′) is quasitransverse to Fs and is homotopic to ϕℓ(α). Also

N∑

i=1

xiαℓi ≤ µs[ϕℓ(α′)] ≤

N∑

i=1

xiαℓi ≤ (

N∑

i=1

xiαℓi) + C

N∑

i=1

xi;

which implies

N∑

i=1

xiαℓi ≤ λℓI(Fs, µs;α) ≤ (

N∑

i=1

xiαℓi) + C

N∑

i=1

xi.

Part a) follows easily from this inequality. Part b) is obtained by interchanging the

roles.

We remark that we haveN∑

j=1

αn+ℓj βℓj ≤ Int(ϕn+ℓ(α′), ϕ−ℓ(β′)). By Lemma 12.12

and Corollary 12.14, for ℓ large enough and n ≥ 0, we have

Int(ϕn+ℓ(α′), ϕ−ℓ(β′)) ≤N∑

j=1

αn+ℓβℓj ,

DR

AFT

13 J

un 2

008


whence by the inequalities written above we have

∑

i,j

αℓia(n)ij β

ℓj ≤ Int(ϕn+ℓ(α′), ϕ−ℓ(β′)) ≤

∑

i,j

(αℓi + C)a(n)ij (βℓj + C).

DR

AFT

13 J

un 2

008


By Proposition 12.15, for ℓ large enough and n ≥ 0, we have

|Int(ϕn+ℓ(α′), ϕ−ℓ(β′))− i(ϕn+ℓ(α), ϕ−ℓ(β))| ≤ P (α′)P (β′).

We also have of course i(ϕn+ℓ(α), ϕ−ℓ(β)) = i(ϕn+2ℓ(α), β).

Combining the preceding inequalities, for ℓ large we obtain

∑

i,j

αℓia(n)ij β

ℓj

− P (α′)P (β′) ≤ i(ϕn+2ℓ(α), β)

≤∑

i,j

(αℓi + C)a(n)ij (βℓj + C) + P (α′)P (β′).

Dividing these inequalities by λn+2ℓ and applying Lemma 12.6, and then letting

n tend to infinity and making (the fixed number) ℓ large enough, we find

∑

i,j

αℓixiyjβℓj

λ2ℓ≤ lim inf

k→∞

i(ϕk(α), β)

λk

≤ lim supk→∞

i(ϕk(α), β)

λk

≤∑

i,j

(αℓi + C)xiyj(βℓj + C)

λ2ℓ.

By Lemma 12.16, if we let ℓ tend to infinity, we obtain:

limk→∞

i(ϕk(α), β)

λk= I(Fs, µs;α)I(Fu, µu;β)

This completes the proof of Theorem 12.2. need end of

proof symbolCorollary 12.3 is an immediate consequence of Theorem 12.2.

Proof. [Proof of Corollary 12.4]

As we have seen in Expose 9 (theorem at the end of Section 9.5), the action of ϕ

cannot have any fixed points in Teichmuller space T (M), and moreover a nontrivial

power of ϕ cannot preserve an isotopy class of simple curves. Supposing then that

we have a fixed point from the action of ϕ on T (M), this fixed point is an element

[F , µ] in PMF(M). In other words, there exists ρ > 0 such that ϕ(F , µ) ∼m (F , ρµ).

It follows that F is arational, because otherwise a nontrivial power of ϕ preserves an

isotopy class of curves. Moreover, ρ is different from 1, because otherwise ϕ would

be isotopic to a periodic diffeomorphism (see Section 9.4). We suppose that ρ > 1; the

case ρ < 1

DR

AFT

13 J

un 2

008


is treated in the same manner. We can then, by Section 9.5, isotope ϕ to a pseudo-

Anosov diffeomorphism ϕ′ that admits (F , µ) for an unstable foliation. Corollary 12.3

applied to ϕ and to ϕ′ gives the following for all α ∈ S(M):

limn→∞

[ϕnα] = [Fu, µu] in PMF(M)

limn→∞

[ϕ′nα] = [F , µ] in PMF(M).

As ϕ and ϕ′ are isotopic, we obtain [Fu, µu] = [F , µ]. The case ρ < 1 would give

[Fs, µs] = [F , µ].

12.5 Proof of Theorem III (uniqueness of pseudo-Anosov

diffeomorphisms)

We begin by proving two lemmas.

Lemma 12.17. Let M be a closed orientable surface of genus g > 1 and ϕ a diffeomorphism

of M isotopic to the identity. If ϕ is periodic, then ϕ is the identity.

Proof.

We have seen (the remark at the end of Section 9.4) that the Uniformization The-

orem implies that ϕ is an isometry for a hyperbolic metric. Since ϕ is isotopic to the

identity, ϕ is in fact the identity (Theorem 3.20).

Lemma 12.18. Let (Fu, µu) be an arational foliation of M and ϕ a diffeomorphism of M ,

isotopic to the identity, that preserves (Fu, µu). Then ϕ is isotopic to the identity through

diffeomorphisms that preserve (Fu, µu).

Proof. Lemma 9.7 says that ϕ is isotopic to a periodic diffeomorphism ϕ′, through

diffeomorphisms that preserve (Fu, µu). The preceding lemma shows that ϕ′ is the

identity.

Letϕ1 andϕ2 be two isotopic pseudo-Anosov diffeomorphisms. Denote by (Fu1 , µu1 )

(resp. (Fu2 , µu2 )) the unstable foliation of ϕ1 (resp. ϕ2), and by (Fs1 , µs1) (resp. (Fs2 , µs2))the stable foliation of ϕ1 (resp. ϕ2). By Corollary 12.4 we have [Fu1 , µu1 ] = [Fu2 , µu2 ] in

P (RS+). Up to multiplying (Fu1 , µu1 ) by a positive nonzero constant,

DR

AFT

13 J

un 2

008


we can thus suppose that (Fu1 , µu1 ) = (Fu2 , µu2 ) inMF . Since these foliations do not

have connections between singularities, there exists a diffeomorphism h isotopic to

the identity such that (Fu1 , µu1 ) = h(Fu2 , µu2 ) where the equality means here that the fo-

liations inM are the same and the transverse measures are the same. Up to replacing

ϕ2 by hϕ2h−1, we are reduced to the case where ϕ1 and ϕ2 have the same unstable

foliation (Fu, µu). We also remark that the expansion constant λ (> 1) is the same

for ϕ1 and ϕ2; this follows for example from the fact that ϕ1(Fu, µu) = ϕ2(Fu, µu)inMF . It follows that ϕ−1

2 ϕ1 preserves (Fu, µu). By Lemma 13, ϕ−12 ϕ1 is isotopic to

the identity through diffeomorphisms that preserve (Fu, µu); we denote by ht one such

isotopy. In particular, for every x in M , ϕ−12 ϕ1(x) and x are on the same Fu-leaf; we

denote by [x, ϕ−12 ϕ1(x)] the segment of the Fu-leaf of x that joins x to ϕ−1

2 ϕ1(x).

Lemma 12.19. We have: D = sup{µs2([x, ϕ−12 ϕ1(x)]) | x ∈M} <∞.

Proof. Let U1, . . . , Uk be a covering of M by charts for the foliation Fu. We denote by

A the subset ofM×M defined by (x, y) ∈ A if there exists a plaque ofFu contained in

one of the Ui and that contains x and y (in particular, since the “plaque” of a singular

point is a single point, if (x, y) ∈ A and x (or y) is a singular point of Fu, then x = y).

If (x, y) ∈ A, we denote by [x, y] the segment of the plaque that contains x and y and

that goes from x to y; the function (x, y)→ µs2([x, y]) is continuous on A. We consider

then the isotopy ht of ϕ−12 ϕ1 to the identity, through homeomorphisms that preserve

Fu. We can find a δ > 0 such that, if |t − t′| < δ, then (ht(x), ht′ (x)) ∈ A; by the

compactness of M and what has been said above, we have

Dt,t′ = sup{µs2([ht(x), ht′(x)])|x ∈M} <∞

We then consider a sequence t0 = 0 < t1 < · · · < tn−1 < tn = 1, such that ti+1−ti < δ.

For all x ∈M , we have

µs2([x, ϕ−12 ϕ1(x)]) ≤

n−1∑

i=0

µs2([hti(x), hti+1 (x)]);

from which we have

D ≤n−1∑

i=0

Dti,ti+1 < +∞

Lemma 12.20. The sequence of homeomorphisms (ϕ−n2 ϕn1 )n≥0 converges uniformly.

DR

AFT

13 J

un 2

008


Proof. Let d be the metric obtained from ds2 = (dµs2)2 + (dµu2 )2. We remark that if

x and y are on the same Fu-leaf, and if [x, y] denotes the segment of this leaf that

goes from x to y, we have d(x, y) ≤ µs2([x, y]) (we do not have equality in general,

because, since the leaves demonstrate recurrence, two points can be close in M with-

out the segment of the leaf that goes from one to the other being small). The uniform

convergence of the sequence follows easily from the following inequality that we are

going to establish:

supx∈M

d(ϕ−(n+1)2 ϕ

(n+1)1 (x), ϕ−n

2 ϕn1 (x)) ≤ λ−nD.

We consider the segment of Fu-leaf [ϕ−12 ϕ1(ϕ

n1 (x)), ϕn1 (x)]; its measure is ≤ D.

The image of this segment under ϕ−n2 is nothing other than the segment of Fu-leaf

[ϕ−(n+1)2 ϕ

(n+1)1 (x), ϕ−n

2 ϕn1 (x)]. Considering the effect of ϕ−n2 on µs2, we have

µs2([ϕ−(n+1)2 ϕ

(n+1)1 (x), ϕ−n

2 ϕn1 (x)]) ≤ λ−nµs2([ϕ−12 ϕ1(ϕ

n1 (x)), ϕn1 (x)]) ≤ λ−nD

from which we obtain

d(ϕ−(n+1)2 ϕ

(n+1)1 (x), ϕ−n

2 ϕn1 (x)) ≤ λ−nD.

We denote by h the uniform limit of (ϕ−n2 ϕn1 )n≥0. We remark that h is invertible

since one shows this in the same manner that one shows that the sequence of inverses

(ϕ−n1 ϕn2 )n≥0 converges uniformly. We also remark that h is isotopic to the identity

since each ϕ−n1 ϕn2 is isotopic to the identity.

We then consider hϕ1; we have:

hϕ1 = (limunifn→∞ ϕ−n2 ϕn1 )ϕ1

= limunifn→∞(ϕ−n2 ϕ

(n+1)1 )

= limunifn→∞(ϕ2(ϕ−(n+1)2 ϕ

(n+1)1 ))

= ϕ2 limunifn→∞(ϕ−(n+1)2 ϕ

(n+1)1 )

= ϕ2h,

so hϕ1 = ϕ2h, which shows that h is a conjugation between ϕ1 and ϕ2.

It remains to check that h is differentiable. Before doing this, we must make the

definition of pseudo-Anosov more precise; that is, we insist that, using a C∞ chart

in the neighborhood of a singularity, the foliations Fs, Fu are given by the absolute

values of the real and imaginary parts of√zp−2 dz2 (p ≥ 3).

Lemma 12.21. A conjugation between two pseudo-Anosov diffeomorphisms is automatically

C∞ differentiable.

DR

AFT

13 J

un 2

008


Proof. [Outline of proof] Denote by h the conjugation, ϕ1 and ϕ2 the two pseudo-

Anosov diffeomorphisms;

DR

AFT

13 J

un 2

008


thus hϕ1h−1 = ϕ2. The first thing we remark is that h sends the (un)stable foliation

of ϕ1 onto the (un)stable foliation of ϕ2 (without talking for the moment about the

transverse measure). This follows for example from the fact that W sx (ϕi) = {y ∈

M | limn→∞

d(ϕni (x), ϕni (y)) = 0} is the leaf of Fsi that passes through x if this leaf does

not arrive at a singularity, and if the leaf of x does reach a singularity x0 of Fsi (or if

x = x0), W sx(ϕi) is the union of x0 and of leaves of Fsi that arrive at x0. Moreover, as

Fs2 (resp. Fu2 ) is uniquely ergodic (Theorem 12.1), h also sends µs1 (resp. µu1 ) onto µs2(resp. µu2 ), up to dividing the measures by a suitable constant.

Considering then a regular point m for Fs1 and Fu1 , its image h(m) is a regular

point for Fs2 and Fu2 . We can find a smooth chart (−ε, ε) × (−ε, ε) ψ→ M , such that

ψ(0) = m and that the foliation (Fs1 , µs1) (resp. (Fu1 , µu1 )) is defined in this chart by the

1-form dx (resp. dy). In the same way, we find one such chart around h(m). When we

read h in these charts, it appears as a homeomorphism of (−ε, ε)× (−ε, ε) on an open

neighborhood of 0 in R2 that sends 0 to 0, the horizontals into the horizontals, the

verticals into the verticals, and that preserves the spacing between two horizontals

or two verticals. It is easy to see that h is the restriction to (−ε, ε)× (−ε, ε) of one of

the following four linear maps of R2: identity, orthogonal symmetry with respect to

the x- (resp. y-) axis, reflection through the origin. It follows that h is C∞ at every

regular point.

We can make an analogous reasoning at a singular point. Recall that we have

made precise the definition of a pseudo-Anosov. This precision implies that in suit-

able charts, h appears as a germ of a homeomorphism at 0 ∈ C that preserves the

absolute values of the real and imaginary parts of√zp−2 dz2 (p ≥ 3). The reader will

verify that such germs, that preserve orientation, are rotations of angles 2kπp , and the

germs that reverse orientation are given by symmetries with respect to the lines that

contain the union of two separatrices.

Remark. One may wonder which are the necessary and sufficient conditions so that

an arational foliation, uniquely ergodic, is the stable foliation of a pseudo-Anosov.

DR

AFT

13 J

un 2

008

Expose 13

Construction of pseudo-Anosov

diffeomorphisms

by F. Laudenbach

13.1 Generalized pseudo-Anosov diffeomorphism

A measured foliation with spines is a measured foliation for which we allow, in ad-

dition to the usual singularities (Expose 5), those of Figure 13.1; the figure represents

two measured foliations with spines, mutually transverse.

Figure 13.1:

A generalized pseudo-Anosov diffeomorphism is a homeomorphism ϕ for which there

exists two measured foliations with spines (Fs, µs), (Fu, µu), mutually transverse,

and a scalar λ > 1, such that ϕ(Fs, µs) = (Fs, λ−1µs) and ϕ(Fu, µu) = (Fu, λµu).The disk admits a measured foliation with spines that is transverse to the bound-

ary (Figure 13.1). It is also possible that, for α ∈ S, one has I(Fs, µs;α) = 0, even num should be

13.2though Fs does not contain any cycles of leaves 1 (since ϕ contracts the µu-lengths,

1We contruct one such example on T 2 by applying the construction of Section III to (α, β), where α is a

“generator” of the torus and where β, isotopic to α, cuts T 2 − α into cells.

255

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008


Fs does not have connections between singularities). A generalized pseudo-Anosov

can fix an element of S; it does not satisfy the lemma on growth of lengths of isotopy

classes of curves (Proposition 9.21 of Expose 9) (see the footnote referenced above).

Therefore, a generalized pseudo-Anosov can be isotopic to the identity (see the ex-

ample on S2 in Section 11.3.6.

Figure 13.2:

Nevertheless, generalized pseudo-Anosovs are still useful because of the follow-

ing remark. If one cuts the surface at the point of the spines, one obtains a pseudo-

Anosov diffeomorphism of the surface with boundary. In particular, Poincare Re-

currence still holds and one can construct a Markov partition. Thus a generalized

pseudo-Anosov is again a Bernoulli process.

13.2 Construction by ramified cover

13.2.1

Let p : N → M be a ramified cover of compact surfaces; let Σ ⊂ M be the locus of

ramification. We suppose that, above M − Σ, the covering is regular, with covering

groupG. Letϕ be a generalized pseudo-Anosov ofM ; by isotopy of p, we can arrange

Σ to be in the infinite set of periodic points of ϕ. Thus, up to replacing ϕ by one of

its powers, we can suppose that ϕ|Σ = Id. The regular covering of p over M − Σ is

classified by an element of H1(M −Σ;G), a finite group on which ϕ acts. Up to again

taking powers of ϕ, we can suppose that ϕ⋆(p : N → M) ≈ (p : N → M); in other

words, ϕ lifts to a diffeomorphism ψ. The local properties of ϕ are the same as those

of ψ; thus ψ is a generalized pseudo-Anosov, with the same dilatation factor.

13.2.2

Every closed orientable surface N , of genus g ≥ 1, is the total space of a ramified

cover with two sheets over T 2 with a locus of ramification Σ satisfying:

cardΣ = 2g − 2.

DR

AFT

13 J

un 2

008


To see this, we puncture T 2 in n = cardΣ points; this open manifold retracts onto

a bouquet of n+ 1 circles ε1, . . . , εn+1; say that each of ε1, . . . , εn−1

DR

AFT

13 J

un 2

008


surrounds a hole and that εn and εn+1 are “generators” of the torus. The last hole,

denoted∞, is surrounded homologically by [ε1]+ · · ·+[εn−1]. We construct the cover

associated to the homomorphism π1(T2 − Σ) → Z/2Z that sends ε1, . . . , εn−1 to 1

and that takes any value on εn and εn+1. As n is even, the covering is nontrivial in

a neighborhood of∞. Thus the compactification gives a covering that is nontrivially

ramified at each point of Σ.

Let ϕ be a (linear) Anosov map of T 2 that is the identity on Σ and that lifts to

ψ in N . The stable foliation of ψ is transversely orientable and its singularities have 4

branches each. Thus ψ is pseudo-Anosov (not generalized) and the stable foliation is

defined as a measured foliation by a closed 1-form ωs; we have

ψ⋆ωs = λωs (λ ∈ R, λ > 1).

In the same way, we have ψ⋆ωu = λ−1ωu, where ωu denotes the unstable foli-

ation. We note that the two equalities together prohibit ψ from being differentiable

at the singularities, but one can approximate ψ by a diffeomorphism ψ′ that satisfies

one of the equalities.

The disadvantage of this construction is that it is unmanageable on the level of

calculation, for example to compute the action of ψ on homology.

13.3 Construction by Dehn twists

We suppose that the surfaceM is orientable and closed. In the case where there is

boundary, one would start by filling the punctures and, at the end of the construction

described below, one would puncture the corresponding number of singularities.

13.3.1 Flat structure on M

Let α and β be two simple curves in M , with transverse intersection, satisfying

the following condition:

(⋆) Each component of M − (α ∪ β) is an (open) cell.

The cellular decomposition induced on M by α ∪ β admits a dual cellular de-

composition: the co-vertices are the centers of the cells of M − (α ∪ β); each arc of

(α ∪ β) − (α ∩ β) is crossed by one co-edge; each point x of α ∩ β is the center of a

co-cell, which is a square since α and β only pass through x once.

By “enlarging” α, in the sense of Expose 5, we construct a measured foliation Fα,

transverse to β and to the co-edges that meet α; we also arrange things

DR

AFT

13 J

un 2

008


so that the co-edges that do not meet α are in the leaves of Fα. Similarly, we construct

Fβ by suitably enlarging β; by isotopies in the interior of the co-cells, we can take Fβto be transverse to Fα. These foliations have their singularities at the co-vertices; in

the complement, they define a flat structure. We understand these foliations better in

the “unrolled” figures below.

If we unroll the co-cells along α, we obtain a band of n = card(α ∩ β) squares,

which we place in R2 in such a way that dy and dx respectively induce the measured

foliations Fα and Fβ .

α-atlas leaf of Fβ

x

y

αβ

In the same way, we construct the β-atlas by unrolling along β (respecting the orien-

tation).

qβ-atlas

leaf of Fα

xy

α

β

The transition maps are isometries of R2 with derivative ±I , according to the

sign of the intersection at the center of the co-cell. [Note that a transition map that

preserves orientation cannot have the diagonal matrix diag(1,−1) for its derivative.]

Consequently, relative to this atlas, the notion of a linear measured foliation is intrin-

sic; moreover, the slope of the foliation is invariant under the transition maps. Once

DR

AFT

13 J

un 2

008


we return to M , the foliation is smooth except at the co-vertices (each point of the

complement is interior to at least one chart). The co-vertices act like singularities (un-

less the corresponding cell is a square). The number of separatrices of the foliation at

a vertex is half the number of sides of the corresponding cell.

DR

AFT

13 J

un 2

008


αp

p

α

β

β

β

β

α

α

α

Figure 13.3:

Remark. If all the points of intersection of α and β are of the same sign, then the tran-

sition maps have +I for their derivative; thus, the orientation of the foliations is in-

variant under the transition maps. In other words, in this case, all linear foliations are

orientable. Moreover, the atlas defines a function M → T 2 which is an n-fold cover-

ing ramified at one point; but the covering is not regular and one cannot control the

singular fiber.

13.3.2 Affine homeomorphisms

A homeomorphism ϕ is said to be affine if it leaves invariant the set of the co-

vertices and if the image of a straight line of the flat structure is a straight line. Let

A(M) be the group of affine homeomorphisms.

The derivative of ϕ, modulo ±I , is independent of the atlas used and of the point

where one calculates it. We thus have a a derivation homomorphism:

D : A(M)→ GL(2,R)/± I.

For example, the positive Dehn twist along α (resp. β) admits an affine repre-

sentative which, in the α-atlas (resp. β-atlas), is induced by the linear transformation

DR

AFT

13 J

un 2

008


with matrix (1 n

0 1

)[resp

(1 0

−n 1

)].

DR

AFT

13 J

un 2

008


The derivatives of these twists are given by the classes of these matrices in PSL(2,Z).

Remark. It is at this point that we use that there are only two curves. In fact, in

this case, the α-atlas covers all of M and the homeomorphism is well-defined by

its description in the α-atlas. Moreover, if there are more than two curves, the twist

along α cannot in general be represented by an affine homeomorphism.

Lemma 13.1. An affine homeomorphism ϕ of M is a generalized pseudo-Anosov if and only

if Dϕ has real eigenvalues (λ, 1λ) with λ 6= 1.

Proof. The condition on Dϕ means that ϕ respects two transverse linear foliations

by contracting the distances on the leaves of one and by stretching one those of the

other, by a constant factor.

thm environ-

ment?

13.3.3 Theorem

Let G(α, β) be the subgroup of A(M) generated by the affine Dehn twists along

curves α and β that satisfy condition (⋆). The derivation map induces a homomor-

phism D : G(α, β) → PSL(2,Z). Thus, ϕ ∈ G(α, β) is a generalized pseudo-Anosov

if and only if Dϕ has real eigenvalues distinct from ±1. If in addition, card(α ∩ β) =

i(α, β) (minimal intersection), then ϕ is pseudo-Anosov.

Remarks. 1) In his announcement, Thurston says that there exists a homomor-

phism G(α, β)→ SL(2,Z); this is a lift of D.

2) A matrix of SL(2,Z), where the trace has a modulus > 2, is Anosov. Thus, if ϕ

is obtained by combining positive twists along α and negatives along β, with at least

one of each, then ϕ is a generalized pseudo-Anosov.

Proof. [Proof of the theorem] By the lemma, it only remains to prove the secondfix beginproof

assertion. Since a linear foliation has a spine if and only if the corresponding co-

vertex is the center of a cell with two sides, the hypothesis of minimal intersection

prohibits this configuration.

13.3.4 Examples

In the first example, we take α to have a connected complement in

DR

AFT

13 J

un 2

008


M = closed surface of genus 2. Let M ′ be the surface obtained by cutting M along α;

α1 and α2 are the two copies of α that form the boundary of M ′. Four arcs joining α1

to α2 are needed to cutM ′ into two octagonal cells. But there is no way to reglue α1 to

α2 so that these arcs make a connected curve; however, this becomes possible if each

arc is doubled (see Figure 13.4). In this example, all the points of intersection are of the

same sign, therefore the linear foliations are orientable; they have two singularities

with 4 branches; they are thus defined by closed 1-forms with Morse saddles for

singularities.

1234 5 6 7 8 1′2′ 3′4

′5′6′

7′8′

α1 α2

Figure 13.4:

Other examples arise from the following lemma.

Lemma 13.2. Let α be a simple curve that is not homotopic to a point on the closed surface

M . Then, there exists a simple curve β such that M − (α ∪ β) is a union of cells. Moreover,

if α is null-homologous, β can be chosen to be null-homologous.

Proof. We find a decomposition of M into pairs of pants by curves Kj such that, for

all j, i(Kj, α) 6= 0 (if we think of α as a measured foliation,

DR

AFT

13 J

un 2

008


we can apply Lemma 6.15. Let β be the curve obtained by applying a (positive) Dehn

twist along each Kj to α. We first prove that, for all γ ∈ S, i(α, γ) 6= 0 or i(β, γ) 6=0. Suppose that i(α, γ) = 0; then, for some j, i(γ,Kj) 6= 0; otherwise γ would be

isotopic to one of the Kℓ and hence would intersect α. Now, applying the inequality

of Appendix A, we have

|i(β, γ)−∑

j

i(α,Kj)i(γ,Kj)| ≤ i(α, γ) = 0.

Thus, i(β, γ) is strictly positive.

This proves that any simple curve γ in M − (α ∪ β) is null-homotopic and there-

fore bounds a disk D in M . We see that D is contained in M − (α ∪ β); otherwise,

intD would contain a piece of β (or of α); as β does not cut the boundary of D, β

would be entirely contained in D, which is absurd. From this, it is easy to see that the

components of M − (α ∪ β) are open disks.

When α and β are null-homolgous, the affine homeomorphisms induce the iden-

tity on homology. However some of them are not, up to isotopy, either periodic or

reducible; this contradicts a conjecture of Nielsen [Nie44b], saying that, if the eigen-

values of the induced automorphism on homology are on the unit circle, then the

diffeomorphism is decomposable into periodic pieces.

Remark. All the preceding constructions lead to pseudo-Anosovs where the dilata-

tion factor is a quadratic integer. The “members of the seminar” do not know how to

construct examples where it is of higher degree.

DR

AFT

13 J

un 2

008

Expose 14

Fibrations of S1 with

Pseudo-Anosov Monodromy

by David Fried

We will develop Thurston’s description of the collection of fibrations of a closed three

manifold over S1. We will then show that the suspended flows of pseudo-Anosov

diffeomorphisms are canonical representatives of their nonsingular homotopy class,

thus extending Thurston’s theorem for surface homeomorphisms to a class of three

dimensional flows. Our proof uses Thurston’s work on fibrations and surface home-

omorphisms and our criterion for cross-sections to flows with Markov partitions. We

thank Dennis Sullivan for introducing Thurston’s results to us. We are also grateful

to Albert Fathi, Francois Laudenbach and Michael Shub for their helpful suggestions.

A smooth fibration f : X → S1 of a manifold over the circle determines a non-

singular (i.e. never zero) closed 1-form f∗(dθ) with integral periods. Conversely if ω

is a nonsingular closed 1-form and X is closed, then the map f(x) =∫ xx0ω from X

to R/periods(ω) will be a fibration over S1 provided the periods of ω have rational

ratios. For since π1X is finitely generated, the periods of ω will be a cyclic subgroup

of R (not trivial since X is compact and f open) and we have R/periods ∼= S1. By

constructing a smooth flow ψ on X with ω(dψdt ) = 1, we see that f is a fibration. The

relation of nonsingular closed 1-forms to fibrations over S1 is very strong indeed, as

the following theorem (which gives strong topological constraints on the existence of

nonsingular closed 1-forms) indicates.

Theorem 14.1 ([Tis70]). For a compact manifold X , the collection C of nonsingular classes,

that is the cohomology classes of nonsingular closed 1-forms onX , is an open cone inH1(X ; R)−{0}. The cone C is nonempty if and only if X fibers over S1.

267

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008


Proof. The openness of C follows easily from de Rham’s Theorem. If η1, . . . , ηd are

closed 1-forms that span H1(X ; R) and if ω0 is a closed 1-form, then the forms ωa =

ωo +

d∑

i=1

aiηi, |ai| < ǫ, represent a neighborhood of [ω0] in H1(X ; R). If ω0 is nonsin-

gular and ǫ sufficiently small, then the ωa are nonsingular. The forms λωa, with λ > 0

represent all positive multiples of [ωa], so C is an open cone.

Choosing a so that the periods of ωa are rationally related, we see that X fibers

over S1. We already noted that 0 /∈ C.

In dimension 3, Stallings characterized the elements of C∩H1(X ; Z) ⊂ H1(X ; R).

We note that ifX is closed, connected and oriented and does fiber over S1 with fibers

of positive genus, then X will be covered by Euclidean space R3. Thus X will be

irreducible, that is, every sphere S2 embedded in X must bound a ball (this follows irreducible

from Alexander’s theorem showing R3 is irreducible). We assume hence-forward that

M is a closed, connected oriented and irreducible 3-dimensional manifold.

Theorem 14.2 ([Sta61]). If u ∈ H1(M ; Z)−{0}, then there is a fibration f : M → S1 with

[f∗(dθ)] = u, if and only if ker(u : π1M → Z) is finitely generated.

We observe that the forward implication holds even for finite complexes since

the homotopy exact sequence identifies the kernel as the fundamental group of the

fiber.

Theorem 14.2 reduces the geometric problem of fibering M to an algebraic prob-

lem, with only two practical complications. First, whenever dimH1(M ; R) > 1, there

are infinitely many u to check. Secondly, it is difficult to decide if keru is finitely

generated. An infinite presentation may be readily constructed by the Reidemeister-

Schreier process; this yields an effective procedure for deciding if the abelianization

of keru is finitely generated (we work out an example of this at the end of this Ex-

pose.)

Thurston’s theorem (Theorem 14.6 below) helps to minimize the first problem

and make Stallings’ criterion more practical. It will be seen that one need only ex-

amine finitely many u, provided one can compute a certain natural seminorm on

H1(M ; R).

As H1(M ; Z) ⊂ H1(M ; R) is a lattice of maximal rank, the seminorm will be de-

termined by its values onH1(M ; Z). Each u ∈ H1(M ; Z) is geometrically represented

by framed surfaces under the Pontrjagin construction [Mil66]. A framed (that is, nor-

mally oriented) surface S represents u whenever there is a smooth map f : M → S1

DR

AFT

13 J

un 2

008


with regular value x so that S = f−1(x) and u = [f∗(dθ)]. By irreducibility of M , any

framed sphere in M represents the 0 class so S may be taken sphereless (that is, allsphereless

components of S have Euler characteristic ≤ 0).

Definition. ‖u‖ = min{−χ(S) | S is a sphereless framed surface representing u}.It is important to observe that a sphereless framed surface S in M , with ‖u‖ =

−χ(S), must be incompressible (that is, for each component Si ⊂ S, π1(Si) → π1Mincompressible

is injective). For (see Kneser’s lemma [Sta71]), one could otherwise attach a 2-handle

to Si so as to lower −χ(S) without introducing spherical components.

The justification for the notation ‖u‖ is the following result.

Theorem 14.3 ([Thu86]). ‖u‖ is a seminorm on H1(M ; Z).

This follows from standard 3-manifold techniques. The triangle inequality fol-

lows from the incompressibility of minimal representatives and some cut and paste

arguments. The homogeneity follows by the covering homotopy theorem for the

cover zn : S1 → S1.

One instance where ‖u‖ is easily computed is where u is represented by the fiber

K of a fibration f : M → S1. We have:

Proposition 14.4 ([Thu86]). If K →Mf−→S1 is a fibration, then

∥∥[f∗(dθ)]∥∥ = −χ(K).

Proof. By homogeneity we may suppose that u = [f∗(dθ)] is indivisible, that is

u(π1M) = π1S1. This implies that K is connected and that K ×R is the infinite cyclic

cover of M determined by u. If K is a torus we are done, so assume −χ(K) > 0. Any

sphereless framed surface S representing u lifts to K × R, since for any component

S0 ⊂ S we have π1S0 ⊂ keru = π1K . If −χ(S) = ‖u‖, then S is incompressible and

π1S0 → π1(K ×R) = π1(K) is injective. Since subgroups of π1K of infinite index are

free, we see that S0 is a finite cover of K , hence ‖u‖ = −χ(S) ≥ −χ(S0) ≥ −χ(K), as

desired.

In fact, we see that any sphereless framed surface S representing u with minimal

−χ(S) is homotopic to the fiber K .

The behaviour of ‖ ‖ is decisively determined by the fact that integral classes have

integral seminorms. We will show:

Theorem 14.5 ([Thu86]). A seminorm ‖ ‖ : Zn → Z extends uniquely to a seminorm

DR

AFT

13 J

un 2

008


‖ ‖ : Rn → [0,∞). A seminorm on Rn takes integer values on Zn ⇔ ‖x‖ = maxℓ∈F|ℓ(x)|,

where F ⊂ Hom(Zn,Z) is finite.

This enables us to state Thurston’s description of the cone C of nonsingular classes,

C ⊂ H1(M ; R)− {0}.We will consistently use certain natural isomorphisms of the homology and coho-

mology groups ofM . By the Universal Coefficient Theorem,H1(M ; Z) ∼= Hom(H1(M ; Z); Z)

and H1(M ; Z)/torsion ∼= Hom(H1(M ; Z); Z). With real coefficients, Hi(M ; R) and

Hi(M ; R) are dual vector spaces for any i. By Poincare Duality, we may identify

H2(M ; Z) with H1(M ; Z). Thus we regard the Euler class χF of a plane bundle F

on M , which is usually taken to be in H2(M ; Z), as an element of H1(M ; Z) and thus

as a linear functional on H1(M ; R).

Theorem 14.6 ([Thu86]). C is the union of (finitely many) convex open cones in int(Ti),

where Ti is a maximal region on which ‖ ‖ is linear. The region Ti containing a given non-

singular 1-form ω is Ti = {u ∈ H1(M ; R) | ‖u‖ = −χF (u)} where χF is the Euler class of

the plane bundle F = kerω.

Note. When ‖ ‖ is a norm, we may say that C is all vectors v 6= 0 such that v‖v‖ belongs

to certain “nonsingular faces” of the polyhedral unit ball. Incidentally, we have that

‖ ‖ is a norm ⇐⇒ all T 2 ⊂M separateM ⇐⇒ all incompressible T 2 ⊂M separate

M .

We give our own analytic proof of Theorem 14.5.

Proof. Clearly ‖ ‖ extends by homogeneity to a seminorm ‖ ‖ on Qn. This function is

Lipschitz, hence has a unique continuous extension to a function Rn → [0,∞). The

triangle inequality and homogeneity follow by continuity.

By convexity, all one-sided directional derivatives of N(x) = ‖x‖ exist. Suppose

τ = (0, 1q p), q ∈ Z+, p = (p2, . . . , pn) ∈ Zn−1 is a rational point. For integral m, we

compute

∂+N

∂x1(τ) = lim

m→∞N(τ + 1/qme1)−N(τ)

1/qm

= limm→∞

(N(1,mp2, . . . ,mpn)−N(0,mp2, . . . ,mpn))

∈ Z,

since Z is closed.

DR

AFT

13 J

un 2

008


By induction on n, we assume that N(0, x), x ∈ Rn−1, is given by the supre-

mum of finitely many functionals ℓ(x) = a2x2 + · · · + anxn, a2, . . . , an ∈ Z, x =

(x2, . . . , xn). By convexity, any supporting line L to graph(N) ⊂ Rn × R lies in a

supporting hyperplane H (supporting means intersects the graph without passingsupporting

above it). We choose x a rational point for which N |0×Rn−1 is locally given by ℓ and

choose L to pass through (0, x, N(0, x)) ∈ Rn ×R in the direction of (1, 0, ∂+N∂x (0, x)).

Then we see that H is uniquely determined as the graph of (∂+N∂x1(0, x))x1 + a2x2 +

· · ·+anxn. So for a dense set of x, the graph ofN has a supporting functional at (0, x)

with integral coefficients.

Reasoning for each integrally defined hyperplane as we have for {x1 = 0}, we

find integral supporting functionals ℓ(x) = a1x1 + · · ·+ anxn, ai ∈ Z, to the graph

of N exist at a dense set in Rn. Since N is Lipschitz, there is a bound |ai| ≤ K, i =

1, . . . , n. Thus the supporting functionals form a finite set F , so S(x) = supℓ∈F|ℓ(x)| is

clearly a seminorm. But S(x) ≤ N(x) and equality holds on a dense set, implying

that S(x) = N(x) by continuity.

Before giving the proof of Theorem 14.6, let us observe one elementary conse-

quence of Theorem 14.5. Since ‖ ‖ is natural, any diffeomorphism h : M →M induces

an isometry h∗ of H1(M ; R). If ‖ ‖ is a norm, then the finite set of vertices of the unit

ball spans H1(M ; R) and is permuted by h∗.

Corollary 14.7. If all incompressible T 2 ⊂ M separate M , then the image of Diff(M) in

GL(H1(M ; R)) is finite.

Proof. Suppose ω, ω′ are nonsingular closed 1-forms that are C0 close. Then the ori-

ented plane fields F = kerω, F ′ = kerω′ are homotopic and so determine the same

Euler class χF ′ = χF ∈ H1(M ; R).

If [ω′] is rational, let q[ω′] = β′ ∈ H1(M ; Z), where 0 < q ∈ Q and β′ is indivisible.

Then if K ′ is the (connected) fiber of the fibration associated to qω′, we have χ(K ′) =

χF ′(K ′) = χF (K ′). Using this and Proposition 14.4, we find∥∥[ω′]

∥∥ = 1q (−χ(K ′)) =

− 1qχF (K ′) = χF [ω′]. Thus for all rational classes [ω′] near [ω], ‖ ‖ is given by the

linear functional −χF . This show that ‖ ‖ agrees with −χF on a neighborhood of any

nonsingular class [ω], as desired.

It only remains to show that every α ∈ int(T ) is a nonsingular class, where

T = {α ∈ H1(M ; R) | ‖α‖ = −χF (α)} is the largest region containing [ω] on which

‖ ‖ is linear.

DR

AFT

13 J

un 2

008


For this, we need a result of Thurston’s thesis [Thu72] concerning the isotopy of

an incompressible surface S ⊂ M when M is foliated without “dead end compo-

nents”. In fact, this result is only explicitly stated for tori, and one must see [Rou73]

for a published account of this case. Restricting our attention to the foliation F de-

fined by ω (F is tangent to kerω = F ), we may state this result as follows: any incom-

pressible, oriented and connected surface S0 ⊂M with −χ(S0) ≥ 0 may be isotoped

so as to either lie in a leaf of F or so as to have only saddle tangencies with F . (We

call a tangency point s of S0 with F a saddle if for some open ball B around s, the saddle

map∫ xs ω : B ∩ S0 → R has a non-degenerate critical point at s which is not a local

extremum.)

Suppose α ∈ T ∩ H1(M ; Z) is not a multiple of [ω]. Represent α by a framed

sphereless surface with −χ(S) = ‖α‖. As S is incompressible, each component of

S may be isotoped (independently) to a surface Si which either lies in a leaf of For has only saddle tangencies with F . If some Si lies in a leaf L of F , then (as in

Proposition 14.4) π1Si would be of finite index in π1L = ker[ω]. Since π1Si ⊂ kerα,

we would find that α is a multiple of [ω]. Thus each Si has only saddle tangencies

with F .

Lemma 14.8. For each i, the normal orientations of Si and F agree at all tangencies.

Proof. We compute ‖α‖ in two ways. First, ‖α‖ = −χ(S) =∑

i

−χ(Si) Choosing

some Riemannian metric on M , we may use the vector field Vi on Si dual to ω|Sito compute −χ(Si). Vi will have only nondegenerate zeroes of index −1, since all

tangencies are saddles. The Hopf Index theorem [Mil66] gives−χ(Si) = ni, where niis the number of tangencies of Si with F . Thus ‖α‖ =

∑ni.

On the other hand, we know that α ∈ T implies ‖α‖ = −χF (α). The natural nor-

mal orientations of F and S gives us preferred orientations on F and Si, for each i.

Each oriented plane bundle F |Si has an Euler class χF (Si)[Si] where [Si] ∈ H2(Si; Z)

is the orientation class. We compute χF (Si) as the self-intersection number of the

zero section of F |Si. For this purpose, look at the field Wi of vectors on Si tangent

to F , which are the projections onto F of the unit normal vectors of Si. Regarding

Wi as a perturbation of the zero section of F |Si, we compute the self-intersection

number using the local orientations of F and Si. When these orientations agree, one

counts the singularity as −1 (just as in the tangent bundle case already considered)

but when the orientations disagree one counts +1. Thus −χF (Si) = n+i − n−

i , where

n+i is the number of tangencies at which the orientations agree and n−

i is the number

of tangencies at which the orientations disagree. Thus

DR

AFT

13 J

un 2

008


‖α‖ =∑n+i − n−

i .

Since ni = n+i + n−

i , we have∑n+i +

∑n−i = ‖α‖ =

∑n+i −

∑n−i , whence all

the nonnegative integers n−i must be zero. This proves the lemma.

Because of the lemma, we may define a framing Ni of Si with ω(Ni) > 0 ev-

erywhere. This framing may be extended to a product neighborhood structure on

Ui ⊃ Si, where h : Si × [−1, 1]→ Ui is a diffeomorphism, h∗(∂∂t ) = Ni on Si = Si × 0

and ω(h∗(∂∂t )) > 0. Let B : [−1, 1] → [0,∞] be a smooth function vanishing on

|x| > 12 with

∫ +1

−1B = +1. Letting ηi = (π2h

−1)∗B dt we find that, for all s > 0,

(ω + sηi)(h∗∂∂t ) > 0 on U . But since ω + sηi = ω away from U , we see that the closed

1-form ω + sηi is nonsingular.

The portion of Theorem 14.6 already proven gives [ω + sηi] ∈ intT . Thus, [ηi] =

lims→∞

[ω+sηi]s ∈ T ∩H1(M ; Z), for all i. So replacing [ω] by [ω]+s1[η1]+ · · ·+si−1[ηi−1],

we see inductively that [ω] + s1[η1] + · · ·+ si[ηi] is nonsingular for all s1, . . . , si ≥ 0.

In particular, for all s ≥ 0, [ω] + sα = [ω] + s∑

[ηi] is nonsingular.

We just showed that if β = [ω] ∈ intT is a nonsingular class, then β + sα is

nonsingular for all α ∈ T ∩ H1(M ; Z) and s ≥ 0. Now consider an arbitrary γ ∈intT, γ 6= β. By convexity we may find v1, . . . , vd ∈ intT , d = dimH1(M ; R), so

that γ is in the interior of the d-simplex spanned by β, v1, . . . , vd. We may choose

v1, . . . , vd rational, say vj = 1Nαj , some N ∈ Z+, αj ∈ intT ∩ H1(M ; Z). We have

γ = t0β+

d∑

j=1

tjαj , with all tj > 0. By induction on k, we see that each β+

k∑

j=1

(tj/t0)αj

is nonsingular. Setting k = d and multiplying by t0 > 0, we see that γ is nonsingular

as well. Thus if one point β ∈ intT is nonsingular, all γ ∈ intT are nonsingular.

We will sharpen Thurston’s Theorem 14.6 in the case when m is atoroidal (con-atoroidal

tains no incompressible imbedded tori) andH1(M ; Z) 6= Z. We show (Theorem 14.11)

that a nonsingular face T (i.e. one containing a nonsingular class) of the unit ‖ ‖-ball

determines a canonical flow φt : M → M such that intT consists precisely of all [ω]

where ω is a closed 1-form with ω(∂φ∂t ) > 0. We must begin by relating the atoroidal

condition to Thurston’s classification of surface homeomorphisms.

DR

AFT

13 J

un 2

008


We suppose f : M → S1 is a fibration. Then flows ψt for which ddtf(ψtm) > 0 (we

will only consider flows having a continuous time derivative) determine an isotopy

class of surface homeomorphisms. For any k ∈ K = f−1(1), we consider the smallest

time T (k) > 0 for which ψT (k)(k) ∈ K . This map T (k) : K → (0,∞) is smooth (since

the flow lines of ψ are transverse to K) and the return map R(k) = ψT (k)(k) is a return map

homeomorphism. By varying ψ, we obtain an isotopy class of homeomorphisms of

the fiber K as return maps; this isotopy class will be called the monodromy of f and monodromy

denoted m(f).

We remark that the monodromy of f is determined algebraically by the cohomol-

ogy class β = f∗[dθ] ∈ H1(M ; Z), or equivalently by the map f∗ : π1M → π1S1. First

assume that β is indivisible. From the exact homotopy sequence 1−→π1K−→π1Mf∗−→π1S

1−→1,

we see that π1M is the semidirect product π1K ×α Z, where α is the outer automor-

phism of π1K determined by the monodromy of f . Thus π1K (= ker f∗) and α are

determined by f∗ alone. Clearly the topological type of K is determined by π1K ; but

Nielsen also showed that isotopy classes in Diff(K) correspond 1-1 to outer automor-

phisms of π1K . In general, β = nβ′ is a positive integer multiple of an indivisible

class β′, and n is determined by cokerf∗ = Z/nZ. We see that the fiber of f con-

sists of n copies of K (where π1K = ker f∗) which are permuted cyclically by the

monodromy. The nth power of the monodromy preserves K and acts on π1K by α

(the outer automorphism of ker f∗.) Thus we may unambiguously speak of the mon-

odromy of a nonsingular class β ∈ H1(M ; Z).

We say that the monodromy m(f) of a fibration f : M → S1 is pseudo-Anosov if pseudo-Anosov

the isotopy class has a pseudo-Anosov representative R. This representative is then

uniquely determined within strict conjugacy, that is for any two pseudo-Anosov rep- strict conjugacy

resentatives R0, R1 ∈ m(f) there will be a homeomorphism g isotopic to the identity

for which R0g = gR1.

Proposition 14.9. Suppose that H1(M ; Z) 6= Z. Given a fibration f : M → S1, M is

atoroidal precisely when the monodromy m(f) is pseudo-Anosov and the fibers of f are not

composed of tori.

Proof. SupposeM contains an incompressible torus S and let F be the foliation ofM

by the fibers of f . Again using the result of Thurston’s thesis discussed in the proof

of Theorem 14.6 [Rou73, Thu72], we may isotope S to either lie in a leaf of F or to

be transverse to F (since χ(S) = 0, the presence of saddle tangencies would force

there to be tangencies of other types.) If S does lie in a leaf, then the fibers of f are

composed of tori parallel to S. If the torus S is transverse toF , then one may define a

DR

AFT

13 J

un 2

008


flow ψ on M that preserves S and satisfies ddt (f ◦ ψt) = 1. Thus the return map

ψ1 : K → K, K = f−1(1), preserves the family of curves S ∩ K . Since S is incom-

pressible, each of those curves is homotopically nontrivial in K . If the monodromy

of f were pseudo-Anosov, these curves would grow exponentially in length under

iteration by ψ1. So we see that when m(f) is pseudo-Anosov and the fibers of f are

not unions of tori, then M must be atoroidal.

Conversely, when the fibers of f are unions of tori, these tori are essential. So we

assume the components of the fibers have higher genus and that the monodromy

is not pseudo-Anosov (hence reducible or periodic) and look for an incompress-

ible torus. If m(f) is reducible, we may construct ψ with ddt (f ◦ ψt) = 1 for which

ψ1 cyclically permutes a family of homotopically nontrivial closed curves C ⊂ K .

Then {ψtC} is an incompressible torus. If m(f) has period n, after Nielsen (see ex-

pose 11), we may choose ψ with ddt(f ◦ ψt) = 1 for which ψn = identity. Thus M

is Seifert fibered. One may easily compute that H1(M ; Z) ∼= Z2g+1, where g is the

genus of the topological surface which is the orbit space of ψ [Orl72]. As we assumed

H1(M ; Z) 6= Z, we must have a homologically nontrivial curve in this orbit space

which corresponds to an incompressible torus in M .

We may consider flows transverse to a fibration over S1 from three viewpoints.

The first is to begin with the fibration and produce transverse flows and an isotopy

class of return maps. The second is to begin with a homeomorphism R : K → K and

produce a fibration over S1 with fiber K and a transverse flow φ with return map

R. This is the well-known mapping torus construction, for which one sets X = K ×[0, 1]/(k, 1) = (R(k), 0), f : X → ([0, 1]/0 = 1) = S1 the natural fibration and defines

ψ to be the flow along the curves k × [0, 1] with unit speed. Clearly ψ1|K × 0 = R is

the return map of ψ, as desired. This flow ψ is called the suspension of R. The thirdsuspension

viewpoint is to begin with a flow ψ onX and to seek a fibration f over S1 to which ψ

is transverse—a fiber K is called a cross-section to ψ. Note that K and ψ determinecross-section

the return map R and an isotopy class of fibrations f .

In general, one has little hope of finding cross-sections, since many manifolds

don’t fiber over S1 at all. But there is a classification of the fibrations transverse to ψ

which is especially concrete in the case of interest to us now.

Suppose that some cross-section K to a flow φ has a return map R : K → K ad-

mitting a Markov partitionM = {S1, . . . , Sm} (see expose 10 — the case we need is

when R is pseudo-Anosov). There is a directed graph with vertices S1, . . . , Sm and

arrows Si → Sj for each i and j for which R(Si) meets int(Sj). A loop ℓloop

DR

AFT

13 J

un 2

008


for M is a cyclic sequence of arrows Si1 → Si2 → · · · → Sik → Si1 . Each loop ℓ

determines a periodic orbit forR and thus a periodic orbit γ(ℓ) for φ. If all of i1, . . . , ikare distinct, we call ℓ minimal. There are only finitely many minimal loops ℓ. minimal

We now discuss the classification and existence of cross-sections to flows. Given

a flow ψ on a compact manifold X there is a nonempty compact set of homology di-

rections Dψ ⊂ H1(X ; R)/R+, where the quotient space is topologized as the disjoint

union of the origin and unit sphere. A homology direction for ψ is an accumulation homology direction

point of the classes determined by long, nearly closed trajectories of ψ. We note that

whenK is a cross-section to ψ,K is normally oriented by ψ and so determines a dual

class u ∈ H1(X ; Z). Let CZ(ψ) = {u ∈ H1(X ; Z) | u is dual to some cross-section K to ψ}.

Theorem 14.10 ([Fri82b, Fri76]). CZ = {u | u(Dψ) > 0}. If φ, as above, has a cross-section

K and the return map R admits a Markov partition M, then CZ(φ) = {u | u(γ(ℓ)) >0 for all minimal loops ℓ forM}.

Thus CZ(ψ) consists of all lattice points in a (possibly empty) open convex cone

CR(ψ) = {u | u(Dψ) > 0} ⊂ H1(X ; R) − {0}. It follows easily from Theorem 14.10

that CR(ψ) = {[ω] | ω is a closed 1-form with ω(dψdt ) > 0}.Returning to our discussion of three-manifolds, we call a flow φ on M pseudo-

Anosov if it admits some cross-section for which the return map is pseudo-Anosov.

We now describe the cross-sections to pseudo-Anosov flows, and show they are

uniquely determined by their homotopy class among nonsingular flows on M .

Theorem 14.11. Suppose M fibers over S1. Then each flow ψ on M that admits a cross-

section determines a nonsingular face T (ψ) for the norm ‖ ‖ on H1(M ; R). Here T (ψ) =

{‖u‖ = −χψ⊥(u)} and ψ⊥ denotes the normal plane bundle to the vector field dψdt . One has

CR(ψ) ⊂ intT (ψ).

For any pseudo-Anosov flow φ on M , CR(φ) = intT (φ).

The face T (φ) (or the class χφ⊥) determines the pseudo-Anosov flow φ up to strict con-

jugacy. Thus any nonsingular face T on an atoroidal M with H1(M ; Z) 6= Z determines a

strict conjugacy class of pseudo-Anosov flows.

Proof. For u ∈ CZ(ψ), there is a cross-sectionK toψ dual to u. We have ‖u‖ = −χ(K),

by Proposition 14.4. Since the restriction ψ⊥|K is the tangent bundle of K , we have

−χ(K) = −χψ⊥(u). Thus −χψ⊥ is a linear functional on H1(M ; R) that agrees with

‖ ‖ on C Z(ψ) and the first paragraph of Theorem 14.11 is shown.

DR

AFT

13 J

un 2

008


We now observe

Lemma 14.12. Any cross-section K to a pseudo-Anosov flow φ on M will have pseudo-

Anosov return map RK .

Proof. By definition, there is some cross-section L to φ with pseudo-Anosov return

map RL, but K and L will generally not be homeomorphic (one calls return maps

to distinct cross-sections flow-equivalent). In any case, any structure on L invariantflow-equivalent

under RL is carried over to a structure on K invariant under RK under the system

of local homeomorphisms between K and L determined by φ. This shows that RKpreserves a pair of transverse foliations FuK and FsK with the same local singularity

structure as a pseudo-Anosov diffeomorphism.

We now show that the closure P of any prong P of FuK or FsK is the component

K0 of K which contains P . By passing to a cyclic cover Mn → M determined by

the composite homeomorphism π1M → (π1M/π1K0) ∼= Z → Z/nZ and restricting

to the cross-section K0 ⊂ Mn we may assume that K is connected and that RKleaves P invariant (choose n so that P is invariant under RnK0

). Consider the closed

RL invariant subset {φtP} ∩ L = I . Since I contains the closure of a prong for the

pseudo-Anosov diffeomorphism RL, we know that I is dense in some component

L0 ⊂ L. As L0 is a cross-section to φ, we find that {φtP} = M . As P is RK invariant,

we find P = K as desired.

Similarly we can check that the foliations FuK and FsK have no closed leaves.

It follows by the Poincare–Bendixson theorem that each leaf closure contains a

singularity, and thus a prong. So we find that all leaves of FsK and FuK are dense in

their component of K .

We may see from this density of leaves and the fact that the local stretching and

shrinking properties of RK are the same as those of RL that the Markov partition

construction of expose 10 works for RK . (It is easiest to construct birectangles for

RK by “analytic continuation” from immersed birectangles in L. This makes sense

because K and L have the same universal cover.) As in the Anosov case [RS75], the

Parry measures for the one-sided subshifts of finite type associated toM push for-

ward to give transverse measures on FuK and FsK that transform underRK by factors

λ−1K and λK , for some λK > 1. As leaves are dense, these measures have positive

values on any transverse interval but vanish on points. Thus RK is pseudo-Anosov.

Now suppose that φ1 and φ2 are pseudo-Anosov flows on M for which CR(φ1)

DR

AFT

13 J

un 2

008


intersects CR(φ2). Then we may choose u ∈ CR(φ1) ∩ CR(φ2) ∩ H1(M ; Z) and find

fibrations fi : M → S1 with ddt (fi ◦ φit) > 0 and u = [f∗

i (dθ)], i = 1, 2.

As discussed earlier, u determines m(fi). This gives a homeomorphism h : M →M such that f1 ◦ h = f2 where h acts on π1M by the identity. Thus h is isotopic to

the identity [Wal68]. Hence, by this preliminary isotopy, we assume f1 = f2 = f and

denote the fiber by K .

Each φi determines a return map Ri : K → K . By the lemma above, these Ri are

pseudo-Anosov. Since the mapsRi are in the same isotopy class h(f), they are strictly

conjugate by the uniqueness of pseudo-Anosov diffeomorphisms (expose 12).

Now suppose that gR1 = R2g, with g isotopic to the identity. Then the map

C0 : M → M defined by C0(φ1sk) = (φ2

sgk), k ∈ K , 0 ≤ s ≤ 1, is a homeomorphism

conjugating flows φ1 and φ2 and f ◦ C0 = f . As C0|K = g is isotopic to the identity,

C0 may be isotoped to C1 where f ◦ Ct = f , for t ∈ [0, 1] and C1 fixes K . Since

Diff K is simply connected [Ham66], we may isotope C1 to the identity C2 (through

Ct satisfying f ◦ Ct = f , t ∈ [1, 2]).

We have shown so far that if φi are pseudo-Anosov flows, i = 1, 2, then either

CZ(φ1) equals CZ(φ2) or is disjoint from it, since conjugating a flow by conjugacy

isotopic to the identity doesn’t affect CZ. It follows easily that the open cones CR(φ1)

and CR(φ2) are either disjoint or equal.

Now suppose that φ is pseudo-Anosov but CR(φ) is a proper subcone of intT (φ).

By Theorem 14.10, CR(φ) is defined by linear inequalities with integer coefficients,

and so there is an integral class u ∈ intT ∩ ∂CR(φ). Then u is nonsingular (Theo-

rem 14.6), the fibration corresponding to u has pseudo-Anosov monodromy (Propo-

sition 14.9) and one obtains an Anosov flow ψ with u ∈ CR(ψ). This shows that CR(ψ)

and CR(φ) are neither disjoint nor equal, contradicting the previous paragraph.

Thus we see that pseudo-Anosov flows satisfy C(φ) = intT (φ).

Theorem 14.11 shows that pseudo-Anosov maps satisfy an interesting extremal

property within their isotopy class. Suppose h0 : K → K has suspension flow ψ0t : M →

M , where we takeK connected and dual to the indivisible class u ∈ H1(M ; Z). Given

an isotopy ht starting at h0, we may deform ψ0 through flows ψt with cross-section

K and return map ht. We regard u−1(1) as a subset of H1(M ; R)/R+ and note that

we always have Dψt ⊂ u−1(1). By the Wang exact sequence:

H1(K; R)h0∗−Id−→ H1(K; R)−→H1(M ; R)

u−→R−→0,

we may identify u−1(1) with u−1(0) = coker(h0∗ − Id) by some fixed splitting of u.

Whenever hs = ht, the simple connectivity of Diff K [Ham66] implies that Dψs =

Dψt .

DR

AFT

13 J

un 2

008


Thus we may unambiguously associate a set of homology directionsDh ⊂ coker(h0∗−Id) to homeomorphisms h isotopic to h0. Now assume h0 is pseudo-Anosov. By The-

orem 14.11, we have CR(ψs) ⊂ intT (ψs) = intT (ψ0) = CR(ψ0). Thus we find, using

Theorem 14.10, that the convex hull ofDhs (which may be identified with the asymp-

totic cycles of ψs in this situation [Fri82a, Sch57]) always contains the convex polygon

determined at s = 0. Thus we may say that pseudo-Anosov diffeomorphisms have

the fewest generalized rotation numbers in their isotopy class.

We may analyze the topological entropy of the return-maps RK of the various

cross-sectionsK to a pseudo-Anosov flow C. We parameterize these cross-sections K

by their dual classes u ∈ H1(M ; Z) and define h : CZ(φ)→ (0,∞) by h([K]) = h(RK),

the topological entropy of RK . We showed in [Fri82a] that 1/h extends uniquely to

a homogeneous, downwards convex function 1/h : CR(φ)→ [0,∞] that vanishes ex-

actly on ∂CR(φ). Thus h(u) may be defined for all u ∈ H1(M ; R) in a natural way.

The smallest value of h on intT ∩ {‖u‖ = 1} defines an interesting measure of the

complexity of φ (or equivalently, by Theorem 14.11, of the face T = T (φ)). The in-

tegral points at which h is the largest give the “simplest” cross-section to the flow φ

(see [Fri82a]).

If one is given a pseudo-Anosov diffeomorphism h : K → K and a Markov par-

titionM for h, Theorems 14.10 and 14.11 give an effective description of the nonsin-

gular face T determined by the suspended flow φt : M → M of h, in terms of the

orbits corresponding to minimal loops. As the computation of minimal loops in a

large graph is difficult, we observe that there is a more algebraic way of usingM to

obtain a system of inequalities defining T . (We refer the reader to [Fri82a] for details,

where we used this method to construct a rational zeta function for axiom A and

pseudo-Anosov flows.) For sufficiently fine M, we may associate to M a matrix A

with entries in H1(M ; Z)/torsion = H . The expression det(I − A), regarded as an

element in the group ring of the free abelian group H , may be uniquely written as

1+∑aigi, gi ∈ H−{0}, ai ∈ Z−{0}, gi distinct. Then T is defined by the inequalities

u(gi) > 0.

To illustrate Thurston’s theory, it is convenient to work on a bounded M3. The

norm considered above can be extended to such M by omitting spheres and discs

before computing the negative Euler characteristic. One should restrict to the case

where ∂M is incompressible, and then Theorems 14.2 and 14.6 and proposition 14.4

extend [Hem76, Thu86].

DR

AFT

13 J

un 2

008


h

α

γ

β

Figure 14.1:

We let K be the quadruply connected planar region and h the indicated compos-

ite of the two elementary braids (Figure 14.1) which fixes the outer boundary compo-

nent. We will let M be the mapping torus of h and compute ‖ ‖. Rather than finding

a pseudo-Anosov map isotopic to h, which would only help compute one face, we

will instead compute ker(u : π1M → Z) for several indivisible u ∈ H1(M ; Z). When

this kernel is finitely generated, Theorem 14.2 shows u is nonsingular and Proposi-

tion 14.4 enables us to compute ‖u‖. From a small collection of values of ‖ ‖, Theo-

rem 14.6 allows us to deduce all the others, indicating the existence of nonsingular

classes that would be hard to detect using only Theorem 14.2.

We first compute π1M = π1K ×Z. Writing π1K as the free group on the loops

α, β and γ shown in the diagram, we find:

π1M =⟨α, β, γ, t

∣∣∣ t−1αt = γ, t−1βt = γ−1αγ, t−1γt = (γ−1αγ)β(γ−1αγ)−1⟩

=⟨α, β, γ, t

∣∣∣ t−1αt = γ, t−1βt = γ−1αγ, γβt = βtβ⟩

=⟨γ, t

∣∣∣ (tγ−1tγt−1γ)2 = γ(tγ−1tγt−1γ)t⟩.

Abelianizing givesH1(M ; Z) = Zγ⊕Zt. Suppose u ∈ H1(M ; Z) is indivisible, so that

a = u(γ) and b = u(t) are relatively prime. The Reidemeister–Schreier process gives

a presentation for ker(u : π1M → Z) (essentially by computing the fundamental

DR

AFT

13 J

un 2

008


group of the infinite cyclic covering corresponding to u) which is very ungainly for

large a. When a = 1, one finds the relatively simple expression:

keru =⟨ti

∣∣∣ ti ti+b−1 t−1i+b ti+b+1 ti+2b t

−1i+2b+1 = ti+1 ti+b t

−1i+b+1 ti+b+2

⟩.

For b > 1, this relation expresses ti in terms of ti+1, . . . , ti+2b+1 and expresses ti+2b+1

in terms of ti, . . . , ti+2b. Thus keru is free on t1, . . . , t2b+1. Similarly, if b < −1, then

keru is free on t1, . . . , t1−2b and if b = 0, then keru is free on t1, t2, t3. If b = ±1,

however, one may abelianize and obtain (keru)ab = Z[t, t−1]/(2t3 − 3t2 + 3t − 2)

which maps onto the collection of all 2n th roots of unity, and so keru is not finitely

generated.

By Theorem 14.2 (Stallings), there is a fibration for u = (1, b) when b 6= ±1, with

fiber Ku satisfying π1(Ku) = keru. By Proposition 14.4, ‖u‖ = −χ(Ku), which is

clearly

−1 + rank(H1(Ku)) =

{|2b|, b > 1, b ∈ Z2, b = 0.

.

We will see that these values determine ‖ ‖ completely. Using the dual basis to

(γ, t), we know that:

‖(1, b)‖ =

{|2b|, b > 1, b ∈ Z2, b = 0.

.

But ‖(1, b)‖ is a convex function f of b by Theorem 14.3 and it takes integer values

at integer points. By convexity, f(1) must be 2 or 3. Were f(1) = 3, convexity would

force

f(x) =

{2 + x for 0 ≤ x ≤ 2

2x for x ≥ 2

and then (1, 2) would not lie in an open face of the unit ball, contradicting Theo-

rem 14.6. Thus one must have f(1) = 2, and likewise, f(−1) = 2. By convexity,

we find f(x) = max(|2x|, 2). Homogenizing shows ‖(a, b)‖ = max(|2a|, |2b|), i.e.

‖u‖ = max(|u(2γ)|, |u(2t)|).By Theorem 14.6, u ∈ H1(M ; R) is nonsingular ⇐⇒ |u(γ)| 6= |u(t)|.This example embeds in a larger one, constructed with the mapping torus M0

of the transformation h3 (M0 is a triple cyclic cover of M ). H1(M0; Z) is free abelian

on α, β, γ, t, so there is a norm on H1(M0; R) whose restriction to H1(M ; R) ∼= {u ∈H1(M0; R) | u(α) = u(β) = u(γ)} is 3‖ ‖. We leave its computation as an exercise.

DR

AFT

13 J

un 2

008

Expose 15

Presentation of the group of

diffeotopies of a compact

orientable surface

1

(A proof of a theorem of A. Hatcher and W. Thurston)

by F. Laudenbach2 (oral exposition by A. Marin)

15.1 Introduction

Let M be a closed compact surface3 of genus g and let G be the group of diffeotopies

of M (the elements of G are the isotopy classes of diffeomorphisms of M preserving

the orientation). Let C be a simple curve of M with g components; we will say that

C is a marking of M if M − C is connected; since g is the genus of M , the compact

manifold with boundary obtained by cutting M along C is the disk ∆ with (2g −1) holes. The group G acts transitively on the right on the set of isotopy classes of

markings by:

C −→ ϕ−1(C)

1In modern terminology, this group is called the mapping class group2I thank A. Marin for his oral exposition and for the clarifications that it brought to me on the work of

Hatcher and Thurston3The case with boundary can be treated in an analogous fashion

283

DR

AFT

13 J

un 2

008


where ϕ is a diffeomorphism of M . Let us choose a base marking C0 and let us de-

note by H the subgroup of G stabilizing C0. The group H is finitely presented: this is

determined by decomposing H into three groups: the pure braid group on (2g − 1)

strands,

DR

AFT

13 J

un 2

008


the group of permutations of components of C0, and the group generated by the

Dehn twists along each of the curves.

A. Hatcher and W. Thurston [HT80] have given a presentation of G modulo H ;

precisely, they have constructed an element σ ofG and words µ1, . . . , µq whose letters

belong to {σk : k ∈ Z} ∪H with the following properties:

1o H and σ generate G;

2o for i = 1, . . . , q, the element mi of G represented by µi belongs to H ;

3o the words µim−1i generate the relations of G, that is, conjugates of these ele-

ments generate the kernel of the natural homomorphism H ∗ Z → G associated to

σ.

In fact, even if one knows a presentation of H , this only says that there exists a

presentation of G, but does not give it unless one knows how to calculate the mi.

It is true that the words µi are given by simple geometric constructions and that a

diffeomorphism of ∆ is entirely determined up to isotopy if one says what it does to

some arcs. Thus, with enough courage, it is possible to make the “implicit relations”

of Hatcher–Thurston into explicit ones.

Although the oral exposition of A. Marin reported faithfully on this work, it

seems inappropriate to copy an article that has appeared. Instead, we will try to

make the arguments of Hatcher–Thurston a little more conceptual; we will see for

example in the proof of Lemma 15.4 a geometric fact particular to dimension 2 that

contributes in an essential way to the finiteness. To simplify, we have chosen not to

guve an explicit presentation of G, except in the case of the torus.

15.2 A method for presenting G

15.2.1

LetX be a simply connected polyhedral complex of dimension 2 (possibly not locally

finite), in which each edge and face is determined by its vertices. Let x0 be a base-

point, let A be the set of edges containing x0, and let F be the set of faces containing

DR

AFT

13 J

un 2

008


x0. Suppose that the groupG acts cellularly onX on the right, thatG acts transitively

on the 0-skeletonX [0] and thatH is the stabilizer of x0; thenH acts onA and F on the

right. We suppose thatA/H and F/H are finite and we choose sets of representatives

of each orbit, a1, . . . , ap, and f1, . . . fq .

Generators

We choose σ1, . . . , σp ∈ G so that the two endpoints of ai are (x0, x0σi). Then the end-

points of any edge can be written (x0g, x0σihg), g ∈ G, h ∈ H . A word σikhk · · ·σi1h1

describes a path of edges leaving from x0 and passing successively through x1 =

x0σi1h1, x2 = x0σi2h2σi1h1, etc. (see Figure 15.1).

b

ef

a

d

c

Figure 15.1:

fix labels in fig-

ure. psfrag? Every path of edges has such a description. The connectedness of X implies that

σi, . . . , σp generate G. Note that a word represents a loop if and only if the product

of the letters belongs to H .

Relations

1) Backtracking: in expressing (x0, x0σi, x0) as a loop, we obtain a relation; that is,

there exists an integer j ∈ [1, p] and h ∈ H such that:

σjhσi ∈ H.

DR

AFT

13 J

un 2

008


2) Different writings of the same edge: one must calculate the stabilizer Ti of the

edge (x0, x0σi) and, for each t ∈ Ti, write

(x0, x0σi)t = (x0, x0σi)

that is

σitσ−1i ∈ H.

3) The boundary of each face fi, i = 1, . . . , q gives a relation.

To see that one thus obtains a presentation of G modulo H in the sense of Sec-

tion 15.1, it suffices to recall that any homotopy of a loop of edges of X is formed

from the following elementary operations: insertion or deletion of the boundary of a

face or of a backtrack. We have therefore shown the following.

Proposition 15.1. If H is finitely presented and if the stabilizers of edges are of finite

type4, then G is finitely presented.

15.2.2

The objective now is to find a complexX on which the group of diffeotopies acts. The

first one that one thinks of is the nerve N of the space of C∞ real valued functions

(of codimension ≤ 2) on M , given by its natural stratification5 [Cer70]; N is simply

connected, but G does not act transitively on N [0]. We can try only watching the

codimension 1 strata that correspond to essential crossings (see Section 15.3.1); we

then find a simply connected nerve where G acts transitively on the vertices, but the

stabilizer of a vertex is bigger thanH and seems difficult to study. We are nevertheless

going to utilize these ideas to exhibit the set of isotopy classes of markings as the 0-

skeleton of a complex whose simple connectivity follows from that of N . The lemma

below is utilized for this purpose.

15.2.3

If Y and Z are two connected complexes, we will say that π : Y → Z is cellular if the

following conditions are satisfied.

4We say that a group is of finite type if it has an Eilenberg–MacLane space with finitely many cells.5The space of C∞ functions with isolated critical points admits a natural stratification. The codimension 0

stratum consists of points where the critical values are all distinct, the codimension 1 stratum consists of

points where exactly two critical values coincide, etc.

DR

AFT

13 J

un 2

008


1) for each cell σ of Y , π(σ) is a cell of Z ;

2) π|int σ is a fibration on its image (intσ denotes an open cell).

For example, if σ is a 2-cell, and π(σ) is a point; or π(σ) is an edge and ∂σ is the

union of four edges τ1, τ2, τ3, τ4, with π(τ1) = 1 point, π(τ3) = 1 point, π(τ2) = π(σ) =

π(τ4); or π(σ) is a 2-cell and π|∂σ is degree 1 onto its image. We very easily obtain the

following.

Lemma 15.2. Let π : Y → Z be a cellular map in the above sense. We suppose that

1) for each x ∈ Z [2] − Z [1], π−1(x) is nonempty ;

2) for each x ∈ Z [1] − Z [0], π−1(x) is connected; and

3) for each x ∈ Z [0], π−1(x) is simply connected.

Then π1(Z) = 0 implies π1(Y ) = 0.

The converse is true as long as π−1(x) is connected for all x ∈ Z [0] and π is surjective

on the 1-skeleton.

15.3 The cell complex of marked functions

15.3.1

We consider the space F of C∞ functions on M , of codimension ≤ 2, with the action

by Diff(M)×Diff(R), and the nerveN of F stratified by the orbits.G acts on the right

by the formula: f 7→ f ◦ ϕ, where f ∈ F , ϕ ∈ Diff(M).

Two functions are said to be isotopic if they are in the same orbit of the identity

component of this group.

To any function f ∈ F , we can associate its graph of level sets Γ(f): the projection

M → Γ(f) identifies two points if they belong to the same connected component of

a level set of f ; Γ(f) is a complex of dimension 1. If f is

DR

AFT

13 J

un 2

008


generic, the Betti number β1(Γ(f)) is equal to the genus g ofM . If f is of codimension

1 and β1(Γ(f)) = g − 1, we say that f belongs to a stratum of essential crossings. Fig-

ure 15.2 shows the critical level set at a crossing as well as the two neighboring level

sets; Figure 15.3 shows the graphs of the functions on a path crossing the stratum.

Figure 15.2:

Figure 15.3:

An edge of N , dual to a stratum of an essential crossing, is said to be of the 1st

type. The other edges are of the 2nd type. A face of N is said to be principal if it is dual principal

to a stratum of equality of 3 critical values belonging to 3 strata of essential crossings;

such a face is a hexagon that alternates 3 edges of the 1st type and 3 edges of the 2nd

type. Figures 15.4, 15.5, and 15.6, respectively, show the (immersed) level sets of a

DR

AFT

13 J

un 2

008


principal function of codimension 2, the corresponding stratification in the space of

functions and the graph of an unspecified neighboring generic function.

DR

AFT

13 J

un 2

008


Figure 15.4:

Figure 15.5:

put labels in fig-

ures

15.3.2

We say that (f, C) is a marked function if C is a marking of M where each component

is contained in a level set of f . Each component of the marking corresponds to an

edge of Γ(f) and the complement of these (open) edges is a maximal subtree. All

generic functions admit a marking, but a function of codimension 1 or 2 belonging to

a stratum of essential crossings only admits an incomplete marking (n−1 components).

For example, if f belongs to a stratum of an essential crossing, we can mark f by

simple curves α1, . . . , αn−1; if we mark the neighboring generic functions f ′ and f ′′,

on both sides of the stratum, by (α1, . . . , αn−1, α′) and (α1, . . . , αn−1, α

′′) respectively,

then the minimal intersection of α′ and α′′ is one point.

We put the following isotopy relation on marked functions: (f, C) is isotopic to

(f ′, C′) if f is isotopic to f ′ and C is isotopic to C′. [This relation is

DR

AFT

13 J

un 2

008


Figure 15.6:

less fine than the relation of isotopy of pairs.] The cell complex Y of marked functions

is constructed with the set of isotopy classes of marked functions as the 0-skeleton.

We have a projection

π : Y [0] → N [0]

by forgetting the marking. The fiber above [f ] ∈ N [0] is formed of all the markings,

up to isotopy, of f .

Lemma 15.3. There exists a bound, independent of f , for the cardinality of Y [0] ∩ π−1([f ]).

Proof. The graph Γ(f) collapses onto an (uncollapsible) reduced subgraph Γred(f).

The number of markings, up to isotopy of f , coincides with the number of markings

of Γred(f); indeed, if two markings C1 and C2 of f mark Γ(f) on both sides of the foot

of a collapsible tree (see Figure 15.3.2), then C1 and C2 are isotopic.

The lemma follows from the fact that, the Betti number being fixed, there are only

a finite number of reduced graphs up to PL isomorphism.

We remark that the lemma is not true if we endow the marked functions with the

finer isotopy relation.

The group G acts on Y [0] on the right by (f, C) → (f ◦ ϕ,ϕ−1(C)). This action is

not transitive. The projection π is equivariant.

DR

AFT

13 J

un 2

008


*C1

C2

Figure 15.7:

15.3.3 The 1-skeleton

Edges of the 1st type: Let y0 and y1 be two vertices of Y represented by (f0, C0) and

(f1, C1), where C0 and C1 have (n − 1) components in common. To each edge of the

1st type of N joining [f0] to [f1] (unique if it exists)6, we associate an edge of Y , said

to be of the 1st type, between y0 and y1.

Observe that if such an edge exists, the incomplete marking common to C0 and

C1 necessarily marks the function of codimension 1 from the essential crossing; the

distinct components intersect in one point.

Edges of the 2nd type: Let (f0, C) and (f1, C) be two vertices having the same mark-

ing. Let ft, t ∈ [0, 1], be a path representing an edge τ of the second type in N . If, up

to isotopy, C is a marking of ft for all t, we lift τ to an edge, said to be of the 2nd type,

from (f0, C) to (f1, C).

Edges of the 3rd type: We join by an edge, said to be of the 3rd type, each pair of

distinct vertices of π−1([f ]).

The projection π and the action of G extends naturally to Y [1].

15.3.4 The 2-skeleton

Faces of type I: By examining the geometric models associated to each stratum of

codimension 2 of the space of functions (see [Cer70]), we verify that, for each face σ′

6If there were two edges of the 1st type from [f0] to [f1], we would have for f1 two markings C1 and C′

1

such that i(C1, C′

1) 6= 0, which is absurd (here i(., .) is intersection in the sense of Thurston; see Expose

4). To see this, we utilize the classification of crossings of critical values, due to J. Cerf [Cer70].

DR

AFT

13 J

un 2

008


of N , there exists a loop γ of Y [1] such that π|γ is an isomorphism of γ onto ∂σ′.

DR

AFT

13 J

un 2

008


This remark being made, we choose one face σ′ in each class moduloG; we choose

one lift γ of ∂σ′ satisfying the preceding condition; we attach to γ one 2-cell σ and we

saturate by the action of G.

Note that condition (1) of Lemma 15.2 is satisfied.

Faces of type II: Let τ and τ ′ be two edges of Y lifting the same edge of the 1st or 2nd

type of N . In joining their endpoints in the fibers of π, we form a square or a triangle,

onto which we attach a face σ. We extend π to σ, with values in π(τ) = π(τ ′), in such

a way that π is cellular.

Condition (2) of the lemma is now satisfied.

Faces of type III: To each triangle of the fiber of π, we attach a face; by brute force;

this makes the fibers π−1([f ]) simply connected for each [f ] ∈ N [0].

Finally, we have constructed Y , which admits a cellular action by G, and which

is simply connected by Lemma 15.2.

Remark. We could have skipped the edges of 3rd type and, by consequence, the

faces of types II and III; in this language, Hatcher and Thurston would only have

put an edge between (f, C0) and (f, C1) if C0 and C1 have (n − 1) components in

common. The advantage of their restrained system is to obtain relations in G that are

all carried by a surface of genus 2 with boundary.

15.4 The marking complex

15.4.1 Construction

The 0-skeleton X [0] is formed from isotopy classes of markings of M . We have an

equivariant projection

P : Y [0] → X [0]

DR

AFT

13 J

un 2

008


by forgetting the function. We recall that the group G acts transitively on X [0] with

H as stabilizer.

Two distinct markings C0 and C1 are joined by an edge whenever there exist

marked functions (f0, C0) and (f1, C1) joined by an edge of Y [1]. The action of G

extends toX [1] and the projection P is extended equivariantly to Y [1]. For example, if

(f0, C) and (f1, C) are connected by an edge (necessarily of the 2nd type), its projection

is a point.

We attach a 2-cell σ to a loop γ of X [1] if there exists a loop γ of Y [1] such that:

1) γ = ∂σ, σ ∈ Y [2],

2) The restriction of P to γ is degree 1 onto γ.

It is then easy to extend P to a homeomorphism int σ → intσ. Further, G acts on X .

By examining the types of faces, we see right away that, for each face σ of Y ,

either P (∂σ) is an edge or a point, or P (∂σ) is a loop in X [1] and P |∂σ is of degree 1

onto its image (this is for example the case for the lifts in Y of the principal faces of

N ). It is then immediate to extend P to Y .

The projection π sends P−1([C]) injectively to the nerve of a convex open set of

the space of functions, namely the open set of functions that admit C for a marking.

Thus P−1([C]) is surely connected (similarly simply connected). Thus X is simply

connected (Lemma 15.2).

15.4.2 Finiteness

Let C0 be a base marking.

Lemma 15.4. Let f be a function marked by C0. The set of cells of Y passing through (f, C0)

projects to a finite subset of the set of cells of X modulo H .

DR

AFT

13 J

un 2

008


Proof. Let Gf be the stabilizer of π(f, C0) = [f ]. We will prove in fact a stronger

lemma where we consider all the cells meeting π−1([f ]) and where we replace H by

the subgroup Gf ∩H .

By Lemma 15.3, there is only a finite number, uniformly bounded, of cells of Y

above a cell of N ; moreover, applied to the 0-cells, this says that Gf ∩ H is of finite

index inGf . The lemma is therefore reduced to the statement that, inN , there are only

a finite number of cells passing through f modulo Gf . This fact does not correspond

to a general property of the stratification of the space of functions on a manifold. In

dimension 2, it suffices to prove it for edges of crossings (double critical values) and

the faces of triple critical values; because the cells passing through [f ] that are dual to

the singularities “at the origin” are finite in number. The general fact is that a dual cell

to a stratum of equality of two or three values is determined by a system of sheets7

adapted to f (see [Cer70]). But precisely, for surfaces, the Dehn twists along curves of

level sets of f represent elements of Gf and acts on the system of sheets adapted to

f , with finitely many orbits.

Definitions. We use the term “small loop” for a loop of Γ(f) that is not null-homotopic

and only passes through 2 branch points.

If m is a local maximum (resp. minimum) of f , we denote by d(m) the minimal

number of edges that one must traverse in order to descend (resp. climb) from the

vertex of Γ(f) corresponding to m to a “small loop”; if there are no small loops, d(m)

is not defined.

A function is said to be minimal if it has only one local maximum and one local

minimum.

A Morse function f , with distinct critical values, is said to be almost minimal in

either of the following cases.

a) Γ(f) does not have a “small loop” (in this case, f is minimal);

b) Γ(f) has at least one “small loop”, and, for each non-absolute extreme

7In Cerf’s original paper, which is in French, the term for sheet is “nappe”.

DR

AFT

13 J

un 2

008


m, we have d(m) ≤ 2.

We remark that the graphs of the almost minimal functions form a finite set. The

almost minimal functions are important because, in general, there are no principal

faces of N passing through a given minimal function.

Lemma 15.5. There exists only a finite number of isotopy classes of almost minimal functions

marked by C0.

Proof. Starting from a minimal function, we can only give birth to a finite number

of pairs of critical points if we want to stay in the space of almost minimal functions;

up to isotopy, there are only a finite number of possible choices for each birth. As

every almost minimal function is obtained by this process starting from a minimal

function, it suffices to prove the lemma for minimal functions. In fact, we are going

to prove that if, in addition to the marking, one is given the graph, endowed with its

height function, and the position of the marking on this graph, then the function is

determined up to isotopy.

For this, we must locate the figure eight critical levels in the disk ∆ obtained

by cutting M along the marking. The marked graph indicates which holes of ∆ are

surrounded by each loop of the figure eight. Thus, starting from the lowest level set,

the critical curves are placed one after the other, in a way that is unique up to isotopy.

From this, the lemma is clear.

If f , marked by C0, is not almost minimal, Γ(f) contains an edge α with a free

endpoint m such that d(m) is either undefined or is greater than 2. Collapsing α

amounts to the elimination of two critical points of f . Let f ′ be the endpoint of this

path.

Lemma 15.6. For each cell σ of Y passing through (f, C0), there exists a cell σ′ passing

through (f ′, C0) such that P (σ) = P (σ′).

DR

AFT

13 J

un 2

008


As an immediate corollary of these three lemmas, we obtain that, modulo H ,

there are only finitely many cells of X passing through C0.

Proof. Since the edge that is collapsed when we pass from f to f ′ is found far from

the “small loops”, the elimination is independent of all the essential crossings or

changes of marking that one can do starting from (f, C0). From this, the lemma is

clear. Note that if σ moves around the saddle corresponding to the branching point

of α, then dimP (σ) < dim(σ) and we take σ′ with dimσ′ = dimP (σ).

Theorem 15.7. The group G of diffeotopies of a surface is finitely presented.

Proof. To apply Proposition 15.1, it remains to prove that the stabilizers of edges are

of finite type.

Let C0 and C1 be two markings of M that we choose in their isotopy classes with

the minimal number of points of intersection. Let H0 and H1 be the stabilizers of [C0]

and [C1] inX . If the two vertices are joined by an edge,H0∩H1 is the stabilizer of the

edge. By Proposition 3.13, this group is identified with the connected components of

the group of diffeomorphisms ofM leavingC0 andC1 invariant. Up to permutations,

it is related to the group of diffeomorphisms of a certain disc with holes, which is

therefore of finite type.

15.4.3 The case of the torus T 2

Proof. We have here the simplification that a function only admits a single marking

up to isotopy. Therefore:

Y ∼= N.

By the Lemmas of Section 15.4.2, we obtain the classes of cells passing through

[C0] in X in the following way: we consider a marked function (f0, C0) that is almost

minimal, that is, where the graph Γ(f0) looks like Figure 15.8; we consider

DR

AFT

13 J

un 2

008


an essential crossing starting from f0 (unique modulo Gf0 ) that we determine by

choosing a curveC1 intersecting C0 in one point; we consider a principal face passing

through this edge that is determined by choosing a curve C2 such that C1∩C2 = 1 pt

and C0 ∩ C2 = 1 pt. In the particular case of the torus, we find that, C0 and C1 being

fixed, there are exactly two possibilities for C2 up to isotopy (Figure 15.9), denoted

respectively C′2 and C′′

2 . Thus modulo H , X has in [C0] an edge (C0, C1) and two

triangles (C0, C1, C′2), (C0, C1, C

′′2 ).fig 15.08 should

be here instead

of 15.07

c1*

c2

Figure 15.8:

We denote by σ the 90o rotation of the torus about the point where C0 and C1

coincide; we have C1 = C0σ,

σ2 ∈ H. (1)

Let ρ be the twist parallel to C0, so that C′2 = C1ρ.

Then, we haveC′′2 = C1ρ

−1. Finally, the geometry of the torus implies that σ takes

the edge (C0, C′′2 ) to the edge (C1, C

′2). Thus the path (C0, C1, C

′2) is described by the

word σρ−1σ, whereas the edge (C0, C′2) is described by σρ. This gives the relation

σρ−1σρ−1σ ∈ H. (2)

We see immediately that the other cell gives

σρσρσ ∈ H. (3)

But (3) follows from (2) and (1). Finally the stabilizer of an edge is trivial; thus

DR

AFT

13 J

un 2

008


we have written a complete system of relations modulo H . To completely determine

the relation (2), we calculate the effect of the written element on C1; it takes C1 onto

C′2; thus (σρ−1)3 stabilizes the edge (C0, C1). Thus

(σρ−1)3 = 1.

Finally, H is generated by ρ and σ2, with the commutation relation [σ2, ρ] = 1.

fig 15.09 should

be here instead

of 15.07

c1*

c2

Figure 15.9:

Final remark. In [McC75], McCool has given a purely algebraic proof of the theo-

rem. In [Bir77], Joan Birman, who was the first to give an explicit presentation in the

case of a surface of genus 2, suggests that it seems difficult to exhibit a presentation

from the proof of McCool.

On the other hand, there is an approach using algebraic geometry (see [Mar77]).

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008

Appendix A

The “pair of pants”

decomposition of a surface.

by A. Fathi

In the first part, we will give a proof of the inequality used to prove Theorem 4.4. In

the second part, we will apply this inequality to case where an entire pants decom-

position of the surface M gets twisted, instead of a single curve.

First part

We have a system α0, . . . , αk of mutually disjoint simple closed curves on M .

Also, γ is a simple closed curve whose intersection with each αj is minimal (among

curves isotopic to γ). We are given positive integers nj . We construct Γ by acting on γ

by a positive twist along αj , for j = 0, . . . , k (the notion of positive twist only depends

on the orientation of M ).

Proposition A.1. For each simple curve β, we have the formula:

∣∣∣∣i([Γ], [β])−∑

j

nji([γ], [αj ])i([αj ], [β])

∣∣∣∣ ≤ i([γ], [β])

where [ ] means “isotopy class of”.

Proof. The curve Γ coincides with γ outside of tubular neighbourhoods of the αj .

The position of Γ and γ at the endpoints of a common arc is given in Figure A.1.

Thus Γ is approximated by a curve denoted by Γ′ that

303

DR

AFT

13 J

un 2

008


γΓ

Γ γ

Figure A.1:

crosses each interval of Γ ∩ γ once. This is due to the fact that all of the twists are

positive. By the criterion of Proposition 3.10, we check that card(γ ∩ Γ′) = i([γ], [Γ]).

We observe that γ∪Γ′ is the image of a continuous map, defined on∑

j

nji([γ], [αj ])

copies of S1, with nji([γ], [αj ]) copies of S1 going to the free homotopy class of [αj ].

Thus, we have the inequality:

card(β ∩ (γ ∪ Γ′)) ≥∑

j

nji([γ], [αj ])i([αj ], [β]).

If β does not pass through the points of intersection of γ with Γ′, we have

card(β ∩ (γ ∪ Γ′)) = card(β ∩ γ) + card(β ∩ Γ′).

If we take for β a geodesic of a metric of curvature−1 for which γ and Γ′ are geodesics

(such a metric exists by Proposition 3.10), we have:

card(β ∩ (γ ∪ Γ′)) = i([Γ], [β]) + i([γ], [β]),

which gives one of the desired inequalities.

It remains to prove that

i([Γ], [β]) ≤∑

j

nji([γ], [αj ])i([αj ], [β]) + i([γ], [β]).

Here, we use the representative Γ rather than Γ′. We chose β to be in minimal position

with respect to the αj and to not pass through the points of intersection of γ with αj .

Each time that β intersects αj , β crosses the corresponding tubular neighbourhood.

It thus gives nji([γ], [αj]) points of intersection with Γ. We therefore have

card(Γ ∩ β) = card(β ∩ γ) +∑

j

nji([γ], [αj ])i([αj ], [β]).

DR

AFT

13 J

un 2

008


If, additionally, β has minimal intersection with γ, we have card(β ∩ γ) = i([γ], [β]);

the left side is always greater than or equal to i([Γ], [β]).

Second part

Let M be a closed surface of genus g > 1. Let K = {K1, . . . ,K3g−3} be a system

of mutually disjoint simple curves on M with the following properties:

1. Kj is has connected complement in M ;

2. If one cutsM along these curves, one obtains (2g−2) pairs-of-pants (disks with

two holes).

DR

AFT

13 J

un 2

008


We can easily construct a simple curve α that cuts every Kj in an essential way:

i([α], [Kj]) 6= 0. Let ϕ be a diffeomorphism of M , which is equal to the identity out-

side of a tubular neighbourhood of α, and coincides with a Dehn twist of one turn in

the tubular neighborhood. We set:

K ′j = ϕ(Kj).

Clearly, the system K′ = {K ′1, . . . ,K

′3g−3} has properties (1) and (2).

Proposition A.2. For all j, k, we have

i([Kj ], [K′k]) 6= 0.

Proof. From the inequality of Proposition A.1, it follows that:

∣∣∣∣i([K′k], [Kj])− i([Kk], [α])i([α], [Kj ])

∣∣∣∣ ≤ i([Kk], [Kj]) = 0.

Remark. We may take αwith i([α], [Kj ]) = 2 for all j. We then obtain i([K ′k], [Kj]) = 4

for all j, k.

DR

AFT

13 J

un 2

008

Appendix B

Spines of manifolds of

dimension 2

by V. Poenaru

LetN be a compact, connected manifold of dimension 2, with a nonempty boundary.

If N is triangulated and if L1 ⊂ L2 ⊂ N are two subcomplexes, we say that we pass

from L1 to L2 by a dilation of dimension n if there exists an n-simplex τ of N and a

face τ ′ of τ such that

L2 − L1 = int τ − int τ ′

(here int designates the open cell.) The inverse passage is called collapsing. If one collapsing

passes from L′ to L′′ by a set of dilations, then one can do so in an ordered way, such

that the set of respective dimensions is nondecreasing.

A slide is a set consisting of collapses and dilations: slide

L′′ = CnDnCn−1Dn−1 · · ·C1D1(L′) (B.1)

where dimL′′ = dimL′ = 1, dimCi = dimDi = 2, and support Ci = support Di.

More generally, if L′ and L′′ ⊂ N are two complexes of dimension 1, we will speak

of a slide L′ =⇒ L′′ if there exists a triangulation of N in which (B.1) is realized.

A subpolyhedron L ⊂ N is a spine if, for a particular triangulation, N collapses spine

to L.

Theorem B.1. Let Σ1,Σ2 be two 1-complexes of N having no free ends. If Σ1 and Σ2 are

two spines of N , we may pass from Σ1 to Σ2 by a set of slides and isotopies.

Proof. The proof decomposes into the following lemmas.

307

DR

AFT

13 J

un 2

008


Lemma B.2. Isotopies and slides transform a spine into a spine.

Lemma B.3. Let Σ be a spine of N and L a simple arc of N that does not meet Σ except at

its endpoints. There exists a continuous map ϕ : D2 → N and a decomposition of ∂D2 into

two segments: ∂D2 = A ∪B, ∂A = ∂B, intA ∩ intB = ∅,

DR

AFT

13 J

un 2

008


such that

1. ϕ|A it a homeomorphism on L;

2. ϕ|intD2 is a smooth immersion in N \ Σ

3. ϕ(B) ⊂ Σ

The proofs of these two lemmas are left as an exercise.

Lemma B.4. For a triangulation of N , consider two sets of sub-complexes

X0 ⊂ X1 ⊂ · · · ⊂ Xn

Y 0 ⊂ Y 1 ⊂ · · · ⊂ Y n

having the following properties:

1. X0 and Y 0 are spines of N ;

2. the transformations X i−1 ⊂ X i, Y i−1 ⊂ Y i are dilations of dimension 2

3. Xn is the same subcomplex of N as Y n. Then there exists a set of subcomplexes Z0 ⊂Z1 ⊂ · · · ⊂ Zn−1 such that:

4. Z0 is obtained from Y 0 by a slide (in particular, Z0 is a spine);

5. Zn−1 = Xn−1;

6. the transformations Zi−1 ⊂ Zi are dilations of dimension 2.

Proof. Let σ be a 2-simplex of N that corresponds to the dilations Xn−1 ⊂ Xn, and

let σ1, σ2, andσ3 be its three faces. Denote by Pi the vertex opposite σi. Suppose also

that σ1 is the free face of the collapse Xn ց Xn−1 and that σj is the free face of the

collapse Y i0 ց Y i0−1:

Y i0 − Y i0−1 = intσ ∪ intσj

If j = 1, the lemma follows immediately; thus suppose that j = 2.

DR

AFT

13 J

un 2

008


This figure is not

yet available

Since σ1 is a free face in Y n = Xn, this intersection is not in the edge of another

2-simplex of Y i0−1. Thus σ1 ⊂ Y 0. Similarly, by 2), the 0-skeleton of Xn = Y n is

contained in X0 and in Y 0. Therefore

∂σ ∩ Y 0 =

σ1 ∪ σ3

or

σ1 ∪ P1.

Let Z0 = (Y 0 − intσ1) ∪ σ2. If ∂σ ∩ Y 0 = σ1 ∪ σ3, it is evident that one can pass from

Y 0 to Z0 by a slide. If ∂σ∩Y 0 = σ1 ∪P1, we may apply Lemma B.3 with Σ = Y 0 and

L = σ3. This therefore permits us to conclude, in this case, that the passage Y0 → Z0

is a slide. Thus point 4) is verified. The constructions of Z1 ⊂ · · · ⊂ Zn−1 to assure 5)

and 6) are left to the reader.

Lemma B.5. Let L1, L2 be two complexes of dimension 1 in N , with no free ends. Let L′1, L

′2

be complexes obtained by dilations of dimension 1 fromL1 andL2, respectively. If it is possible

to pass from L′1 to L′

2 by slides and isotopies, then the same is true for L1 and L2.

This is an easy exercise. From these four lemmas, we can deduce the theorem

without difficulty.

DR

AFT

13 J

un 2

008


This figure is not

yet available

Figure B.1:

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008

Appendix C

Explicit formulas for Measured

Foliations

by A. Fathi

On the double pair of pants, we consider the curves K , K ′, K ′′ (Figure C.1).

yellow arc yellow arc

T ′S′

T

S

AA

K ′ ≃ arc jaune ∪ S ∪ arc jaune ∪ S′

K ′′ ≃ arc jaune ∪ T ∪ arc jaune ∪ T ′

A and A are on the same side with respect to K ′

K × [−1,+1]

Figure C.1:

For a foliation in normal form with respect to this decomposition, we have de-

fined three numbers (m, s, t), in addition to the four measures of the curves of the

313

DR

AFT

13 J

un 2

008


boundary (see Section 6.4).num 6.4.5

Proposition C.1. There exist continuous formulas, positively homogeneous of degree 1, cal-

culating s and t as a function of the minimal measures of the isotopy classes [K], [K ′], and

[K ′′] and of the curves of the boundary of the double pair of pants.

Proof. We use the following notation: m is the length of K , s, t, s′, t′, a and a are the

lengths of the arcs S, T , S′, T ′, A, and A, defined in 6.4 and recalled in Figure C.1.num 6.4.5

Claim 1. If m 6= 0, we can calculate s and t as a functions of α = s+ s′, β = t + t′, m,

a and a.

DR

AFT

13 J

un 2

008


We trivialize the annulus K × [−1,+1] in such a way that the projection onto K

foliates like the given foliation. In the covering R× [−1,+1] of the annulus, the group

acts as translation by m; we have a picture like Figure C.2, where we have drawn the

(line) segments realizing the minimum lengths of the arcs lifting S, T , etc. Obvious

t′

s′t

s

a

aa m− a

Figure C.2:

geometric reasons imply that the upper endpoints of these arcs always appear in the

indicated order. We also recall something visible in the figure:

(m, s, t) ∈ ∂(≤ ∇), (m, s′, t′) ∈ ∂(≤ ∇).

From this it follows that (2m,α, β) ∈ (≤ ∇). Thus we have: reverse notation

for tri ineq.?α ≤ β + 2m

β ≤ α+ 2m

2m ≤ α+ β

and, of course, m ≥ a, and m ≥ a. Moreover, (s, s′, a, a) and (t, t′, a, a) are the lengths

of the sides of degenerate quadrilaterals; thus, we have:

α , β ≥ |a− a|.

We describe the possible configurations in terms of the angle that each strand makes

with the horizontal in the universal cover. We exclude some configurations by re-

marking that if S makes an angle ≤ π/2, then T ′ cannot make an angle > π/2; other-

wise we would have a > m.

DR

AFT

13 J

un 2

008


I

II

III

IV

V

VI

VII

t′

t′

t′

t′

t′

t′

t′s′

s′

s′

s′

s′

s′

s′

t

t

t

t

t

t

t

s

s

s

s

s

s

s

Figure C.3:

DR

AFT

13 J

un 2

008


Configuration I is characterized by: β = α+ 2m; further we have:

β ≥ α

s+ s′ = α

s− s′ = a− at = s+m

t′ = s′ +m.

In fact, taking into account that (m, s, t) ∈ ∂(≤ ∇), and that (m′, s′, t′) ∈ ∂(≤ ∇), we

see that β = α+2m implies t = s+m and t′ = s′+m, which determines configuration

I. ab in fig should

be a-barConfiguration II is characterized by: α = a− a; further we have:

β ≥ α

s+ s′ = α

s− s′ = β − 2m

t = s+m

t′ = m− s′.

In fact, s + s′ + a = a determiness’

a

ab

s

; as a < m, the angle of T ′ must be

smaller than π/2.

Analogous reasonings allow one to establish characterizations of the other cases.

Configuration III is characterized by: α = a− a; further, we have:

β ≥ α

s+ s′ = α

s− s′ = 2m− βt = m− st′ = s′ +m.

Configuration IV is characterized by α+ β = 2m; further we have:

s+ s′ = α

s− s′ = a− at = m− st′ = m− s′.

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008


Configuration V is characterized by: β = a− a; further we have:

α ≥ β

s+ s′ = α

s− s′ = a− at = m− st′ = m− s′.

Configuration VI is characterized by: β = a− a; further we have:

α ≥ β

s+ s′ = α

s− s′ = a− at = s−mt′ = m− s′.

Configuration VII is characterized by: α = β + 2m; further we have:

α ≥ β

s+ s′ = α

s− s′ = a− at = s−mt′ = s′ −m. �

By a small calculation, we see that in cases I, II, III, IV we have:

(⋆)

{s = |m+ a−a−β

2 |t = a−a+β

2 ,

and that in cases IV, V, VI, VII, we have:

(⋆⋆)

{s = α+a−a

2

t = |m+ a−a−α2 | .

DR

AFT

13 J

un 2

008


We introduce a closed positive cone in R5+

C = {(α, β,m, a, a) ∈ R5+|(α, β, 2m) ∈ (≤ ∇),m ≥ a,m ≥ a,

α ≥ |a− a|, β ≥ |a− a|; one of the following equalities is satisfied:

α = |a− a|, β = |a− a|, α = β + 2m,α+ β = 2m,β = α+ 2m}.

DR

AFT

13 J

un 2

008


a a

aa

J

Figure C.4:

We see that α = |a− a|. By analyzing in an analogous manner what happens with

J , we obtain the proof of the claim.

If m = 0, the preceding formulas become k′ = α+ j + j and k′′ = β + j + j. If we

look at the models, we see that they agree on the geometry; in this observation, do

not forget that the case where one of the pants is not in the support of the foliation;

in this case, the three measures of the boundary are zero as well as the lengths of the

“arcs jaune”?

Fundamental remark. This appendix is universal! Precisely, we can change the arc

jaune for each type of foliation of the standard pair of pants to any other arc that has

the following properties:

1) It stays in the same isotopy class;

2) It realizes the minimum transverse length in this class.

A new choice of arcs on the models leads to a new classifying homeomorphism

θ : I⋆(MF) → B − {0}. This will be built from the formulas of the appendix which

stay exactly the same. The only change is in the expression of the length of the arc A

associated to each arc jaune.

DR

AFT

13 J

un 2

008


We define φ : C → R2 by the formulas (⋆) if β ≥ α and by the formulas (⋆⋆) if

β ≤ α. It is easy to see that the two formulas coincide if α = β. On the other hand,

if (s, t) are the coordinates of φ, we see that (s, t) belongs to R2+ and that (m, s, t)

belongs to ∂(≤ ∇).

The interest in introducing C is to show that the function θ extends to a closed

subcone of RS+.

We remark that if m = 0 (and as a consequence, a = 0, a = 0), we obtain for the

above formulas:

s = t =α

2=β

2

which coincides with what the geometry says. �

We set k′ = I(F , µ; [K ′]), k′′ = I(F , µ; [K ′′]), j and j the lengths of the arcs jaune

J and J of the pairs of pants containing A and A, respectively.

Claim 2. If m 6= 0, we have α = sup(|a− a|, k′− j− j) and β = sup(|a− a|, k′′− j− j).First of all, by definition of k′, we have α+ j + j ≥ k′, and we have already seen

α ≥ |a − a|. If J and J are the nonzero lengths (which means that the chosen arcs

pass through singularities), we easily replace J ∪ S ∪ J ∪ S′ with a quasitransverse

curve of the same length; in this case, k′ = α + j + j. If J is of zero length (piece of

a smooth leaf) and if S and S′ leave from different sides of J , we replace S ∪ J ∪ S′

with a transversal of the same measure (Figure C.5).

S

S′

J

Figure C.5:

If S and S′ leave from the same side of J , we have one of the two configurations

of Figure C.4.

DR

AFT

13 J

un 2

008

Appendix D

Estimates of hyperbolic

distances

by A. Fathi

We consider the Poincare half-plane H2 = {z ∈ C|z = x + iy, y > 0}, endowed

with the metric ds2 = dx2+dy2

y2 . The distance between two points z and z′ is denoted

d(z, z′).

D.1 Hyperbolic distance of i to z0.

cosh(d(i, z0)) =|z0 + i|2 + |z0 − i|2|z0 + i|2 − |z0 − i|2

.

Proof. Let f : H2 → D2 be the isomorphism z 7→ z−iz+i . Let gz0 be the automorphism

of D2 which is multiplication by f(z0)|f(z0)| ; we verify that the automorphism of H2, f−1 ◦

gz0 ◦f , has i for a fixed point and sends z0 to the purely imaginary point i |z0+i|+|z0−i||z0+i|−|z0−i| .

As we have the formula d(i, iy) = | log y|, it follows that

ed(i,z0) =|z0 + i|+ |z0 − i||z0 + i| − |z0 − i|

.

323

DR

AFT

13 J

un 2

008


D.2

Corollary D.1. If z0 = ai+bci+d with ad− bc = 1, we have:

cosh(d(i, z0)) =1

2[a2 + b2 + c2 + d2].

DR

AFT

13 J

un 2

008


D.3 Hyperbolic translation along the imaginary axis

The transformation(ek/2 0

0 e−k/2

)sends the geodesic iy to itself and the points of this

line are displaced by a distance k.

D.4 Translation along the hyperbolic geodesic of com-

plex numbers of modulus 1

If z is of modulus 1 and if z′ satisfies z′+1z′−1 = ek z+1

z−1 , then z′ is of modulus 1. The

transformation z 7→ z′ that is given by the matrix of SL(2,R)

(cosh(k2 ) sinh(k2 )

sinh(k2 ) cosh(k2 )

)

displaces the points of this “line” by a distance of k, to the right (real positive part) if

k > 0, and to the left if k < 0.

D.5 Relations between the sides of a right hyperbolic

hexagon

ℓ

D2 D1

M2

M1

t′

t

s

k′ k

Figure D.1:

DR

AFT

13 J

un 2

008


In Figure D.1, s, k, and k′ are given. We want to calculate ℓ, which is the shortest

distance between the lines D1 and D2. We are going to establish the formula:

cosh ℓ = cosh(s) sinh(k) sinh(k′)− cosh(k) cosh(k′).

Proof. We calculate the distance from an arbitrary point M1, of abcissa t on D1 (ori-

ented), to an arbitrary point M2, of abcissa t′ on D2 (as in Figure D.1).

We place M2 at i and we try to obtain M1 = f(i), where f is composed of hyper-

bolic translations along the axis iy and along the unit circle.

We easily see that one can take f ∈ SL(2,R) as a product:

f =

(e−t

′/2 0

0 et′/2

)F

(e−t/2 0

0 et/2

)=

(a b

c d

)

where

F =

(cosh(k

′

2 ) sinh(k′

2 )

sinh(k′

2 ) cosh(k′

2 )

)(es/2 0

0 e−s/2

)(cosh(k2 ) − sinh(k2 )

− sinh(k2 ) cosh(k2 )

)=

(α β

γ δ

)

In performing these calculations, we see:

a = αe−t+t′

2 , b = βet−t′

2 , c = γet′−t

2 , d = δet+t′

2 ,

α = es/2 cosh(k′

2) cosh(

k

2)− e−s/2 sinh(

k′

2) sinh(

k

2),

β = e−s/2 sinh(k′

2) cosh(

k

2)− es/2 cosh(

k′

2) sinh(

k

2),

γ = es/2 sinh(k′

2) cosh(

k

2)− e−s/2 cosh(

k′

2) sinh(

k

2),

δ = e−s/2 cosh(k′

2) cosh(

k

2)− es/2 sinh(

k′

2) sinh(

k

2).

We have: 2 cosh(d(M1,M2)) = a2 + b2 + c2 + d2

= α2e−(t+t′) + β2et−t′

+ γ2et′−t + δ2et+t

′

.

DR

AFT

13 J

un 2

008


In solving for the critical point of this function of the variables t and t′ (unique

because the function is convex), we get:

et+t′

= |αδ|, et−t

′

= |γβ|.

The critical value is therefore: cosh ℓ = |αδ|+ |βγ|.

αδ =1

2[1 + cosh(k) cosh(k′)− cosh(s) sinh(k) sinh(k′)]

βγ =1

2[cosh(k) cosh(k′)− 1− cosh(s) sinh(k) sinh(k′)].

We moreover verify that αδ − βγ = 1. We thus find:

cosh ℓ = sup(1, | cosh(k) cosh(k′)− cosh(s) sinh(k) sinh(k′)|).

And, we see geometrically that if, starting from the hexagon, we augment s, we ob-

tain a new hexagon for which ℓ is definitely nonzero (cosh(ℓ) > 1); moreover, ℓ is an

increasing function of s (see Section 3.2). Thus:

cosh(ℓ) = cosh(s) sinh(k) sinh(k′)− cosh(k) cosh(k′).

D.6 Distance between two points at equal distance from

a line

ℓ

M2M1

m m

Figure D.2:

In the situation of Figure D.2, we have the formula:

cosh(d(M1,M2)) =1

2[cosh(ℓ) + 1 + (cosh(ℓ)− 1) cosh(2m)].

DR

AFT

13 J

un 2

008


Proof. We place M1 at i and we write M2 = f(i), where f is the following product

in SL(2,R):

f =

(e−m/2 0

0 em/2

)(cosh( ℓ2 ) sinh( ℓ2 )

sinh( ℓ2 ) cosh( ℓ2 )

)(em/2 0

0 e−m/2

)

=

(cosh( ℓ2 ) e−m sinh( ℓ2 )

em sinh( ℓ2 ) cosh( ℓ2 )

)

We then apply Formula 2.

D.7 Bounding distances in pairs of pants

We consider on the pair of pants P 2 a hyperbolic metric for which the components

of the boundary are geodesics of respective lengths 2m1, 2m2, 2m3 (attention! this is

not the usual notation). Let gij be the simple geodesic orthogonal to ∂iP2 and ∂jP

2;

if i = j, it cuts P 2 into two annuli. We set ℓ3 = length g12, ℓ2 = length g13, and

ℓ1 = length g23.

M2

M1

m2

m1

m3

g12 (length = ℓ3)

Figure D.3:

DR

AFT

13 J

un 2

008


We have: cosh(m3) = cosh(ℓ3) sinh(m1) sinh(m2)− cosh(m1) cosh(m2).

Thus

cosh(ℓ3) =cosh(m3) + cosh(m1) cosh(m2)

sinh(m1) sinh(m2).

Proposition D.2. Let M1 (resp. M2) be a point of abscissa m ≤ inf(m1,m2) on ∂1P2

(resp. ∂2P2), where the origin is the point of intersection with g12 and the orientation is that

of Figure D.3. For any ǫ > 0, there exists a constant, only depending on ǫ, which bounds

d(M1,M2) from above, provided that m, m1, m2, and m3 satisfy the inequalities (i), (ii) or

(i), (iii):(i) m1,m2,m3 > ε;

(ii) (m1,m2,m3) ∈ (≤ ∇) (triangle inequality) and |m| ≤ m1+m2−m3

2 ;

(iii) m1 ≥ m2 +m3 and |m| ≤ m2.

Proof.

By Formula 6, we have to bound the quantity (cosh(ℓ3)+1)+(cosh(ℓ3)−1) cosh(2m).num

For this, it suffices to bound Q = [cosh(ℓ3)− 1] cosh(2m) because we have cosh(ℓ3) +

1 ≤ Q+ 2.

1o Suppose (i) and (ii) are true. We have align

Q =

[cosh(m3) + cosh(m1 −m2)

sinh(m1) sinh(m2)

]cosh(2m) ≤


sinh(m1) sinh(m2)

]cosh(m1+m2−m3).

Since ∂1P2 and ∂2P

2 play symmetric roles here, we can suppose m1 −m2 ≥ 0. Then,

we have:

cosh(m1 −m2) cosh(m1 +m2 −m3) ≤ cosh(2m1 −m3)

cosh(m3) cosh(m1 +m2 −m3) ≤ cosh(m1 +m2).

Further |m1 − m3| ≤ m2; thus 0 ≤ |2m1 − m3| ≤ m1 + m2; from which we have:

cosh(2m1 −m3) ≤ cosh(m1 +m2).

Finally, we have: Q ≤ 2 cosh(m1+m2)sinh m1 sinh m2

= 2 + 2coth(m1)coth(m2).

DR

AFT

13 J

un 2

008


The right hand side is bounded by (i).

2o Suppose (i) and (iii) are true.

Q =


sinh(m1) sinh(m2)

]cosh(2m2).

We have:

cosh(m3) cosh(2m2) ≤ cosh(m3 + 2m2) ≤ cosh(m1 +m2),

cosh(m1 −m2) cosh(2m2) ≤ cosh(m1 +m2).

We conclude as in the first case.

Corollary D.3. Let M , M ′ be two distinct points of ∂1P2, equidistant from the geodesic g11

and on the same side of it. Then d(M,M ′) is bounded by a constant that only depends on ǫ,

provided we have:

(i) m1,m2,m3 > ε,

(ii) m1 ≥ m2 +m3

(iii) M ∈ AA′ (see Figure D.4).

M

M ′

B B′

AA′

CC′

g11

g23 g12

m

m2m3

(all the lines in this figure are geodesics)

Figure D.4:

DR

AFT

13 J

un 2

008


Proof. By the preceding proposition, the quantities d(A,B) = d(C,B) and d(A′, B′) =

d(C′, B′) are bounded. By the formula of Section D.6,

d(M,M ′) ≤ sup(d(A,C), d(A′, C′)).

By the triangle inequality, d(A,C) ≤ 2d(A,B) and d(A′, C′) ≤ 2d(A′, B′).

DR

AFT

13 J

un 2

008


DR

AFT

13 J

un 2

008

Appendix E

Errata

Expose 2, Theorem 2, page ???

G. Mess has informed us that the inclusions Diff(S2) ← G(S2) and DiffP 2 ← G(P 2)

are not homotopy equivalences.Theorem 2 is not used in the sequel.

Expose 5, Proposition 5, page ???

The argument for properness is incomplete. It fails to consider the case of a sequence

of metrics where the length of one of the curves Ki tends to zero. The inequality

(∗) cosh(l(m, [K ′i])) sinh(l(m, [Ki])) ≥ 1

shows that, in this case, the length of K ′i (and of K ′′

i ) tends to infinity.We can establish the inequality (∗) from the formula on the sides of a right-angled

hexagon (top of page 153):With the notation of Figure E, the length of K ′

i is larger than that of the bridge

Pi and the length of Ki is larger than the length of Xi and that of Yi. If we cut along

Pi and Hi, we obtain a right-angled hexagon which admits Xi, Pi, Yi as consecutive

sides. The above formula from page 153 shows then that

cosh(l(m,Pi)) sinh(l(m,Xi)) sinh(l(m,Yi)) ≥ 1 + cosh(l(m,Xi)) cosh(l(m,Yi));

from which it follows that

cosh(l(m,Pi)) sinh(l(m,Xi)) ≥ 1.

The formula (∗) follows. It is clear from this proof that this formula is not optimal. We

remark that this is a particular case of the “collar theorem” or the Margulis inequality.

333

DR

AFT

13 J

un 2

008


i

i

i

i

i

Xi

Yi

Pi

Hi

Ki

Expose 11, S3.6. pseudo-Anosov for a manifold with bound-num

ary

B.J. Jiang has informed us that, in the definition of a pseudo-Anosov “diffeomorph-

ism” of a manifold with boundary, one cannot require it to be the identity on the

boundary. In particular, the last phrase on page ??? is incorrect.

DR

AFT

13 J

un 2

008

References

At the end of each entry, the pages on which that entry is cited are listed in

parentheses.

[Abi72] W. Abikoff. Topics in the real analytic theory of Teichmuller space, volume 820

of Lecture Notes in Mathematics. Springer Verlag, New York, 1972. Cited

on page(s) i

[AKM65] R. L. Adler, A. G. Konhein, and M. H. McAndrew. Topological entropy.

Trans. Amer. Math. Soc., 114:309–319, 1965. Cited on page(s) 188, 190

[Ano67] D. V. Anosov. Geodesic flows on compact riemannian manifolds of

negative curvature. Proceedings of the Steklov Mathematics Institute, 90,

1967.

[Ber78] Lipman Bers. An extemal problem for quasi-conformal mappings and a

theorem by Thurston. Acta Mathematica, 141:73–98, 1978.

[Bir75] Joan Birman. Braids, links, and mapping class groups, volume 82 of Annals of

Mathematics Studies. Princeton University Press, Princeton, NJ, 1975.

Cited on page(s) 20

[Bir77] Joan S. Birman. The algebraic structure of surface mapping class groups.

In Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975),

pages 163–198. Academic Press, London, 1977. Cited on page(s) 301

[BM77] Rufus Bowen and Brian Marcus. Unique ergodicity for horocycle

foliations. Israel J. Math., 26(1):43–67, 1977. Cited on page(s) 234

[BO69] L. Bishop and B. O’Neill. Manifolds of negative curvature. Trans. A.M.S.,

145:1–48, 1969. Cited on page(s) 134

[Bou85] J. P. Bourguinon. Sur la structure du π, Seminaire de geometrie

riemannienne 1970–71, varietes a courbure negative. In Seminaire de

335

DR

AFT

13 J

un 2

008


geometrie riemannienne 1970–71, varietes a courbure negative, 1985.

Publications de l’Universite Paris VII. Cited on page(s) 134

[Bow70] R. Bowen. Entropy for group endomorphism and homogeneous spaces.

Trans. Amer. Math. Soc., 153:401–413, 1970. Cited on page(s) 189, 190,

197, 212

[Bow78] R. Bowen. Entropy and the fundamental group. In Structure of attractors,

volume 668 of Lecture Notes in Mathematics. Springer Verlag, 1978. Cited

on page(s) 197

[BS85] F. Bonahon and L. Siebenmann. The classification of Seifert fibred

3-orbifolds. In Roger Fenn, editor, Lecture Notes in Mathematics,

volume 95, pages 19–85. London Mathematical Society, 1985.

[CE75] J. Cheeger and D. G. Ebin. Comparison theorems in Riemannian geometry.

North Holland, 1975. Cited on page(s) 36, 42

[Cer68] J. Cerf. Sur les diffeomorphismes de la sphere de dimension trois (Γ4 = 0),

volume 53 of Lecture Notes in Mathematics. Springer Verlag, New York,

1968. Cited on page(s) 19

[Cer70] Jean Cerf. La stratification naturelle des espaces de fonctions

differentiables reelles et le theoreme de la pseudo-isotopie. Inst. Hautes

Etudes Sci. Publ. Math., 39:5–173, 1970. Cited on page(s) 287, 293, 297

[CM80] H.S.M. Coxeter and W.O.J. Moser. Generators and Relations for Discrete

Groups. Springer Verlag, Heidelberg, fourth edition, 1980.

[Cox69] H. S. M. Coxeter. Introduction to Geometry. John Wiley & Sons, Inc., New

York, second edition, 1969.

[Din71] E. F. Dinaburg. On the relations of various entropy characteristics of

dynamical systems. Math. of the USSR, Izvestija, 35:p. 337 (English

translation), p. 324 (Russian), 1971. Cited on page(s) 189

[Dun81] W. Dunbar. Fibred orbifolds and crystallographic groups. PhD thesis,

Princeton, 1981.

[DV 6] Douady and Verdier. Sur les formes de Strebel. In Seminaire de l’E.N.S.

Asterisque, 1975–6. Cited on page(s) 132

[EE69] C. J. Earle and J. Eells. A fiber bundle description of Teichmuller theory.

Journal of Differential Geometry, 3:19–43, 1969. Cited on page(s) 133, 229

DR

AFT

13 J

un 2

008


[Eps66] D. B. A. Epstein. Curves on 2-manifolds and isotopies. Acta Mathematica,

115:83–107, 1966. Cited on page(s) 2, 6, 45

[Eps72] D. B. A. Epstein. Periodic flows on 3-manifolds. Annals of Mathematics,

95:66–82, 1972.

[Fen50] W. Fenchel. Bemaerkninger om endelige grupper af afbildnigngsklasser.

Mat. Tidsskr. B., pages 90–95, 1950. Cited on page(s) 229

[Fin76] R. Fintushel. Local S1-actions on 3-manifolds. Pacific Journal of

Mathematics, 66:111–118, 1976.

[FK98] R. Fricke and F. Klein. Vorlesungen uber die Theorie der automorphen

Funktionen. B. G. Teubner, Leibzig, 1898. Cited on page(s) 133

[FN] W. Fenchel and J. Nielsen. Discontinuous groups of non-euclidean

motions. Unpublished manuscript. Cited on page(s) 229

[Fri76] David Fried. Cross-sections to flows. PhD thesis, Berkeley, 1976. Cited on

page(s) 277

[Fri82a] David Fried. Flow equivalence, hyperbolic systems, and a new zeta

function for flows. Comment. Math. Helv., 57:237–259, 1982. Cited on

page(s) 280

[Fri82b] David Fried. Geometry of cross-sections to flows. Topology, 21:353–371,


[Gan98] F. R. Gantmacher. The theory of matrices. Vol. 1. AMS Chelsea Publishing,

Providence, RI, 1998. Translated from the Russian by K. A. Hirsch,

Reprint of the 1959 translation. Cited on page(s) 235

[Gau27] K. F. Gauss. Disquisitiones generales circa superfieces curvas. Societe

mathematique de France, Paris, 1827. Cited on page(s) 1

[GH78] Phillip Griffiths and Joseph Harris. Principles of Algebraic Geometry. Wiley

Interscience, New York, 1978.

[Goo71] T. N. T. Goodman. Relating topological entropy with measure theoretic

entropy. Bull. London Math. Soc., 3:176–180, 1971. Cited on page(s) 212

[Gre67] Marvin L. Greenberg. Lectures on Algebraic Topology. W. A. Benjamin, Inc.,

Reading, Massachusetts, 1967.

[Gro] M. Gromov. Three remarks on geodesic dynamics and fundamental

group. Cited on page(s) 197

DR

AFT

13 J

un 2

008


[GS87] Branko Grunbaum and G.C. Shephard. Tilings and Patterns. W. H.

Freeman and Company, New York, 1987.

[Ham66] M.-E. Hamstrom. Homotopy groups of the space of homeomorphisms on

a 2-manifold. Ill. J. Math., 10:563–573, 1966. Cited on page(s) 279

[Har77] W. J. Harvey, editor. Discrete groups and automorphic functions. Academic

Press, New York, 1977. Cited on page(s) 217

[Hem76] J. Hemple. 3-Manifolds. Princeton University Press, Princeton, 1976.

Cited on page(s) 280

[HM79] John Hubbard and Howard Masur. Quadratic differentials and foliations.

Acta Math., 142(3-4):221–274, 1979. Cited on page(s) 132

[HT80] A. Hatcher and W. Thurston. A presentation for the mapping class group

of a closed orientable surface. Topology, 19(3):221–237, 1980. Cited on

page(s) 285

[JN83] Mark Jankins and Walter D. Neumann. Lectures on Seifert manifolds.

Technical Report 2, Brandeis University Mathematics Department, March

1983.

[Kar66] Samuel Karlin. A first course in stochastic processes. Academic Press, New

York, 1966. Cited on page(s) 235

[Kau87] Louis H. Kauffman. On Knots. Princeton University Press, Princeton, New

Jersey, 1987.

[Kel55] John L. Kelley. General Topology. Springer Verlag, New York, 1955.

[Ker80] Steven P. Kerckhoff. The asymptotic geometry of Teichmuller space.

Topology, 19(1):23–41, 1980. Cited on page(s) 132

[KR85] R. Kulkarni and F. Raymond. 3-dimensional Lorentz Space-forms and

Seifert fiber spaces. Technical Report 521285, Mathematical Sciences

Research Institute, 1985.

[Mal66] B. Malgrange. Ideals of differentiable functions. Oxford University Press,

Oxford, 1966. Cited on page(s) 43

[Man74] A. Manning. Topological entropy and the first homology group. In

Dynamical Systems, Warwick 1974, volume 468 of Lecture Notes in

Mathematics. Springer Verlag, 1974. Cited on page(s) 196

DR

AFT

13 J

un 2

008


[Mar77] A. Marden. Geometrically finite Kleinian groups and their deformation

spaces. In Discrete groups and automorphic functions (Proc. Conf., Cambridge,

1975), pages 259–293. Academic Press, London, 1977. Cited on

page(s) 301

[Mas67] William S. Massey. Algebraic Topology: An Introduction. Harcourt, Brace &

World, Inc., New York, 1967.

[McC75] James McCool. Some finitely presented subgroups of the automorphism

group of a free group. J. Algebra, 35:205–213, 1975. Cited on page(s) 301

[Mil66] John Milnor. Topology from the differentiable viewpoint. The University Press

of Virgina, Charlottesville, 1966. Cited on page(s) 269, 273

[Mil68] John Milnor. A note on curvature and the fundamental group. Journal of

Diff. Geometry, 2:1–70, 1968. Cited on page(s) 193

[Mon87] Jose M. Montesinos. Classical Tessellations and Three-Manifolds. Springer

Verlag, Berlin-Heidelberg, 1987.

[Nie44a] J. Nielsen. Abbildungsklassen endlicher Ordnung. Acta Mathematica,

75:23–115, 1944. Cited on page(s) 227

[Nie44b] Jakob Nielsen. Surface transformation classes of algebraically finite type.

Danske Vid. Selsk. Math.-Phys. Medd., 21(2):89, 1944. Cited on page(s) 266

[NS87] V.V. Nikulin and I.R. Shafarevich. Geometries and Groups. Springer Verlag,

Berlin-Heidelberg, 1987.

[OR69] Peter Orlik and F. Raymond. On 3-manifolds with local SO(2) action.

Quarterly Journal of Mathematics Oxford, 20:143–160, 1969.

[Orl72] Peter Orlik. Seifert Manifolds, volume 291 of Lecture Notes in Mathematics.

Springer Verlag, New York, 1972. Cited on page(s) 276

[Orn74] D. Ornstein. Ergodic theory, randomness and dynamical systems, volume 5 of

Yale Mathematical Monographs. Yale University Press, New Haven, 1974.

Cited on page(s) 211, 212

[OVZ67] P. Orlik, E. Vogt, and H. Zieschang. Zur Topologie gefaserter

dreidimensionaler Mannigfaltigkeiten. Topology, 6:49–64, 1967.

[Pal60] R. Palais. Local triviality of the restriction map for embeddings. Comm.

Math. Helv., 34:305–312, 1960. Cited on page(s) 135

DR

AFT

13 J

un 2

008


[Par64] W. Parry. Intrinsic markov chains. Trans. Amer. Math. Soc., 112:55–66, 1964.

Cited on page(s) 211

[Poe78] V. Poenaru. Travaux de Thurston sur les diffeomorphismes des surfaces

et l’espace de Teichmuller. In Roger Fenn, editor, Seminaire Bourbaki,

volume 529. Seminaire Bourbaki, 1978. Cited on page(s) 1

[Ray68] F. Raymond. Classification of the actions of the circle on 3-manifolds.

Trans. Amer. Math. Soc., 131:51–78, 1968.

[Rol90] Dale Rolfsen. Knots and Links. Publish or Perish, Inc., Houston, Texas,

1990. Corrected Edition.

[Rou73] R. Roussarie. Plongements dans les varietes feuilletees. Publ. Math.

I.H.E.S., 43:143–168, 1973. Cited on page(s) 273, 275

[RS75] D. Ruelle and D. Sullivan. Currents, flows and diffeomorphisms.

Topology, 14:319–327, 1975. Cited on page(s) 278

[RS82] C.P. Rourke and B.J. Sanderson. Introduction to Piecewise-Linear Topology.

Springer Verlag, New York, 1982.

[Rus73] T.B. Rushing. Topological embeddings. Academic Press, New York &

London, 1973. Cited on page(s) 160

[Sch57] S. Schwartzman. Asymptotic cycles. Ann. Math, 66:270–284, 1957. Cited

on page(s) 280

[Sch78] D. Schattschneider. The plane symmetry groups: Their recognition and

notation. American Mathematical Monthly, 85:439–450, 1978.

[Sco83] Peter Scott. The geometries of 3-manifolds. Bulletin of the London Math.

Society, 15:401–487, 1983.

[See64] R. Seeley. Extension of C∞ functions defined in a half-space. Proc. Amer.

Math. Soc., 15:625–626, 1964. Cited on page(s) 43

[Sei33] H. Seifert. Topologie dreidimensionaler gefaserter Raume. Acta

Mathematica, 60:147–288, 1933.

[Sha77] I. R. Shafaraevich. Basic Algebraic Geometry. Springer Verlag, New York,

1977.

[Sin76] Y. G. Sinai. Introduction to ergodic theory, volume 18 of Mathematical notes.

Princeton University Press, Princeton, NJ, 1976. Cited on page(s) 9, 182,

211

DR

AFT

13 J

un 2

008


[Sma59] Steve Smale. Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc.,

10:621–626, 1959. Cited on page(s) 19

[Sma67] Steve Smale. Differentiable dynamical systems. Bulletin of the American

Mathematical Society, 73:747–817, 1967.

[Smi34] P. Smith. A theorem on fixed points for periodic transformations. Ann. of

Math., 35:572–578, 1934. Cited on page(s) 229

[Spr66] G. Springer. Introduction to Riemann surfaces. Addison-Wesley, New York,


[ST80] H. Seifert and W. Threlfall. A Textbook of Topology. Academic Press, New

York, 1980.

[Sta61] John Stallings. On fibering certain 3-manifolds. In Topology of 3-manifolds

and related topics. Prentice Hall, New York, 1961. Cited on page(s) 269

[Sta71] John Stallings. Group theory and three-dimensional manifolds. Yale Univesity

Press, Yale, 1971. Cited on page(s) 270

[Ste51] Norman Steenrod. The Topology of Fibre Bundles. Princeton University

Press, Princeton, New Jersey, 1951.

[Ste69] S. Sternberg. Celestial mechanics, part II. Benjamin, New York, 1969.

Cited on page(s) 13

[Sti80] John C. Stillwell. Classical Topology and Combinatorial Group Theory.

Springer Verlag, New York, 1980.

[Sul76] D. Sullivan. Cycles for the dynamical study of foliated manifolds and

complex manifolds. Inventiones Mathematicae, 26:225–255, 1976.

[Thu] William Thurston. The geometry and topology of 3-manifolds. Princeton

University Math. Department. Cited on page(s) 1

[Thu67] William Thurston. On the geometry and dynamics of diffeomorphisms of

surfaces. Bulletin of the American Mathematical Society, 73:747–817, 1967.

Cited on page(s) 1

[Thu72] William P. Thurston. Foliations of 3-manifolds that are circle bundles. PhD

thesis, Berkeley, 1972. Cited on page(s) 273, 275

[Thu86] William Thurston. A norm for the homology of 3-manifolds. Mem. Amer.

Math. Soc., 59(339):i–vi and 99–130, 1986. Cited on page(s) 270, 271, 280

DR

AFT

13 J

un 2

008


[Tis70] D. Tischler. On fibering certain foliated manifolds over S1. Topology,

9:153–154, 1970. Cited on page(s) 267

[Wal67] F. Waldhausen. Eine Klasse von 3-dimensionalen Mannigfaltigkeiten II.

Invent. Math., 4:501–504, 1967.

[Wal68] F. Waldhausen. On irreducible 3-manifolds which are sufficiently large.

Ann. Math., 87:56–88, 1968. Cited on page(s) 279

[Wey55] Hermann Weyl. Die Idee der Riemannscher Flache. Teubner, Leibzig, fourth

edition, 1955. Cited on page(s) 6

DR

AFT

13 J

un 2

008

Index

(n, ǫ)-separated, 185

(n, ǫ)-spanning, 185

(F , µ)-rectangle, 163

(FS ,FU )-rectangle, 197

S(M) = S, 2

T (M) = T , 6

m-equivalent, 5

T (P 2), 35

Anosov, 8

Dehn twist, 8, 134

Markov partition, 199

Markov pre-partition, 170

Schwartz equivalent, 5

Teichmuller space, 131


Teichmuller surface, 6


Thurston compactification, 7

Whitehead equivalence, 212

Whitehead equivalent, 5

adapted, 133

algebraic intersection number, 2

arational, 162, 216

atoroidal, 270

big diagonal, 20

birectangle, 197

birectangle, good, 198

braid group, 20

canonical, 57

collapsing, 301

cross-section, 272

displacement, 140

double, 165

equivalent, 6

faces, 68

flow-equivalent, 274

geometric intersection, 212

geometric intersection number, 2

good system of transversals, 164

homology direction, 273

incompressible, 266

irreducible, 265

length, 187

loop, 272

measured foliation, 4

metric covering map, 185

metric entropy, 9

minimal, 273

monodromy, 271

normal form, 59, 113

passage, 242

principal, 285

pseudo-Anosov, 8, 219, 271

pure braid group, 20

quasitransverse, 75, 235

rays, 3

reducible, 216

return map, 271

saddle, 269

separatrices, 68

shift, 193

sides, 68

343

DR

AFT

13 J

un 2

008


simple, 90

slide, 301

sphereless, 266

spine, 301

stable, 176

strict conjugacy, 271

supporting, 268

suspension, 272

topological entropy, 9, 184

total variation, 4

unglue, 160

uniquely ergodic, 230

unstable, 176

width, 199

draft - home - math › ~margalit › flp › flp.pdf · 2008-06-13 · phisms of surfaces” by...

Documents