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Advanced Lectures in Mathematics Volume 31

Handbook of Group Actions Volume I

Companion to the volume Handbook of Group Actions, Volume II

International Presswww.intlpress.com

edited by

Lizhen Ji Athanase Papadopoulos Shing-Tung Yau

HIGHER EDUCATION PRESS

Copyright © 2015 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publica-tion is permitted only under license from International Press. Excluded from these pro-visions is material in articles to which the author holds the copyright. (If the author holds copyright, notice of this will be given with the article.) In such cases, requests for per-mission to use or reprint should be addressed directly to the author. ISBN: 978-1-57146-300-5 Printed in the United States of America. 19 18 17 16 15 1 2 3 4 5 6 7 8 9

Advanced Lectures in Mathematics, Volume 31 Handbook of Group Actions, Volume I Companion to the volume Handbook of Group Actions, Volume II Volume Editors: Lizhen Ji (University of Michigan, Ann Arbor) Athanase Papadopoulos (Université de Strasbourg, France) Shing-Tung Yau (Harvard University)

ADVANCED LECTURES IN MATHEMATICS

Executive Editors

Editorial Board

Shing-Tung Yau Harvard University Cambridge, Mass., U.S.A. Lizhen Ji University of Michigan, Ann Arbor Ann Arbor, Mich., U.S.A.

Kefeng Liu University of California at Los Angeles Los Angeles, Calif., U.S.A.

Chongqing Cheng Nanjing University Nanjing, China Zhong-Ci Shi Institute of Computational Mathematics Chinese Academy of Sciences (CAS) Beijing, China Zhouping Xin The Chinese University of Hong Kong Hong Kong, China Weiping Zhang Nankai University Tianjin, China Xiangyu Zhou Institute of Mathematics Chinese Academy of Sciences (CAS) Beijing, China

Tatsien Li Fudan University Shanghai, China Zhiying Wen Tsinghua University Beijing, China Lo Yang Institute of Mathematics Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Sun Yat-sen University Guangzhou, China

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Foreword to Volumes I and II

The decision of editing this Handbook came after an international conference weorganized in Kunming (the capital of the Yunnan Province, China) on July 21–29,2012, whose theme was “Group Actions and Applications in Geometry, Topologyand Analysis”.Kunming is a wonderful place for meetings and for mathematical discussions,

especially in the summer, when the weather is most favorable. The conference wasa success, from the mathematical and the human point of view. The city is warm,and the landscape is beautiful. There is a big lake, and a mountain behind thelake. Mathematicians like beauty. Hermann Weyl said: “My work always triedto unite the truth with the beautiful, but when I had to choose one or the other,I usually chose the beautiful.” (Quoted in Hermann Weyl’s Legacy, Institute forAdvanced Study.)The first two volumes of this Handbook are a record of the Kunming conference,

but above all, we want them to be a convenient source for people working on orstudying group actions. In spite of the fact that there were 63 talks, we coveredat Kunming only a small part of this broad subject. In fact, group actions areso important that it is surprising that there was no available handbook on thatsubject so far. It is certainly the ubiquity of group actions that makes such aproject so vast and therefore difficult to attain, and our aim for the time beingis to start it. The present two volumes are the first on this important subject,and more volumes in the same series will appear in the future. Other conferenceson the same subject are also planned in the future; the next one will be in Sanya(Hainan Province).This Handbook will serve as an introduction and a reference to both beginners,

non-experts, experts and users of group actions.The conference in Kunming would not have gone so smoothly without the

generous and devoted help of the local organizers, namely, Provost Ailing Gong,Dean Xianzhi Hu, Party Secretary Fengzao Yang, Deputy Dean Yaping Zhang,Youwei Wen, and Jianqiang Zhang from the Kunming University of Science andTechnology. We would like to thank them for their work and hospitality.Many people have also helped with refereeing and reviewing the papers in this

Handbook, and we would like to thank them all for their help.

L. Ji, A. Papadopoulos, and S.-T. YauAnn Arbor, Strasbourg, and Cambridge MANovember, 2014

Introduction

The subject of this Handbook is groups and group actions. Although groups areomnipresent in mathematics, the notion of group was singled out relatively re-cently. We recall that the first definition of a (finite) group was formulated byCayley around the middle of the nineteenth century. But the concept itself is in-herent in the work of Galois (as a group of permutations of solutions of polynomialequations), and it is also contained — at least implicitly — in works of Ruffini,Lagrange and Gauss.On the other hand, the idea of group is closely related to that of symmetry, or

rather, to the mathematics behind symmetry, and the use of groups, seen as sym-metries, can be traced back to antiquity. In fact, the notion of symmetry reflects agroup action, not only in mathematics, but also in other sciences, including chem-istry, biological physics, and the humanities. Symmetry is also one of the mostfundamental concepts in art.In mathematics, the notion of abstract group is at the heart of the formulation

of many problems. Still, it is usually the concept of transformation group, or of agroup acting on a space, rather than that of group alone, which is of fundamentalimportance. A group action brings in an additional notion to the group at hand,coming from the space on which the group acts. It is also the group action whichmakes groups interesting, useful and understandable. The precise identification ofa group with a group of symmetries of a space is made through the action of thegroup on that space. But as the same group can act on different spaces, this groupcan be realized in several different ways as a group of symmetries.The notion of transformation group was inherent in eighteenth-century geome-

try, in particular in projective geometry. But it was Klein, in his Erlangen programmanifesto, and mathematicians like Lie, Poincare and others who worked in thespirit of this program (some of them without being aware of the program) whohighlighted the importance of a transformation group as a basic concept associatedto a geometry, with the view that a geometry is characterized (and, in a certainway, it is defined) by a transformation group rather than by a space.Central to contemporary research is the study of discrete group actions on

homogeneous spaces, in particular on manifolds of constant curvature and locallysymmetric spaces of finite volume. The most famous of such groups are probablythe Fuchsian groups, the Kleinian groups and the arithmetic subgroups of semi-simple Lie groups, where not only the groups are studied individually, but theirdeformation theory is also very rich.Other interesting classes of examples of infinite groups are, on the one hand,

the automorphism groups Aut(Fn) and the outer automorphism groups Out(Fn)

II Introduction

of a free group Fn on n generators (n ≥ 2), and on the other hand, the auto-morphism groups Aut(π1(Sg)) and the outer automorphism groups Out(π1(Sg))of fundamental group of (say, closed) surfaces Sg of genus g (g ≥ 2), i.e., themapping class groups of Sg. One can also mention the Coxeter groups.It can easily be argued that the free group is much simpler to apprehend

than a surface group; for instance, one can easily visualize the Cayley graph ofthe free group Fn, a regular tree with vertices of order 2n, and hence, one canhave a good picture of the geometry and combinatorics of that group, whereasthe Cayley graph of the fundamental group π1(Sg) is more complex. However,it turns out that the theory of automorphism and of outer automorphisms of thefree group Fn is much less understood than that of the automorphism (and theouter automorphism) group of the surface group Sg. The reason is that manyactions of Aut(π1(Sg)) and Out(π1(Sg)) arising naturally from the geometry andthe topology of surfaces have been studied, whereas for the groups Aut(Fn) andOut(Fn), there are not as many actions on geometric or topological spaces. TheCoxeter groups are understood via to their action on Coxeter complexes.The reader can refer to the beginning of the article by L. Ji in this volume,

where many group actions are listed.

The present volume of the Handbook of Group Actions is more especially con-cerned with discrete group actions. It consists of 12 chapters, and it is dividedinto four parts. Each part emphasizes special discrete groups and their actions.

Part I: Geometries and General Group ActionsThis part contains 2 chapters.Chapter 1 is by S.-T. Yau. It is a record of the talk that the author gave at

the Kunming conference, whose main theme was a view of a generalized geometrybased on the notion of operators rather than on that of space. The relation withphysics is also discussed. The concept of group is essential here, as a group ofoperators and as a gauge group. Several constructions of Riemannian geometrycan be done in this setting, including the definitions of the Dirac and the Laplaceoperators, the differential topology of operator geometry, Hodge theory, Yang-Mills theory and conformal field theory. There is also a version of that theory fordiscrete spaces.Chapter 2 is by L. Ji, and it is a summary of group actions that arise in math-

ematics. It attempts to cover all the major fields where group actions play animportant role and to convey a sense of how broad group actions are in mathe-matics and other sciences. Hopefully it will give some content to the statementthat group actions and symmetry, which are the same thing, are everywhere.

Part II: Mapping Class Groups and Teichmuller SpacesThis part concerns mapping class groups and Teichmuller spaces. The two

topics are related, because the action of the mapping class group of a surfaceon the Teichmuller space of that surface constitutes one of the most interesting(and may be the most interesting) action of that group, in terms of the richnessand the developments of the underlying theory, and also in terms of applications.Furthermore, Teichmuller spaces equipped with actions of mapping class groupsare the primary source of holomorphic group actions in high dimensions, including

Introduction III

infinite dimensions. Teichmuller spaces are also related in other ways to the subjectof group actions; for instance, an element of a Teichmuller space can be seen as aFuchsian group acting on hyperbolic 2-space. This makes a relation between thepresent section and the section in Volume II of this handbook which deals withrepresentations and deformations of subgroups of Lie groups.In Chapter 3, A. Papadopoulos surveys some actions of mapping class groups.

The latter admit actions which are of very different natures on spaces associ-ated to surface: group-theoretic, holomorphic, combinatorial, topological, metric,piecewise-linear, etc. The author reviews in more detail actions on spaces of folia-tions and laminations, namely, measured foliations, unmeasured foliations, generalgeodesic laminations and the reduced Bers boundary. The chapter also contains asection on perspectives and open questions on actions of mapping class groups.In Chapter 4, W. Su surveys two horofunction compactifications of Teichmuller

space which are also spaces on which the mapping class group naturally acts. Thehorofunction boundary of a space is defined relatively to a certain metric. Thetwo horofunction spaces that are studied in this chapter are associated to theTeichmuller metric and to the Thurston metric. The relation between these com-pactifications with Thurston and Gardiner-Masur’s compactifications is reviewed(results of Walsh and of Lui and Su), and the isometry groups of Teichmuller spaceequipped with the two metrics are considered.In Chapter 5, F. Herrlich studies Teichmuller disks, that is, embeddings of the

hyperbolic disk in Teichmuller space that are holomorphic and isometric. Moreprecisely, the author studies the stabilizers of these discs in the Schottky spaceSg of a closed Riemann surface of genus g, a quotient of the Teichmuller spaceTg by a certain (non-normal) torsion-free subgroup of the mapping class group.The Schottky space is an infinite orbifold covering of Riemann’s moduli space. Thestabilizer of a Teichmuller disk is sometimes a lattice in PSL(2,R). Schottky spaceis, like Teichmuller space, a complex manifold. The author studies in particularthe stabilizers in Schottky space of the Teichmuller disks and more generally thebehavior of these disks under the covering map Sg → Tg.Chapters 6 and 7 concern infinite-dimensional Teichmuller spaces.Chapter 6 by E. Fujikawa concerns actions of mapping class groups of sur-

faces of infinite type. There are various groups which play the role of a mappingclass groups, and various spaces which play the role of Teichmuller spaces, in thisinfinite-dimensional setting, and the author considers some of them. In particular,she considers the action of the so-called asymptotically trivial mapping class groupon the asymptotic Teichmuller space, a space which was first introduced by Sul-livan. The main result she describes in this context is that for surfaces satisfyinga condition of bounded geometry (a quasi-isometry invariant condition which in-volves lower and upper bounds on certain classes of geodesics, when the Riemannsurface is equipped with a hyperbolic metric), the asymptotically trivial mappingclass group coincides with that of the so-called stable quasiconformal mappingclass group, that is, the subgroup of conformal mapping classes which have repre-sentatives which are the identity outside a compact subset. She then introducesanother Teichmuller space, which is called the intermediate Teichmuller space,which is the quotient of the classical (quasiconformal) Teichmuller space by the

IV Introduction

asymptotically trivial mapping class group. Under the same bounded geometrycondition, this space inherits a complex structure from that of the quasiconfor-mal Teichmuller space. In general, the asymptotically trivial Teichmuller modulargroup is a proper subgroup of the group of holomorphic automorphisms of theasymptotic Teichmuller space. The author then studies the dynamics of the vari-ous actions that arise, and conditions under which such group actions are properlydiscontinuous. She also gives an asymptotic version of the Nielsen realizationproblem.Chapter 7 by K. Matsuzaki is a survey of the complex analytic theory of the

universal Teichmuller space and of some of its subspaces. Roughly speaking, theuniversal Teichmuller space is the space of equivalence classes of hyperbolic met-rics on the unit disc, where two structures are considered equivalent if they differby an isotopy which induces the identity on the boundary S1 of the disc. Thisspace can also be defined as a certain quotient of the group of diffeomorphisms ofthe unit circle. It is termed universal because it contains naturally the Teichmullerspaces of all hyperbolic surfaces. In this theory, the representation of the elementsof a Teichmuller space by Fuchsian groups is useful if not essential. One of theimportant concepts that are studied in detail in this chapter is a natural subsetof the universal Teichmuller space which is not the Teichmuller space of a sur-face, namely, a space of equivalence classes of diffeomorphisms of the circle withHolder continuous derivatives. The author shows that this space is equipped witha complex structure modeled on a complex Banach space. This complex structureis described through a careful study of the Bers embedding of the space in thespace of Schwarzian derivatives. The diffeomorphisms of the circle with Holdercontinuous derivatives are characterized by certain properties of their quasiconfor-mal extensions to the unit disc, and the theory bears relations with the space ofasymptotically conformal maps studied by Carleson.Chapter 8 by T. Satoh concerns mapping class groups of surfaces, and it has a

more algebraic nature. It is a survey of the Johnson homomorphisms associated tomapping class groups. These are homomorphisms associated to graded quotientsof a certain descending filtration of these groups. Johnson defined in the 1980s thefirst homomorphism in the sequence, as a tool to study the Torelli group. A similartheory for automorphisms of free groups was developed before, by Andreadakis,in his thesis in the 1960s. The Johnson homomorphism for surface mapping classgroups was generalized later on to the so-called Johnson homomorphisms of higherdegrees, and several people did extensive work on them, including Morita, Hain,Satoh and others, and there are recent results on the same subject by Kawazumi-Kuno and by Massuyeau-Turaev.

Part III: Hyperbolic Manifolds and Locally Symmetric SpacesChapter 9 by G. J. Martin is a survey on the various aspects of the geometry

and arithmetic of Kleinian groups. The author examines the geometry of Kleiniangroups and he gives geometric conditions on isometry groups of hyperbolic 3-space in order to be discrete. He studies in detail the two-generator groups, givingseveral generalizations of Jørgensen’s inequality for discreteness, and he discussesthe classification of arithmetic generalized triangle groups. One motivation for thisstudy is a problem which Siegel posed in 1943, namely, to identify the minimal co-

Introduction V

volume lattices of isometries of hyperbolic n-space, and more generally of rank-onesymmetric spaces.Chapter 10 by G. Prasad and A. S. Rapinchuk is a survey on several results

related to the basic question: Can you hear the shape of a drum? They concernlocally symmetric spaces of finite volume. The problem asks whether two Rieman-nian manifolds having the same spectrum, i.e., the same set of eigenvalues, areisometric. A closely related question is the so-called iso-length spectrum problemfor locally symmetric spaces: if two Riemannian manifolds have the same lengthspectrum, i.e., the same set of lengths of closed geodesics, are they isometric orat least commensurable? The major portion of this paper deals with this latterquestion and with related problems on algebraic groups and their maximal alge-braic tori and the authors give a fairly complete and detailed survey of results inthis direction.

Part IV: Knot GroupsThis part contains two chapters on representations of knot groups and twisted

Alexander polynomials. The twisted Alexander polynomial is defined as a pairconsisting of a group and a representation of that group. It generalizes the classicalAlexander polynomial. The twisted Alexander polynomial is naturally defined forlinks in S3 and more generally for finitely presentable groups. In some instancesit can easily be calculated.Chapter 11 by T. Morifuji is a survey on representations of knot groups and

twisted Alexander polynomials, with a special focus on the twisted Alexanderpolynomial for finitely presentable groups introduced by Wada. This polynomialis associated to a representation into SL(2,C). There are applications to fiberingand genus detecting problems of knots in S3. The twisted Alexander polynomialof a knot is seen as a C-valued rational function on the character variety of theknot group, and it is also expressed in terms of Reidemeister torsion. The chapteralso contains a comprehensive introduction to the classical Alexander polynomi-als and to the algebraic theory which is behind it (presentations of knot groups,Wirtinger presentations, Tietze transformations and Fox derivatives), as well ason representations of knot groups into SL(2,C), their character varieties and theirdeformations. The author focuses on the deformation of an abelian representationto a nonabelian one and of a reducible representation to an nonreducible one.Chapter 12 by M. Suzuki is devoted to the study of the existence of epimor-

phisms between knot groups. The author indicates by some examples how todetect the existence of a meridional epimorphism (that is, an epimorphism thatpreserves meridians) between knot groups and he gives explicit descriptions ofsome non-meridional epimorphisms. He shows that the existence of an epimor-phism between finitely presentable groups implies that their twisted Alexanderpolynomials are divisible. He makes connections with other works on the subject,and in particular with the so-called Simon conjecture (a problem in Kirby’s list)whose general case was settled recently by Agol and Liu. The result says thatevery knot group admits an epimorphism onto at most finitely many knot groups.

Contents

Part I: Geometries and General Group Actions

Geometry of Singular Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Shing-Tung Yau

1 The development of modern geometry that influenced our

concept of space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Geometry of singular spaces . . . . . . . . . . . . . . . . . . . . . . . 5

3 Geometry for Einstein equation and special holonomy group . . . . . 5

4 The Laplacian and the construction of generalized Riemannian

geometry in terms of operators . . . . . . . . . . . . . . . . . . . . . 6

5 Differential topology of the operator geometry . . . . . . . . . . . . . 9

6 Inner product on tangent spaces and Hodge theory . . . . . . . . . . 10

7 Gauge groups, convergence of operator manifolds and Yang-Mills

theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8 Generalized manifolds with special holonomy groups . . . . . . . . . 13

9 Maps, subspaces and sigma models . . . . . . . . . . . . . . . . . . . 14

10 Noncompact manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 16

11 Discrete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

13 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A Summary of Topics Related to Group Actions . . . . . . . . . . . . . . . . . . . . . 33

Lizhen Ji

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Different types of groups . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Different types of group actions . . . . . . . . . . . . . . . . . . . . . 56

4 How do group actions arise . . . . . . . . . . . . . . . . . . . . . . . 59

5 Spaces which support group actions . . . . . . . . . . . . . . . . . . 65

6 Compact transformation groups . . . . . . . . . . . . . . . . . . . . . 70

II Contents

7 Noncompact transformation groups . . . . . . . . . . . . . . . . . . . 74

8 Quotient spaces of discrete group actions . . . . . . . . . . . . . . . . 80

9 Quotient spaces of Lie groups and algebraic group actions . . . . . . 86

10 Understanding groups via actions . . . . . . . . . . . . . . . . . . . 87

11 How to make use of symmetry . . . . . . . . . . . . . . . . . . . . . 95

12 Understanding and classifying nonlinear actions of groups . . . . . 101

13 Applications of finite group actions in combinatorics . . . . . . . . . 103

14 Applications in logic . . . . . . . . . . . . . . . . . . . . . . . . . . 104

15 Groups and group actions in algebra . . . . . . . . . . . . . . . . . 105

16 Applications in analysis . . . . . . . . . . . . . . . . . . . . . . . . . 105

17 Applications in probability . . . . . . . . . . . . . . . . . . . . . . . 107

18 Applications in number theory . . . . . . . . . . . . . . . . . . . . . 107

19 Applications in algebraic geometry . . . . . . . . . . . . . . . . . . 110

20 Applications in differential geometry . . . . . . . . . . . . . . . . . 111

21 Applications in topology . . . . . . . . . . . . . . . . . . . . . . . . 112

22 Group actions and symmetry in physics . . . . . . . . . . . . . . . . 114

23 Group actions and symmetry in chemistry . . . . . . . . . . . . . . 121

24 Symmetry in biology and the medical sciences . . . . . . . . . . . . 123

25 Group actions and symmetry in material science and

engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

26 Symmetry in arts and architecture . . . . . . . . . . . . . . . . . . . 126

27 Group actions and symmetry in music . . . . . . . . . . . . . . . . 126

28 Symmetries in chaos and fractals . . . . . . . . . . . . . . . . . . . 128

29 Acknowledgements and references . . . . . . . . . . . . . . . . . . . 130

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Part II: Mapping Class Groups and Teichmuller Spaces

Actions of Mapping Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Athanase Papadopoulos

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

2 Rigidity and actions of mapping class groups . . . . . . . . . . . . . 192

3 Actions on foliations and laminations . . . . . . . . . . . . . . . . . . 196

4 Some perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Contents III

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

The Mapping Class Group Action on the Horofunction

Compactification of Teichmuller Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Weixu Su

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

3 Thurston’s compactification of Teichmuller space . . . . . . . . . . . 257

4 Compactification of Teichmuller space by extremal length . . . . . . 262

5 Analogies between the Thurston metric and the Teichmuller

metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

6 Detour cost and Busemann points . . . . . . . . . . . . . . . . . . . 269

7 The extended mapping class group as an isometry group . . . . . . . 273

8 On the classification of mapping class actions on Thurston’s

metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

9 Some questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Schottky Space and Teichmuller Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Frank Herrlich

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

2 Schottky coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

3 Schottky space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

4 Schottky and Teichmuller space . . . . . . . . . . . . . . . . . . . . . 295

5 Schottky space as a moduli space . . . . . . . . . . . . . . . . . . . . 298

6 Teichmuller disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

7 Veech groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

8 Horizontal cut systems . . . . . . . . . . . . . . . . . . . . . . . . . . 303

9 Teichmuller disks in Schottky space . . . . . . . . . . . . . . . . . . . 305

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Topological Characterization of the Asymptotically Trivial

Mapping Class Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Ege Fujikawa

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

3 Discontinuity of the Teichmuller modular group action . . . . . . . . 318

IV Contents

4 The intermediate Teichmuller space . . . . . . . . . . . . . . . . . . . 319

5 Dynamics of the Teichmuller modular group . . . . . . . . . . . . . . 321

6 A fixed point theorem for the asymptotic Teichmuller modular

group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

7 Periodicity of asymptotically Teichmuller modular transformation . . 327

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

The Universal Teichmuller Space and Diffeomorphisms of the

Circle with Holder Continuous Derivatives . . . . . . . . . . . . . . . . . . . . . . 333

Katsuhiko Matsuzaki

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

2 Quasisymmetric automorphisms of the circle . . . . . . . . . . . . . . 336

3 The universal Teichmuller space . . . . . . . . . . . . . . . . . . . . . 340

4 Quasisymmetric functions on the real line . . . . . . . . . . . . . . . 344

5 Symmetric automorphisms and functions . . . . . . . . . . . . . . . . 347

6 The small subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7 Diffeomorphisms of the circle with Holder continuous derivatives . . 357

8 The Teichmuller space of circle diffeomorphisms . . . . . . . . . . . . 365

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

On the Johnson Homomorphisms of the Mapping Class Groups of

Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Takao Satoh

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

2 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . 377

3 Mapping class groups of surfaces . . . . . . . . . . . . . . . . . . . . 377

4 Johnson homomorphisms of AutFn . . . . . . . . . . . . . . . . . . . 381

5 Johnson homomorphisms ofMg,1 . . . . . . . . . . . . . . . . . . . . 397

6 Some other applications of the Johnson homomorphisms . . . . . . . 401

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Part III: Hyperbolic Manifolds and Locally Symmetric Spaces

The Geometry and Arithmetic of Kleinian Groups . . . . . . . . . . . . . . . . . . 411

Gaven J. Martin

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

Contents V

2 The volumes of hyperbolic orbifolds . . . . . . . . . . . . . . . . . . 415

3 The Margulis constant for Kleinian groups . . . . . . . . . . . . . . . 420

4 The general theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

5 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

6 Two-generator groups . . . . . . . . . . . . . . . . . . . . . . . . . . 427

7 Polynomial trace identities and inequalities . . . . . . . . . . . . . . 443

8 Arithmetic hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . 452

9 Spaces of discrete groups, p, q ∈ {3, 4, 5} . . . . . . . . . . . . . . . . 46510 (p, q, r)-Kleinian groups . . . . . . . . . . . . . . . . . . . . . . . . . 475

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

Weakly Commensurable Groups, with Applications to Differential

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

Gopal Prasad and Andrei S. Rapinchuk

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

2 Weakly commensurable Zariski-dense subgroups . . . . . . . . . . . . 501

3 Results on weak commensurability of S-arithmetic groups . . . . . . 505

4 Absolutely almost simple algebraic groups having the same

maximal tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

5 A finiteness result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

6 Back to geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

Part IV: Knot Groups

Representations of Knot Groups into SL(2,C) and Twisted

Alexander Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Takayuki Morifuji

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

2 Alexander polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 530

3 Representations of knot groups into SL(2,C) . . . . . . . . . . . . . 538

4 Deformations of representations of knot groups . . . . . . . . . . . . 544

5 Twisted Alexander polynomials . . . . . . . . . . . . . . . . . . . . . 548

6 Twisted Alexander polynomials of hyperbolic knots . . . . . . . . . . 560


VI Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Meridional and Non-meridional Epimorphisms between Knot

Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

Masaaki Suzuki

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

2 Some relations on the set of knots . . . . . . . . . . . . . . . . . . . 579

3 Twisted Alexander polynomial and epimorphism . . . . . . . . . . . 580

4 Meridional epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . 584

5 Non-meridional epimorphisms . . . . . . . . . . . . . . . . . . . . . . 591

6 The relation ≥ on the set of prime knots . . . . . . . . . . . . . . . . 5967 Simon’s conjecture and other problems . . . . . . . . . . . . . . . . . 597


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

Advanced Lectures in Mathematics Volume 32

Handbook of Group Actions Volume II

Companion to the volume Handbook of Group Actions, Volume I

International Presswww.intlpress.com

edited by

Lizhen Ji Athanase Papadopoulos Shing-Tung Yau

HIGHER EDUCATION PRESS

Copyright © 2015 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publica-tion is permitted only under license from International Press. Excluded from these pro-visions is material in articles to which the author holds the copyright. (If the author holds copyright, notice of this will be given with the article.) In such cases, requests for per-mission to use or reprint should be addressed directly to the author. ISBN: 978-1-57146-301-2 Printed in the United States of America. 19 18 17 16 15 1 2 3 4 5 6 7 8 9

Advanced Lectures in Mathematics, Volume 32 Handbook of Group Actions, Volume II Companion to the volume Handbook of Group Actions, Volume I Volume Editors: Lizhen Ji (University of Michigan, Ann Arbor) Athanase Papadopoulos (Université de Strasbourg, France) Shing-Tung Yau (Harvard University)

ADVANCED LECTURES IN MATHEMATICS

Executive Editors

Editorial Board

Shing-Tung Yau Harvard University Cambridge, Mass., U.S.A. Lizhen Ji University of Michigan, Ann Arbor Ann Arbor, Mich., U.S.A.

Kefeng Liu University of California at Los Angeles Los Angeles, Calif., U.S.A.

Chongqing Cheng Nanjing University Nanjing, China Zhong-Ci Shi Institute of Computational Mathematics Chinese Academy of Sciences (CAS) Beijing, China Zhouping Xin The Chinese University of Hong Kong Hong Kong, China Weiping Zhang Nankai University Tianjin, China Xiangyu Zhou Institute of Mathematics Chinese Academy of Sciences (CAS) Beijing, China

Tatsien Li Fudan University Shanghai, China Zhiying Wen Tsinghua University Beijing, China Lo Yang Institute of Mathematics Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Sun Yat-sen University Guangzhou, China

Foreword to Volumes I and II

The decision of editing this Handbook came after an international conference weorganized in Kunming (the capital of the Yunnan Province, China) on July 21–29,2012, whose theme was “Group Actions and Applications in Geometry, Topologyand Analysis”.

Kunming is a wonderful place for meetings and for mathematical discussions,especially in the summer, when the weather is most favorable. The conference wasa success, from the mathematical and the human point of view. The city is warm,and the landscape is beautiful. There is a big lake, and a mountain behind thelake. Mathematicians like beauty. Hermann Weyl said: “My work always triedto unite the truth with the beautiful, but when I had to choose one or the other,I usually chose the beautiful.” (Quoted in Hermann Weyl’s Legacy, Institute forAdvanced Study.)

The first two volumes of this Handbook are a record of the Kunming conference,but above all, we want them to be a convenient source for people working on orstudying group actions. In spite of the fact that there were 63 talks, we coveredat Kunming only a small part of this broad subject. In fact, group actions areso important that it is surprising that there was no available handbook on thatsubject so far. It is certainly the ubiquity of group actions that makes such aproject so vast and therefore difficult to attain, and our aim for the time beingis to start it. The present two volumes are the first on this important subject,and more volumes in the same series will appear in the future. Other conferenceson the same subject are also planned in the future; the next one will be in Sanya(Hainan Province).

This Handbook will serve as an introduction and a reference to both beginners,non-experts, experts and users of group actions.

The conference in Kunming would not have gone so smoothly without thegenerous and devoted help of the local organizers, namely, Provost Ailing Gong,Dean Xianzhi Hu, Party Secretary Fengzao Yang, Deputy Dean Yaping Zhang,Youwei Wen, and Jianqiang Zhang from the Kunming University of Science andTechnology. We would like to thank them for their work and hospitality.

Many people have also helped with refereeing and reviewing the papers in thisHandbook, and we would like to thank them all for their help.

L. Ji, A. Papadopoulos, and S.-T. YauAnn Arbor, Strasbourg, and Cambridge MANovember, 2014

Introduction

This is the second volume of the Handbook of Group Actions, which will containseveral volumes.

The present volume is divided into five parts, where the chapters are organizedaccording to the nature of the groups concerned or to the applications of theactions involved.

Part I: Geometric Topology

This part consists of 5 chapters.Discrete groups such as fundamental groups of manifolds and their actions on

topological spaces are essential in topology. In geometric topology, interactionsbetween topology and the fundamental group (or its group ring) are particularlyimportant, and this is reflected in various far-reaching conjectures and problemssuch as the Borel conjecture for aspherical manifolds, the Novikov conjecture, theBaum-Connes conjecture, the Farrell-Jones conjecture and the space form problem.

Chapter 1 by F. T. Farrell, A. Gogolev and P. Ontaneda discusses how ex-otic differentiable structures on higher-dimensional spheres and related unexpectednontrivial topology of the stable pseudo-isotopy space led to solutions of longstand-ing problems on the topology of the space of negatively curved metrics on Rie-mannian manifolds and the topology of the space of Anosov diffeomorphisms. Thespace of negatively curved metrics is a generalization of the Teichmuller space ofhyperbolic surfaces. The authors discuss several results showing that in dimensionshigher than ten, this space is disconnected and has nontrivial homotopy groups, insharp contrast with the Teichmuller spaces of surfaces. They also describe an anal-ogy between metrics of negative curvature and Anosov diffeomorphisms and theypresent several results on the existence of these diffeomorphisms, on the rigidityof manifolds admitting them and on the nontrivial topology of spaces of Anosovdiffeomorphisms.

Chapter 2 by J. Guaschi and D. Juan-Pineda constitutes a comprehensive in-troduction to surface braid groups and a survey of results on the Farrell-Jonesisomorphism conjecture and its variants, e.g. the fibered Farrell-Jones conjectureon the lower algebraic K-theory of group rings of surface braid groups. One pur-pose of these isomorphism conjectures is to compute the algebraic groups explicitly,though it is very difficult to carry them out in general. This paper also containsexplicit computational results for surface braid groups. The combination of thesetwo kinds of results makes surface braid groups special.

Groups acting on trees often have particular properties, generally related todecompositions of these groups, and results about the whole groups can be as-

II Introduction

sembled from those of vertex and edge stabilizers. Chapter 3 by S. K. Roushonis a survey on the Farrell-Jones fibered isomorphism conjecture in both K- andL-theories and its relation with the vanishing of the Whitehead groups for groupsacting on trees. The author presents both the classical material and the recentdevelopments of the lower K-theory and the surgery L-theory of groups acting ontrees. The paper ends with a list of open problems on the (fibered) isomorphismconjecture.

Motivated by Kazhdan’s property T of groups acting on Hilbert spaces, somerecent activity is concerned with actions of groups on Banach spaces. Affine actionsof a group G on a Banach space E can be translated into properties of the firstcohomology group of G with coefficients in the G-module formed by the Banachspace E together with a representation of G. Fixed point properties of such groupactions are equivalent to vanishing of associated cocycles. The existence of properactions on Hilbert spaces has implications in the setting of the Baum-Connesconjecture for groups. Chapter 4 by P. W. Nowak is a survey on recent progresson group actions on Banach spaces and their fixed point properties, motivated byproperty T, but with a stress on actions on Banach spaces which are not Hilbertspaces. The author also thoroughly discusses the property known in the case ofHilbert spaces as a-T-menability, or the Haagerup property, that is, the existenceof metrically proper affine isometric actions on Banach spaces. Applications tothe dimension theory of boundaries of hyperbolic groups are mentioned.

One basic problem on spherical space forms asks which finite groups can actfreely on spheres by homeomorphisms – or diffeomorphisms. In dimension three,this corresponds to the question of which finite groups can occur as fundamentalgroups of compact topological or smooth manifolds. In Riemannian geometry, thespherical space form problem amounts to the classification of compact Riemannianmanifolds with constant sectional curvature, and it was solved quite satisfactorily.On the other hand, the topological spherical space form problem is more subtle,and there has been a lot of work on it. Chapter 5 by I. Hambleton concernstopological spherical space forms and it is an updated short survey on this problem,starting from the nineteenth century work, and ending with the work done recentlybased on Perelman’s solution of the Poincare conjecture. Some new directions ofresearch in this field are also mentioned.

Part II: Representations and Deformations

This part consists of 5 chapters.For a discrete subgroup Γ of a semisimple Lie group G, the most natural and

obvious action is probably the action of Γ on the associated symmetric spaceG/K,where K is a maximal compact subgroup of G. Particular examples are the actionof a Fuchsian group on the upper half-plane and the action of a Kleinian group onthree-dimensional hyperbolic space. There are at least two ways of generalizingthese examples. The first one is to consider representations of discrete groupsinto semisimple Lie groups and their actions on associated symmetric spaces, andthe second one is to consider many actions of a given type, that is, families, ordeformations of such actions, simultaneously.

Chapter 6 by R. D. Canary is a survey on the dynamics of the action of the

Introduction III

outer automorphism group Out(Γ) of a word hyperbolic group Γ on the charactervariety of representations of Γ in a semisimple Lie group G. The stress is on twoaspects of this theory: (1) the setting started by Labourie and developed later onby Guichard, Labourie and Wienhard on the proper discontinuity of the action onspaces of Anosov representations; (2) the works of Canary, Gelander, Lee, Magid,Minsky and Storm on the special case where Γ is the fundamental group of acompact-orientable 3-manifold with boundary and G = PSL(2,C).

In Chapter 7, J. Maubon surveys the Higgs bundle theory of Hitchin and Simp-son and he shows how these bundles can be used in the representation theory ofcomplex hyperbolic lattices into Lie groups of Hermitian type. He proves in par-ticular the rigidity of the so-called maximal representations. The survey coversrepresentations of surface groups but the main part of the survey concerns repre-sentations of higher-dimensional lattices. For this reason, the main Lie group thatis involved is the Hermitian Lie group SU(p; q), since this is the only simple Liegroup of Hermitian type into which maximal representations of higher-dimensionalcomplex hyperbolic lattices are expected to exist.

Chapter 8 by K. Ohshika concerns deformation spaces of Kleinian groups,and limits of such groups. Deformation spaces are equipped with several differ-ent topologies (this was highlighted by Thurston and others), deformation spaceshave several boundary structures, with complicated structure. Ohshika studiesin particular the Bers boundaries for quasi-Fuchsian groups. He presents recentresults that he obtained, which provide a complete classification of the geometriclimits of quasi-Fuchsian groups. Combined with work of Kerckhoff and Thurston,these results explain why the action of the mapping class group on Teichmullerspace does not extend continuously to a Bers boundary, and the author describesa quotient of this boundary, the so-called reduced Bers boundary, which admits acontinuous action of the mapping class group.

Teichmuller spaces are parameter spaces which describe deformation of (equiv-alence classes of) complex structures on surfaces. Several chapters of the first vol-ume of this handbook deal with Teichmuller spaces and their associated mappingclass groups. In principle, complex structures are defined abstractly by charts.But there is another point of view, namely, to consider deformation of surfacesembedded in higher-dimensional manifolds, and in particular, four-dimensionalmanifolds. Diffeomorphism groups and hence mapping class groups of surfaces acton the space of embeddings of the surfaces. One question is how different theseembeddings are when viewed from the point of view of the diffeomorphism groupof the ambient space. Chapter 9 by S. Hirose is a survey of results on deformationsof smooth surfaces embedded in four-dimensional manifolds with respect to thesediffeomorphism groups and to the mapping class groups of the surfaces. Given anembedding e of the surface in the 4-manifold, the main question which is inves-tigated is to what extent a diffeomorphism φ of an embedded surface extends toa diffeomorphism Φ of the ambient 4-manifold respecting the embedding e, thatis, satisfying Φ ◦ e = e ◦ φ. The author surveys some flexibility and rigidity re-sults on this question. Knotted and unknotted embeddings of closed surfaces inthe 4-sphere and algebraic curves in CP 2 are considered. Results on the Rokhlinquadratic form and on the Arf invariant are used.

IV Introduction

One important and effective method to deform surfaces in four-dimensionalsymplectic manifolds is to consider Lefschetz fibrations. This subject was moti-vated by the study of Lefschetz fibrations in algebraic geometry, a basic tool inthat area, and in particular in the study of algebraic surfaces. The works of Don-aldson and Gompf showed that these fibrations give a fairly complete descriptionof symplectic 4-manifolds. Motivated by the question of the geography of algebraicsurfaces in terms of the ratio (or the slope) of two invariants of algebraic surfaces,in Chapter 10, N. Monden gives an introduction to Lefschetz fibrations with aparticular stress on the relation with mapping class groups. The author presentsseveral methods of construction of Lefschetz fibrations and he gives a summaryof some results on the geography of symplectic 4-manifolds and, in particular, onthe construction of some Lefschetz fibrations over the sphere S2 which violate theso-called slope inequality.

Part III: Geometric Groups

This part consists of 3 chapters.Among all infinite discrete groups, the modular group SL(2,Z) is special and

it plays the role of a model for several classes of groups. One reason for itsimportance comes from its actions on very different spaces. Chapter 11 by A. M.Uludag is a survey on several actions of SL(2,Z), starting from the very classicalones and ending with the recently discovered ones. The examples include theactions on Farey trees, on binary quadratic forms, on planar lattices, on the unitcircle considered as the boundary of the upper half plane, and on dessins d’enfants,introduced by Grothendieck in the 1980s.

Simplicial sets are basic objects in algebraic topology and they are a general-ization of simplicial complexes. They are related to the notion of simplicial group.Chapter 12 by J. Wu gives an introduction to simplicial groups and simplicialstructures on geometric groups such as braid groups, link groups and mappingclass groups.

By definition, a Lie group is a combination of a group and a compatible smoothmanifold. If we ignore the underlying smooth structure, then we obtain a discretegroup, or an abstract group. One basic question in Lie group theory is to whatextent the smooth structure of a Lie group can be recovered from the group struc-ture. For example, when is a group homomorphism a Lie group homomorphism?Similar questions can be raised for rational points of linear algebraic groups overgeneral fields. Chapter 13 by I. A. Rapinchuk is a survey on known results onthese questions and on their applications such as to character varieties of elemen-tary subgroups of Chevalley groups over finitely generated commutative rings.

Part IV: Geometric Invariants and Growth of Discrete Groups

This part consists of 2 chapters.One important question in the field of group actions in algebraic geometry

concerns the actions of algebraic groups on algebraic varieties. The behavior oforbits of such actions is a basic question. This theory is one of the basic objects ofgeometric invariant theory and it is closely related to the problem of constructingmoduli spaces in algebraic geometry.

Introduction V

In Chapter 14, D. P. Bac and N. Q. Thang give a summary of classical geomet-ric invariant theory over algebraically closed fields as well as a survey of geometricinvariant theory over non-algebraically closed fields, i.e., actions of algebraic groupson affine varieties over non-algebraically closed fields. The authors discuss ques-tions related to the notion of stability, of various types and closures that are akinto this algebraic geometry setting and of other topological properties of orbits.

One basic point of view in geometric group theory is to consider a finitely gen-erated group as a metric space. One way to do this is to endow the group witha word metric (which depends on the choice of a generating set). One geometricobject associated to such a group (or a word metric) is the growth series, whosecoefficients are the number of group elements at fixed distance from the identity el-ement. This gives rise to the important notion of spherical growth series. Chapter15 by M. Fujii gives an introduction to a method of computing spherical growthseries of finitely generated groups via finite-state automata. The theory is illus-trated by several explicit examples of growth series related to the pure Artin groupof dihedral type. The paper also contains background material on Artin groupsas well as a survey on the known groups for which the growth series is rational orirrational, for some or for an arbitrary generating set.

Part V: Music and Group Actions

Chapter 16 by A. Papadopoulos, concerns applications of groups and groupactions in arts, and more precisely in music theory and music composition. Sev-eral examples are presented, especially from the compositions and the theoreticalwork of the French composer Olivier Messiaen. Since most readers of this hand-book are not familiar with the details on the relation between mathematics andmusic — although most of them know that such a relation exists — the chapteralso contains an introduction, with a historical overview, on this subject.

Contents

Part I: Geometric Topology

Exotic Topology in Geometry and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

F. Thomas Farrell, Andrey Gogolev, and Pedro Ontaneda

1 The space of negatively curved metrics . . . . . . . . . . . . . . . . . 4

2 The Teichmuller and the moduli spaces of negatively curved

metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Bundles with negatively curved fibers . . . . . . . . . . . . . . . . . 13

4 Exotic topology and hyperbolic dynamical systems . . . . . . . . . . 16

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

A Survey of Surface Braid Groups and the Lower Algebraic

K -Theory of Their Group Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

John Guaschi and Daniel Juan-Pineda

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Basic definitions of surface braid groups . . . . . . . . . . . . . . . . 25

3 Some properties of surface braid groups . . . . . . . . . . . . . . . . 31

4 Braid groups of the sphere and the projective plane . . . . . . . . . . 43

5 K-theory of surface braid groups . . . . . . . . . . . . . . . . . . . . 55

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

The Isomorphism Conjecture for Groups with Generalized Free

Product Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Sayed K. Roushon

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2 Lower K-theory and surgery L-theory of groups acting on trees . . . 83

3 The isomorphism conjecture for groups acting on trees . . . . . . . . 90


II Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Group Actions on Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Piotr W. Nowak

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4 Metrically proper actions . . . . . . . . . . . . . . . . . . . . . . . . 139

5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Topological Spherical Space Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Ian Hambleton

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

2 First steps: pre-1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3 Swan complexes: 1960–1970 . . . . . . . . . . . . . . . . . . . . . . . 154

4 Semicharacteristic obstructions and 3-manifolds . . . . . . . . . . . . 156

5 Poincare complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6 Surgery theory and stable existence . . . . . . . . . . . . . . . . . . . 158

7 The finiteness obstruction . . . . . . . . . . . . . . . . . . . . . . . . 159

8 Space forms in the period dimension . . . . . . . . . . . . . . . . . . 160

9 Topological Euclidean space forms . . . . . . . . . . . . . . . . . . . 161

10 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Part II: Representations and Deformations

Dynamics on Character Varieties: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . 175

Richard D. Canary

1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

3 Teichmuller space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4 Convex cocompact representations . . . . . . . . . . . . . . . . . . . 182

5 Hitchin representations . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6 Anosov representations . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7 PSL(2,C)-character varieties . . . . . . . . . . . . . . . . . . . . . . 186

Contents III

8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Higgs Bundles and Representations of Complex Hyperbolic

Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Julien Maubon

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

2 The symmetric space SU(p, q)/S(U(p)×U(q)) . . . . . . . . . . . . . 206

3 Representations, flat bundles, Higgs bundles . . . . . . . . . . . . . . 209

4 Proof of the Milnor-Wood type inequality . . . . . . . . . . . . . . . 230

5 Maximal representations . . . . . . . . . . . . . . . . . . . . . . . . . 236

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Geometric Limits of Kleinian Groups and Their Applications . . . . . 245

Ken’ichi Ohshika

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

3 Examples of geometric limits . . . . . . . . . . . . . . . . . . . . . . 251

4 Classification of geometric limits . . . . . . . . . . . . . . . . . . . . 254

5 Boundaries of deformation spaces . . . . . . . . . . . . . . . . . . . . 262

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Deformations of Surfaces Embedded in 4-dimensional Manifolds

and Their Mapping Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Susumu Hirose

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

3 Surfaces standardly embedded in S4 . . . . . . . . . . . . . . . . . . 274

4 Flexible surfaces in 4-manifolds . . . . . . . . . . . . . . . . . . . . . 289

5 Knotted embeddings in S4 . . . . . . . . . . . . . . . . . . . . . . . . 290

6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

On the Geography of Lefschetz Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Naoyuki Monden

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

IV Contents

2 Lefschetz fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

3 Examples of Lefschetz fibrations . . . . . . . . . . . . . . . . . . . . 302

4 Signatures of Lefschetz fibrations . . . . . . . . . . . . . . . . . . . . 304

5 Fundamental groups of Lefschetz fibrations . . . . . . . . . . . . . . 306

6 Some methods to construct Lefschetz fibrations . . . . . . . . . . . . 307

7 Sections of Lefschetz fibrations . . . . . . . . . . . . . . . . . . . . . 310

8 The geography problems of symplectic 4-manifolds and Lefschetz

fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

9 Holomorphic and non-holomorphic Lefschetz fibrations . . . . . . . . 321

10 The slopes of Lefschetz fibrations with b+2 = 1 . . . . . . . . . . . . 324


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Part III: Geometric Groups

The Modular Group and Its Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

A. Muhammed Uludag

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

2 Rise of the modular group . . . . . . . . . . . . . . . . . . . . . . . . 335

3 The bipartite Farey tree F and its boundary . . . . . . . . . . . . . 343

4 The projective general linear group over Z . . . . . . . . . . . . . . . 348

5 Action on the upper half plane H . . . . . . . . . . . . . . . . . . . . 352

6 Modular graphs and dessins . . . . . . . . . . . . . . . . . . . . . . . 355

7 The action on binary quadratic forms . . . . . . . . . . . . . . . . . 362

8 Et cetera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Appendix: Gosper’s algorithm for continued fraction arithmetic . . . . . 363


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

Groups with Structures and Homotopy Groups . . . . . . . . . . . . . . . . . . . . . 371

Jie Wu

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

2 Simplicial groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

3 Simplicial structure on geometric groups . . . . . . . . . . . . . . . . 385


Contents V

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Abstract Homomorphisms of Algebraic Groups and

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Igor A. Rapinchuk

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

2 On algebraic rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

3 K-theoretic background . . . . . . . . . . . . . . . . . . . . . . . . . 410

4 Rigidity results over commutative rings . . . . . . . . . . . . . . . . 415

5 Rigidity results over noncommutative rings . . . . . . . . . . . . . . 429

6 Applications to character varieties and deformations of

representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Part IV: Geometric Invariants and Growth of Discrete Groups

Some Topics in Geometric Invariant Theory over Non-algebraically

Closed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

Dao Phuong Bac and Nguyen Quoc Thang

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

2 An overview of geometric invariant theory. Observability and

related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

3 Stability in geometric invariant theory over non-algebraically

closed fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

4 Topology of relative orbits for actions of algebraic groups over

completely valued fields . . . . . . . . . . . . . . . . . . . . . . . . . 470


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Computation of the Spherical Growth Series of Finitely Generated

Groups and Monoids by Using Automata . . . . . . . . . . . . . . . . . . . . . . . 479

Michihiko Fujii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

2 Finite-state automata and the subset construction . . . . . . . . . . 482

3 Unique-geodesic word acceptors for groups and monoids,

and growth series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

VI Contents

4 Examples of rational growth series and irrational growth series . . . 491

5 The spherical growth series of Artin groups and Artin monoids . . . 493

6 The pure Artin group PI2(k) and the monoid P+I2(k)

of dihedral

type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

7 Automata for geodesic representatives of P+I2(k)

. . . . . . . . . . . . 503

8 Unique-geodesic word acceptors constructed by subset

construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

9 The spherical growth series of P+I2(k)

. . . . . . . . . . . . . . . . . . 513

10 The spherical growth series of PI2(k) . . . . . . . . . . . . . . . . . . 515


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

Part V: Music and Group Actions

Mathematics and Group Theory in Music . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Athanase Papadopoulos

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

2 A brief overview of the interaction between mathematics

and music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

3 The music of Olivier Messiaen . . . . . . . . . . . . . . . . . . . . . . 540

4 Rhythm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

5 Counterpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

6 Modes of limited transposition . . . . . . . . . . . . . . . . . . . . . 559

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

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