contents · the disc graph is hyperbolic for m = 0 and if m− g ≥ 3. if m= 1 and g≥ 2 then the...

GEOMETRY OF GRAPHS OF DISCS IN A HANDLEBODY

URSULA HAMENSTADT

Abstract. For a handlebody H of genus g ≥ 1 with m ≥ 0 marked points on

the boundary we investigate the geometry of two graphs whose vertex set isthe set of isotopy classes of essential discs in H. In the disc graph, two discs areconnected by an edge of length one if they can be realized disjointly. The elec-

trified disc graph is obtained from the disc graph by adding an edge connectingany two discs whose boundaries are disjoint from a common essential simpleclosed curve in ∂H. We show that if m 6= 1 then the electrified disc graph ishyperbolic, and we determine its Gromov boundary. If m = 1 and if g = 2n

is even then the electrified disc graph contains quasi-isometrically embedded

copies of Z2. The disc graph is hyperbolic for m = 0 and if m − g ≥ 3. If

m = 1 and g ≥ 2 then the disc graph contains a quasi-isometrically embedded

copies of Z2. For m = 2 and g ≥ 2 the disc graph contains for every n > 0 aquasi-isometrically embedded copy of Zn.

Contents

1. Introduction 12. Distance and intersection I: Handlebodies without spots 63. Distance in the curve graph 154. Quasi-geodesics in the superconducting disc graph 185. Hyperbolic thinnings of hyperbolic graphs 226. Decreasing conductivity: The electrified disc graph for m = 0 277. Decreasing conductivity: The disc graph for m = 0 368. The electrified disc graph for a handlebody with a single spot 409. The disc graph of a handlebody with a single spot 4510. Distance and intersection II: The case m ≥ 2 4811. The disc graph for handlebodies with at least two spots 54References 59

1. Introduction

The curve graph CG of an oriented surface S of genus g ≥ 0 withm ≥ 0 puncturesand 3g − 3 +m ≥ 2 is the graph whose vertices are isotopy classes of essential (i.e.non-contractible and not homotopic into a puncture) simple closed curves on S.Two such curves are connected by an edge of length one if and only if they can berealized disjointly. The curve graph is a locally infinite hyperbolic geodesic metricspace of infinite diameter [MM99].

Date: March 25, 2013.Partially supported by the Hausdorff Center Bonn and ERC Grant Nb. 10160104

AMS subject classification:57M99.

1

2 URSULA HAMENSTADT

The mapping class group Mod(S) of all isotopy classes of orientation preservinghomeomorphisms of S acts on CG as a group of simplicial isometries. This action iscoarsely transitive, i.e. the quotient of CG under this action is a finite graph. Curvegraphs and their geometric properties turned out to be an important tool for theinvestigation of the geometry of Mod(S) [MM00]. More recently they were used toconstruct a finite product of hyperbolic spaces into which the mapping class groupquasi-isometrically embeds [BBF10]. Using the fact that the asymptotic dimensionof the curve graph is finite [BF08], one deduces that the asymptotic dimension ofthe mapping class group is finite as well [BBF10].

For mapping class groups of other manifolds, much less is known. Examplesof manifolds with large and interesting mapping class groups are handlebodies. Ahandlebody of genus g ≥ 1 is a compact three-dimensional manifold H which canbe realized as a closed regular neighborhood in R

3 of an embedded bouquet of gcircles. Its boundary ∂H is an oriented surface of genus g. We allow that ∂H isequipped with m ≥ 0 marked points (punctures) which we call spots in the sequel.The group Map(H) of all isotopy classes of orientation preserving homeomorphismsof H preserving the spots is called the handlebody group of H. The homomorphismwhich associates to an orientation preserving homeomorphism of H its restrictionto ∂H induces an embedding of Map(H) into Mod(∂H). We refer to [HH12] formore and for references.

For a number L > 1, a map Φ of a metric space X into a metric space Y is calledan L-quasi-isometric embedding if

d(x, y)/L− L ≤ d(Φx,Φy) ≤ Ld(x, y) + L for all x, y ∈ X.

It is called a quasi-isometry if in addition there is a number R > 0 such that theR-neighborhood of Φ(X) is all of Y . The restriction homomorphism Map(H) →Mod(∂H) in not a quasi-isometric embedding [HH12] and hence understanding thegeometry of the mapping class group does not lead to an understanding of thegeometry of Map(H).

Instead one can try to develop tools which are similar to the tools used for theinvestigation of the mapping class group. In particular, there is an analog of thecurve graph for a handlebody which is defined as follows.

An essential disc in H is a properly embedded disc (D, ∂D) ⊂ (H, ∂H) whoseboundary ∂D is an essential simple closed curve in ∂H.

Definition 1. The disc graph DG of H is the graph whose vertices are isotopyclasses of essential discs in H. Two such discs are connected by an edge of lengthone if and only if they can be realized disjointly.

The handlebody group acts coarsely transitively as a group of simplicial isome-tries on the disc graph. Paralleling the strategy established in [MM00], the discgraph of H and of its subhandlebodies may be used to obtain a geometric under-standing of the handlebody group. However, it turns out that the geometry of discgraphs is more complicated than the geometry of curve graphs.

Theorem 1. Let H be a handlebody of genus g ≥ 1 with m ≥ 0 spots and 3g− 3+m ≥ 2.

(1) If m = 0 then the disc graph of H is hyperbolic.

(2) For m = 1 the disc graph of H contains quasi-isometrically embedded copies

of R2.

GEOMETRY OF GRAPHS OF DISCS IN A HANDLEBODY 3

(3) For m = 2 and g = 1 the disc graph of H is a tree.

(4) For m = 2 and g ≥ 2, the disc graph of H contains for every n ≥ 1 a

quasi-isometrically embedded copy of Rn.

(5) If m− g ≥ 3 then the disc graph of H is hyperbolic.

The first part of the theorem was earlier established by Masur and Schleimer[MS13], with a different proof. Our argument is self-contained and only dependson the results in [MM99, MM00, MM04].

Part (4) of the theorem implies that the asymptotic dimension of the disc graph ofa handlebody of genus g ≥ 2 with two spots on the boundary is infinite. We do notknow whether the asymptotic dimension of the disc graph is finite for m = 0. Themethods of Bell and Fujiwara [BF08] used to establish finite asymptotic dimensionfor the curve graph do not apply in a straightforward way in the situation at hand.

Disc graphs in handlebodies of genus g ≥ 2 with m spots and 3 ≤ m ≤ g+2 aremore complicated. Although we believe that our methods can be used to obtain afairly complete understanding (see the remark at the end of this paper), we do notcarry out this analysis here.

Theorem 1 indicates that unlike in the case of curve graphs and the mappingclass group of a surface of finite type [MM00], disc graphs can not be used in anobvious way to establish a distance formula for the handlebody group. Rather, thegeometry of the handlebody group Map(H) for a handlebody H of genus g ≥ 2without spots seems to resemble the geometry of the outer automorphism groupof the fundamental group of H (which is a free group of rank g). More explicitly,we conjecture that for g ≥ 3 the Dehn function of Map(H) is exponential (see[BV10, HM10] for the corresponding result for the outer automorphism group of afree group, and see [HH12] for a proof of which is closer related to disc graphs).

There is another meaningful Map(H)-graph whose vertices are discs. This graphis the analog of the free factor graph of a free group [KL09] and is defined as follows.

Definition 2. The electrified disc graph is the graph EDG whose vertices are iso-topy classes of essential discs in H and where two vertices D1, D2 are connected byan edge of length one if and only if there is an essential simple closed curve on ∂Hwhich can be realized disjointly from both ∂D1, ∂D2.

Since for any two disjoint essential simple closed curves c, d on ∂H there is asimple closed curve on ∂H which can be realized disjointly from c, d (e.g. one ofthe curves c, d), the electrified disc graph is obtained from the disc graph by addingsome edges. We show

Theorem 2. (1) If m 6= 1 then the electrified disc graph EDG of H is hyper-

bolic.

(2) If m = 1 and if the genus g = 2n of H is even then EDG contains quasi-

isometrically embedded copies of R2.

(3) If m = 0 and g = 2n+ 1 is odd then the map which associates to a disc its

boundary extends to a quasi-isometric embedding EDG → CG.(4) If m ≥ 2 then the map which associates to a disc its boundary is a quasi-

isometry EDG → CG.

We do not know whether for m = 1 and g = 2n+1 odd the electrified disc graphis hyperbolic. We refer to Section 8 for a detailed comment.

Form = 0 and even genus g = 2n the natural vertex inclusion EDG → CG is not aquasi-isometric embedding. However, we can construct a graph SDG whose vertices

4 URSULA HAMENSTADT

are discs and such that the vertex inclusion defines a quasi-isometric embeddingSDG → CG. To define this graph, call a simple closed curve c on ∂H discbusting ifc has an essential intersection with the boundary of every disc.

If H does not have spots then define an I-bundle generator for H to be a dis-cbusting simple closed curve c on ∂H with the following property. There is acompact surface F with connected boundary ∂F , and there is a homeomorphism ofthe orientable I-bundle I(F ) over F onto H which maps ∂F to c. Such an I-bundlegenerator c exists if and only if g = 2n is even (see the remark in Section 2). Thecurve c is separating and decomposes ∂H into two surfaces of genus g/2 = n withconnected boundary. The base surface F of the I-bundle can naturally be identifiedwith one of the two components of ∂H − c. The handlebody group preserves theset of I-bundle generators.

Definition 3. If m = 0 and g = 2n is even then the super-conducting disc graph isthe graph SDG whose vertices are isotopy classes of essential discs in H and wheretwo vertices D1, D2 are connected by an edge of length one if and only if one of thefollowing two possibilities holds.

(1) There is a simple closed curve on ∂H which can be realized disjointly fromboth ∂D1, ∂D2.

(2) There is an I-bundle generator c for H which intersects both ∂D1, ∂D2 inprecisely two points.

In particular, the superconducting disc graph is obtained from the electrified discgraph by adding some edges.

Since the distance in the curve graph CG of ∂H between two simple closed curveswhich intersect in two points does not exceed 3 [MM99], the natural vertex inclusionextends to a coarse 6-Lipschitz map SDG → CG. We show

Theorem 3. If m = 0 and g = 2n is even then the natural vertex inclusion extends

to a quasi-isometric embedding SDG → CG. In particular, SDG is hyperbolic of

infinite diameter.

In contrast, if g is even and m = 0 then the inclusion EDG → CG is not aquasi-isometric embedding. More precisely, we can estimate distances in EDG asfollows.

By an arc in a surfaceX with connected boundary ∂X we mean a homotopy classrelative to ∂X of an embedded arc in X with both endpoints in ∂X. Let c ⊂ ∂Hbe an I-bundle generator and let X1, X2 be the two components of ∂H − c. Definethe arc and curve graph C′(Xi) of Xi to be the graph whose vertices are eitheressential simple closed curves in Xi or arcs with both endpoints on the boundaryof Xi. Two such arcs or curves are connected by an edge of length one if and onlyif they can be realized disjointly.

Every disc D ⊂ H intersects c in at least two points. Define πci (∂D) to be the

union of the intersection components ∂D∩Xi viewed as points in C′(Xi) (i = 1, 2).For a collection A of discs in H we then write diam(πc(A)) to denote the maximumof the diameters of the sets ∪D∈Aπ

ci (D) in the arc and curve graphs C′(Xi) (i = 1, 2).

For numbers C > 0, R > 0 define RC = R if R > C and let RC = 0 otherwise.Let dE be the distance in the electrified disc graph of H, and let dCG be the distancein the curve graph of ∂H. We obtain the following estimate for dE .


Theorem 4. If m = 0 and g = 2n is even then there is a number C > 0 such that

dE(D,E) ≍ dCG(∂D, ∂E) +∑

c

diam(πc(∂D ∪ ∂E))C

where the sum is over all I-bundle generators c for H and ≍ means equality up to

a fixed multiplicative and additive constant.

In analogy to the results of [MS13] we also obtain a distance formula for the discgraph DG for m = 0 which is formulated in Corollary 7.7.

A hyperbolic geodesic metric space admits a Gromov boundary. The Gromovboundary of the curve graph of ∂H can be described as follows [K99, H06].

The space L of geodesic laminations on ∂H can be equipped with the coarse

Hausdorff topology. In this topology, a sequence (λi) of geodesic laminations con-verges to a geodesic lamination λ if every accumulation point of (λi) in the usualHausdorff topology contains λ as a sublamination. This topology is not T0, but itsrestriction to the subset ∂CG of all minimal geodesic laminations which fill up ∂H(i.e. which have an essential intersection with every simple closed curve in ∂H)is Hausdorff. The Gromov boundary of the curve graph can naturally be iden-tified with the subspace ∂CG of L equipped with the coarse Hausdorff topology[K99, H06].

Let

∂H ⊂ ∂CG

be the closed subset of all points which are limits in the coarse Hausdorff topologyof sequences of boundaries of discs in H. As a fairly immediate consequence ofTheorem 3 and the third part of Theorem 2 we obtain

Corollary 1. If m = 0 and g = 2n is even then ∂H with the coarse Hausdorff

topology is the Gromov boundary of SDG. If m = 0 and g = 2n+1 is odd then ∂His the Gromov boundary of EDG.

We can also determine the Gromov boundary of the electrified disc graph in thecase m = 0 and g = 2n even. Namely, for an I-bundle generator c ⊂ ∂H let ∂E(c)be the set of all geodesic laminations which fill up ∂H − c and which are limits inthe coarse Hausdorff topology of boundaries of discs intersecting c in precisely twopoints. Define

∂EDG = ∂H ∪⋃

c

∂E(c) ⊂ L

where the union is over all I-bundle generators and where ∂EDG is equipped withthe coarse Hausdorff topology.

Corollary 2. If m = 0 and g = 2n is even then ∂EDG is the Gromov boundary of

EDG.

In a similar way we can also determine the Gromov boundary of the disc graphin the case m = 0. To this end call a connected essential subsurface X of ∂H thick

if the following properties are satisfied.

(1) No boundary component of X bounds a disc.(2) Every disc intersects X.(3) X is filled by boundaries of discs.

6 URSULA HAMENSTADT

We allow X = ∂H.If X ⊂ ∂H is a thick subsurface then we can consider the space of all discs

whose boundaries are contained in X. As before, these discs can be organized intographs which are defined in the same way as before. In particular, let EDG(X) bethe electrified disc graph of X. Analogous to Theorem 2, if m = 0 then for everythick subsurface X of ∂H the graph EDG(X) is hyperbolic (see Section 6 and 10).Let ∂EDG(X) be its Gromov boundary viewed as a subspace of L which consistsof laminations supported in X. We have

Corollary 3. If m = 0 then the Gromov boundary of DG can be identified with

∪X∂EDG(X) where the union is over all thick subsurfaces X of ∂H.

The organization of this paper is as follows. In Section 2 we use surgery of discsto relate the distance in the superconducting disc graph of a handlebody H withoutspots to intersection numbers of boundary curves.

In Section 3 we estimate the distance in the curve graph using train tracks. Thistogether with a construction of [MM04] is used in Section 4 to show Theorem 3,the third part of Theorem 2 and Corollary 1.

In Section 5 we establish a condition for hyperbolicity of a graph which can beobtained from a hyperbolic graph by deleting some edges. This condition is closelyrelated to the work of Farb [F98], however his results do not directly apply to thesituation at hand. We also need precise information on the behavior of geodesics,so we use a different approach. In Section 6 we deduce the first part of Theorem 2for m = 0, Theorem 4 and Corollary 2 from the results in Section 4 and Section 5.

Section 7 contains the proof of the first part of Theorem 1. The argument usesTheorem 2, the results of Section 6 and an induction over subsurfaces of ∂H. Alongthe way we construct 3g − 5 additional hyperbolic metric Map(H)-graphs whosevertex sets are the set of all discs in H. These graphs lie geometrically betweenEDG and the disc graph DG, i.e. they can be obtained from DG by adding edgesand from EDG by deleting edges (with a slight abuse of notation). Note that ourapproach is opposite to the strategy used in [KR12] to deduce hyperbolicity of thefree factor graph for a free group from hyperbolicity of the free splitting graph.

In Section 8 we show that the electified disc graph of a handlebody of evengenus and a single spot contains quasi-isometric embedded copies of R2. Section9 contains the proof that for m = 1 and g ≥ 2 the disc graph is contains quasi-isometric embedded copies of R2.

The electrified disc graph for handlebodies with at least two spots on the bound-ary is analyzed in Section 10. Finally in Section 11 we show the third, fourth andfifth part of Theorem 1.

Acknowledgement: I am indebted to Saul Schleimer for making me aware ofa missing case in the surgery argument in Section 2 in a first draft of this paperand for sharing his insight in the disc graph with me.

2. Distance and intersection I: Handlebodies without spots

In this section we consider a handlebody H of genus g ≥ 2 without spots on theboundary. We use surgery of discs to establish first estimates for the distance inthe electrified disc graph EDG and in the superconducting disc graph SDG of H.We begin with introducing the basic surgery construction needed later on.


D

Figure A

α

By a disc in the handlebody H we always mean an essential disc in H. TwodiscsD1, D2 are in normal position if their boundary circles intersect in the minimalnumber of points and if every component of D1∩D2 is an embedded arc in D1∩D2

with endpoints in ∂D1 ∩ ∂D2. In the sequel we always assume that discs are innormal position; this can be achieved by modifying one of the two discs with anisotopy.

Let D be any disc and let E be a disc which is not disjoint from D. A componentα of ∂E −D is called an outer arc of ∂E relative to D if there is a component E′

of E − D whose boundary is composed of α and an arc β ⊂ D. The interior ofβ is contained in the interior of D. We call such a disc E′ an outer component ofE −D. An outer component of E −D intersects ∂H in an outer arc α relative toD, and α intersects ∂D in opposite directions at its endpoints.

For every disc E which is not disjoint from D there are at least two distinct outercomponents E′, E′′ of E − D. There may also be components of ∂E − D whichleave and return to the same side of D but which are not outer arcs. An exampleof such a component is a subarc of ∂E which is contained in the boundary of arectangle component of E − D leaving and returning to the same side of D. Theboundary of such a rectangle consists of two subarcs of ∂E with endpoints on ∂Dwhich are homotopic relative to ∂D, and two arcs contained in D.

Let E′ ⊂ E be an outer component of E−D whose boundary is composed of anouter arc α and a subarc β = E′ ∩D of D. The arc β decomposes the disc D intotwo half-discs P1, P2. The unions Q1 = E′ ∪ P1 and Q2 = E′ ∪ P2 are embeddeddiscs in H which up to isotopy are disjoint and disjoint from D. For i = 1, 2 we saythat the disc Qi is obtained from D by simple surgery at the outer component E′

of E −D (see e.g. [S00] for this construction). Since D,E are in minimal position,the discs Q1, Q2 are essential. If X ⊂ ∂H is an essential connected subsurface andif the boundaries of the discs D,E are contained in X then the same holds true forthe boundaries of the discs Qi. In particular, if no boundary component of X is

8 URSULA HAMENSTADT

discbounding then the boundaries of the discs Qi are essential simple closed curvesin X, i.e. curves which are neither contractible nor homotopic into the boundary.

Each disc in H can be viewed as a vertex in the disc graph DG, the electrifieddisc graph EDG and the superconducting disc graph SDG. We will work with allthree graphs simultaneously. Denote by dD (or dE or dS) the distance in DG (or inEDG or in SDG). Note that for any two discs D,E we have

dS(D,E) ≤ dE(D,E) ≤ dD(D,E).

We will also use relative versions of the superconducting disc graph and theelectrified disc graph. For this we first introduce some more terminology (compare[S00, MS13]). Namely, call an arbitrary connected essential subsurface X of ∂Hthick if the following conditions are satisfied.

(1) The boundary of X does not contain a discbounding component.(2) Every disc intersects X.(3) X is filled by boundaries of discs.

An example of a thick subsurface is the complement in ∂H of a suitably chosensimple closed curve which is not discbounding. The entire boundary surface ∂H isthick as well.

For a thick subsurface X of ∂H define EDG(X) to be the graph whose verticesare discs with boundary contained in X. By property (1) in the definition of a thicksubsurface, the boundary of each such vertex is an essential simple closed curve inX. Two such discs D,E are connected by an edge of length one if and only if thereis an essential simple closed curve γ in X which can be realized disjointly from bothD,E (e.g. the boundary of D if the discs D,E are disjoint). Denote by dE,X thedistance in EDG(X). The disc graph DG(X) of X is defined in the obvious way,and we denote by dD,X its distance function.

In the sequel we always assume that all curves and multicurves on X ⊂ ∂H areessential. For two simple closed multicurves c, d on ∂H let ι(c, d) be the geometricintersection number between c, d. The following lemma [MM04] implies that forevery thick subsurface X of ∂H the graph DG(X) is connected. We provide theshort proof for completeness.

Lemma 2.1. Let X ⊂ ∂H be a thick subsurface. Let D,E ⊂ H be discs with

boundaries in X. Then D can be connected to a disc E′ which is disjoint from Eby at most ι(∂D, ∂E)/2 simple surgeries. In particular,

dD,X(D,E) ≤ ι(∂D, ∂E)/2 + 1.

Proof. Let D,E be two discs in normal position with boundary in X. Assumethat D,E are not disjoint. Then there is an outer component of E − D. A discD′ obtained by simple surgery of D at this component is essential in ∂H and itsboundary is contained in X, i.e. D′ ∈ EDG(X). Moreover, D′ is disjoint from D,i.e. we have dD,X(D′, D) = 1, and

(1) ι(∂E, ∂D′) ≤ ι(∂D, ∂E)− 2.

The lemma now follows by induction on ι(∂D, ∂E). �

Lemma 2.1 implies that a thick subsurface X of ∂H can not be a four-holedsphere or a one-holed torus. Namely, if X is a four-holed sphere or a one-holedtorus and if X contains the boundaries of two distinct discs D,E then these discsintersect. Surgery of D at an outer component of E − D results in an essential


disc D′ which is disjoint from the disc D and whose boundary is contained in X.Since any two essential simple closed curves in X intersect, the boundary of D′ isperipheral in X which violates the assumption that no boundary component of Xis discbounding.

Lemma 2.2. Let X ⊂ ∂H be a thick subsurface and let D,E ⊂ H be discs with

boundary in X. If there is an essential simple closed curve α ⊂ X which intersects

∂E in at most one point and which intersects ∂D in at most k ≥ 1 points then

dE,X(D,E) ≤ log2 k + 3.

Proof. Let D,E ⊂ H be discs with boundary in X as in the lemma. If D,E aredisjoint then there is nothing to show, so assume that ∂D∩ ∂E 6= ∅. Let α ⊂ X bea simple closed curve which intersects ∂E in at most one point and which intersects∂D in k ≥ 0 points. If α is disjoint from both D,E then dE,X(D,E) ≤ 1 bydefinition of the electrified disc graph. Thus by perhaps exchanging D and E wemay assume that k ≥ 1. Via a small homotopy we may moreover assume that α isdisjoint from D ∩ E.

We modifyD as follows. There are at least two outer components of E−D. Sinceα intersects ∂E in at most one point, one of these components, say the componentE′, is disjoint from α. The boundary of E′ decomposes D into two subdiscs P1, P2.Assume without loss of generality that P1 contains fewer intersection points withα than P2. Then P1 intersects α in at most k/2 points. The disc D′ = P1 ∪E

′ hasat most k/2 intersection points with α, and up to isotopy, it is disjoint from D. Inparticular, we have dE,X(D,D′) = 1.

Repeat this construction with D′, E. After at most log2 k + 1 such steps weobtain a disc D1 with boundary in X which either is disjoint from E or is disjointfrom α. The distance between D and D1 in the graph EDG(X) is at most log2 k+1.

If D1 and E are disjoint then dE,X(D1, E) ≤ 1 and dE,X(D,E) ≤ log2 k+ 2 andwe are done. Otherwise apply the above construction to D1, E but with the rolesof D1 and E exchanged. We obtain a disc E1 which is disjoint from both E and α,in particular it satisfies dE,X(E1, E) = 1. The discs D1, E1 are both disjoint fromα and therefore dE,X(D1, E1) ≤ 1 by the definition of the electrified disc graph.Together this shows that

dE,X(D,E) ≤ log2 k + 1 + dE,X(D1, E) ≤ log2 k + 3.

�

A simple closed multicurve γ in a thick subsurface X of ∂H is called discbusting

if γ intersects every disc with boundary in X.

Definition 2.3. An I-bundle generator in ∂H is a discbusting simple closed curveγ ⊂ ∂H with the following property. There is an oriented I-bundle J (F ) overa compact surface F with connected boundary ∂F , and there is an orientationpreserving homeomorphism of J (F ) onto H which maps ∂F to γ.

We call the surface F the base of the I-bundle generated by γ.We claim that the base F of the I-bundle generated by γ is orientable. Namely,

otherwise there is a two-sheeted orientation cover F of F and an orientation re-versing involution ϕ of F with F /ϕ = F . The map ϕ extends to an orientation

preserving involution ϕ : F × [0, 1] → F × [0, 1] which exchanges the endpoints of

the interval [0, 1] and preserves F × { 12}. Its restriction to F × { 1

2} ∼ F is just the

10 URSULA HAMENSTADT

map ϕ. Moreover, we have H = F × [0, 1]/ϕ. As a consequence, F is a surface ofgenus g− 1 ≥ 1 with two boundary components and hence the first integral homol-ogy group of the non-orientable quotient F contains an element of order two. Onthe other hand, H = F × [0, 1]/ϕ admits F = F × { 1

2}/ϕ as a deformation retract(see [He80]). Since the fundamental group of H is free, the first integral homologygroup of H does not have an element of order two which yields a contradiction.

To summarize, if ∂H admits an I-bundle generator γ with base surface F thenthe genus g of ∂H is even and equals twice the genus of F . The I-bundle J (F ) =F × [0, 1] is trivial. The I-bundle over every essential arc in F with endpointsin ∂F is an embedded disc in H. There is an orientation reversing involutionΦ : H → H whose fixed point set intersects ∂H precisely in γ. This involution actsas a reflection in the fiber. The union of an essential arc γ in F with endpoints in∂F with its image under Φ is the boundary of a disc in H (there is a small abuse ofnotation here- since the fixed point set of Φ intersects ∂H in a subset of the fibreover ∂F ). This disc is just the I-bundle over γ.

We can also define I-bundle generators for thick subsurfaces X of ∂H. To thisend recall that the boundary ∂J (F ) of an oriented I-bundle J (F ) over a compact(not necessarily orientable) surface F with (not neceessarily connected) boundary∂F decomposes into the horizontal boundary and the vertical boundary. The verticalboundary is the interior of the restriction of the I-bundle to ∂F and consists of acollection of pairwise disjoint open incompressible annuli. The horizontal boundaryis the complement of the vertical boundary.

For a given boundary component α of F , the union of the horizontal boundaryof J (F ) with the I-bundle over α is a compact connected orientable surface Fα ⊂∂J (F ), possibly with boundary. If the boundary of F is not connected then Fα isproperly contained in the boundary ∂J (F ) of J (F ).

Define an I-bundle generator in X to be an essential simple closed curve γ ⊂X with the following property. There is a compact surface F with non-emptyboundary ∂F , there is a boundary component α of F , and there is an orientationpreserving embedding Ψ of the oriented I-bundle J (F ) over F into H which mapsα to γ and which maps Fα onto the complement in X of a tubular neighborhoodof the boundary ∂X of X. The image of ∂J (F )−Fα is contained in the interior ofH. In particular, every boundary circle of X is freely homotopic into a componentof the vertical boundary of J (F ) which is disjoint from α.

An example can be obtained as follows. Let F be an oriented surface of genusk ≥ 1 with two boundary components. The oriented I-bundle J (F ) = F×[0, 1] overF is homeomorphic to a handlebody H of genus 2k+1. A boundary component β ofF is neither discbounding nor discbusting in H, and the subsurface X = ∂H −β ⊂∂H is thick. The second boundary component γ of F intersects every disc withboundary in X and is an I-bundle generator for X whose base is the surface F . Theimage of F × [0, 1] = J (F ) under the embedding J (F ) → H is the complement ofa neighborhood of β in H which is homeomorphic to a solid torus.

Now assume that the base F of the I-bundle J (F ) determined by γ is non-orientable. Up to isotopy, the thick subsurface X of ∂H is the intersection ofthe boundary ∂J (F ) of the bundle J (F ) ⊂ H with ∂H. It can be obtained

from the orientation cover F of F by glueing an annulus to the two preimages ofthe preferred boundary component α of F with a homeomorphism reversing theboundary orientations. The generator γ ⊂ X of the I-bundle is a non-separating


simple closed curve in X. The I-bundle over every essential arc β in F withendpoints on α is a disc in H. Its boundary is the preimage of β in Fα ⊂ ∂J (F )viewed as the orientation cover of F and where the two preimages of the boundarycomponent α are identified (here we use the same small abuse of terminology asbefore).

If D,E ⊂ H are discs in normal position then each component of D − E is adisc and therefore the graph dual to the cell decomposition of D whose two-cellsare the components of D − E is a tree. If D − E only has two outer componentsthen this tree is just a line segment. The following lemma analyzes the case thatthis holds true for both D − E and E −D.

Lemma 2.4. Let X ⊂ ∂H be a thick subsurface and let D,E ⊂ H be discs in

normal position, with ∂D, ∂E ⊂ X. If D − E and E − D only have two outer

components then one of the following two possibilities is satisfied.

(1) dE,X(D,E) ≤ 4.(2) D,E intersect some I-bundle generator γ in X in precisely two points.

Proof. Let X ⊂ ∂H be a thick subsurface and let D,E be two discs in normalposition with boundaries in X. Assume that D−E and E−D only have two outercomponents. Then each component of D −E,E −D either is an outer componentor a rectangle, i.e. a disc whose boundary consists of two components of D∩E andtwo arcs contained in ∂D ⊂ ∂H or ∂E ⊂ ∂H, respectively. Since dE,X(D,E) = 1if there is an essential simple closed curve in X which is disjoint from ∂D ∪ ∂E, wemay assume without loss of generality that ∂D ∪ ∂E fills up X. This means thatX − (∂D ∪ ∂E) is a union of discs and peripheral annuli.

Choose tubular neighborhoods N(D), N(E) of D,E in H which are homeomor-phic to an interval bundle over a disc and which intersect ∂H in an embeddedannulus. We may assume that the interiors A(D), A(E) of these annuli are con-tained in the interior of X. Then ∂N(D) − A(D), ∂N(E) − A(E) is the union oftwo properly embedded disjoint discs in H isotopic to D,E. We may assume that∂N(D)−A(D) is in normal position with respect to ∂N(E)−A(E) and that

S = ∂(N(D) ∪N(E))− (A(D) ∪A(E))

is a compact surface with boundary which is properly embedded in H. Since H isassumed to be oriented, the boundary ∂(N(D) ∪ N(E)) of N(D) ∪ N(E) has aninduced orientation which restricts to an orientation of S.

Let (P, σ) be a pair consisting of an outer component P of D−E and an orienta-tion σ of D which induces an orientation of P . The orientation σ together with theorientation of H determines an orientation of the normal bundle of D and hence σdetermines a side of D in N(D), say the right side. We claim that the componentQ of the surface S containing the copy of P in ∂N(D) to the right of D is a discwhich contains precisely one other pair (P ′, σ′) of this form, i.e. P ′ is an outercomponent of D − E or of E −D, and σ′ an orientation of P ′.

Namely, by construction, each component of ∂N(D) − (A(D) ∪ N(E)) eithercorresponds to an outer component of D − E and the choice of a side, or it corre-sponds to a rectangle component of D − E and a choice of a side. A componentcorresponding to a rectangle is glued at each of its two sides which are contained inthe interior of H to a component of ∂N(E)− (A(E) ∪N(D)). In other words, upto homotopy, the component Q of S can be written as a chain of oriented discs be-ginning with P and alternating between components of D−E and E−D equipped


with one of the two possible orientations. Since Q is embedded in H and containsP , this chain can not be a cycle and hence it has to terminate at an oriented outercomponent of D − E or E −D which is distinct from (P, σ).

To summarize, each pair (P, σ) consisting of an outer component P of D−E orE−D and an orientation σ ofD or E determines a unique component of the orientedsurface S. This component is a properly embedded disc in H which is disjoint fromD ∪E. Each such disc corresponds to precisely two such pairs (P, σ), so there is atotal of four such discs. Denote these discs by Q1, . . . , Q4. If one of these discs isessential, say if this holds true for the disc Qi, then Qi is an essential disc disjointfrom both D,E with boundary in X and hence dE,X(D,Qi) ≤ 1, dE,X(E,Qi) ≤ 1and we are done.

Otherwise define a cycle to be a subset C of {Q1, . . . , Q4} of minimal cardinalityso that the following holds true. Let Qi ∈ C and assume that Qi contains a pair(B, ζ) consisting of an outer component B of D−E (or of E−D) and an orientationζ of D (or E). If Qj is the disc containing the pair (B, ζ ′) where ζ ′ is the orientationof D (or E) distinct from ζ then Qj ∈ C. Note that two distinct cycles are disjoint.The length of the cycle is the number of its components.

For each cycle C we construct a properly embedded annulus (A(C), ∂A(C)) ⊂(H,X) as follows. Remove from each of the discs Qi in the cycle the subdiscswhich correspond to outer components of D − E,E − D and glue the discs alongthe boundary arcs of these outer components.

To be more precise, let B be an outer component of D − E corresponding to asubdisc of Qi and let β = B ∩ E. Then the complement of B in Qi (with a smallabuse of notation) contains the arc β in its boundary, and the orientation of Qi

defines an orientation of β. There is a second disc Qj in the cycle which contains Band which induces on β the opposite orientation (Qj is not necessarily distinct fromQi). Glue Qi−B to Qj −B along β and note that the resulting surface is oriented.Doing this with each of the outer components of D − E and E − D contained inthe cycle yields a properly embedded annulus A(C) ⊂ H with boundary in X asclaimed.

If there is a cycle C of odd length then for one of the two discs D,E, say thedisc D, the cycle contains precisely one outer component (with both orientations).Since the discs Qj are disjoint from D ∪E, this means that a boundary curve γ ofA(C) intersects the disc D in precisely one point, and it intersects the disc E in atmost two points. In particular, γ is an essential curve in X. Lemma 2.2 now showsthat dE,X(D,E) ≤ 4.

Similarly, if there is a cycle C of length two then there are two possibilities. Thefirst case is that the cycle contains both an outer component of D−E and an outercomponent of E − D. Then a boundary curve γ of A(C) intersects each of thediscs D,E in precisely one point. In particular, γ is an essential curve in X anddE,X(D,E) ≤ 3.

If C contains both outer components of say the disc D then a boundary curveγ of A(C) intersects D in precisely two points, and it is disjoint from E. Let D′

be a disc obtained from D by a simple surgery at an outer components of E −D.Then either D′ is disjoint from both D,E (which is the case if D′ is composed ofan outer component of E−D and an outer component of D−E) or D′ intersects γin precisely one point and is disjoint from D. As before, we conclude from Lemma2.2 that dE,X(D,E) ≤ 4.


We are left with the case that there is a single cycle C of length four. Let γ1, γ2be the two boundary curves of A(C). Then γ1, γ2 are simple closed curves in Xwhich are freely homotopic in the handlebody H. We claim that γ1, γ2 are freelyhomotopic in X. Namely, assume that the discs Qi are numbered in such a waythat Qi and Qi+1 share one outer component of D − E or E −D. Glue the discsQ1, . . . , Q4 successively to a single disc Q with the surgery procedure describedabove (namely, if P is the outer component of D − E or E −D contained in bothQ1, Q2 then remove P from Q1, Q2 and glue Q1 to Q2 along the resulting boundaryarc to form a disc Q1, glue Q1 to Q3 to form a disc Q2, and glue Q2 to Q4 to obtainthe disc Q). Since by assumption none of the discs Qi is essential, the disc Q iscontractible. In particular, Q is homotopic with fixed boundary to an embeddeddisc in X ⊂ ∂H. Now the annulus A(C) is obtained from Q by identifying twodisjoint boundary arcs and hence A(C) is homotopic into X. Assume from now onthat A(C) ⊂ X.

By construction, each of the simple closed curves ∂D, ∂E intersects A(C) inprecisely two arcs connecting the two boundary components of A(C). These areexactly the boundary arcs of the outer components of D − E,E −D.

The intersection arcs ∂D∩A(C), ∂E∩A(C) decompose A(C) into four rectangles.The annulus A(C) has a natural structure of an I-bundle over one of its boundarycircles, with ∂D ∩A(C), ∂E ∩A(C) as a union of fibres.

Let ζ ⊂ ∂D be a component of ∂D − A(C). Then ζ is a union of boundarycomponents of rectangles embedded in D. Two opposite sides of such a rectangleR are contained in ∂D. There is a unique side ρ of R which is contained in ζ, andthe side opposite to ρ is contained in the component ζ ′ of ∂D−A(C) disjoint fromζ. This decomposition of D into rectangles determines for D the structure of anI-bundle over ζ. Each component of D ∩E is a fibre of this I-bundle, and the twocomponents of ∂D ∩A(C) are fibres as well. Similarly, E is an I-bundle over eachcomponent ξ of ∂E −A(C).

Since ∂D ∪ ∂E fills up X, each component of X − (∂D ∪ ∂E) which does notcontain a boundary component of X is a polygon, i.e. a disc bounded by finitelymany subarcs of ∂D, ∂E. Such a polygon P is contained in the boundary of acomponent V ofH−(D∪E). The boundary ∂V of V has two connected componentscontained in ∂H. One of these components is the polygon P , the other componentP ′ either is a polygon component of X − (∂D ∪ ∂E), or it contains a boundarycomponent of X.

The complement of P ∪P ′ in ∂V is a finite collectionW of fibred rectangles. Thebase of such a rectangle is an edge of the boundary ∂P of the polygon P . Usingagain the fact that ∂D∪∂E fills up X, if P ′ is a polygon in X then V is a 3-ball. IfP ′ contains a boundary component α of X then the boundary of P ′ is homotopicto α, and ∂V − P ′ is an embedded disc in H whose boundary is homotopic to α.This violates the assumption that no boundary component of X is discbounding.

As a consequence, each component V of H− (D∪E) which contains a polygonalregion P of X−(∂D∪∂E) in its boundary is a ball whose boundary consists of P , afinite union R of fibred rectangles with base ∂P and a second polygonal componentP ′ of X−(∂D∪∂E). The I-bundle structure on R naturally extends to an I-bundlestructure on V . Therefore the union of the components of H − (D ∪ E) whichcontain a polygonal region in X− (∂D∪∂E) in their boundary is an I-bundle. Theinvolution of the I-bundle which exchanges the endpoints of the interval I preserves


each component V of H − (D ∪E) whose boundary intersects X, and it exchangesthe two components of V ∩X.

If the annulus A(C) is separating in X then each of the two components ofX − A(C) contains precisely one component of ∂D − A(C), ∂E − A(C). The baseof the I-bundle can be identified with a component F of X − A(C), and this baseis orientable. Up to homotopy, each boundary component of X is contained in theI-bundle over a boundary component of F which is distinct from γ. Vice versa,each boundary component of F distinct from γ corresponds to two boundary circlesof X in this way. These boundary circles are homotopic in H, and they may behomotopic in ∂H as well (see the example preceding this proof). A core curve γ ofA(C) is a generator of the I-bundle as defined above.

If the curve γ is non-separating then the base F of the I-bundle is non-orientable.As before, there is an orientation reversing involution ϕ of X which preserves γpointwise and exchanges the two components of ∂D − γ and of ∂E − γ. In otherwords, each of the discs D,E is the I-bundle over an arc with boundary on thedistinguished boundary component of the base F . This completes the proof of thelemma. �

For a thick subsurface X of ∂H let SDG(X) be the graph whose vertices arediscs with boundaries contained in X and where two such discs D,E are connectedby an edge of length one if one of the following two possibilities is satisfied.

(1) There is an essential simple closed curve α ⊂ X (i.e. which is essential as acurve in the subsurface X of ∂H) which is disjoint from D∪E (for example,∂D if D,E are disjoint).

(2) There is an I-bundle generator γ ⊂ X which intersects both D,E in pre-cisely two points.

We denote by dS,X the distance in SDG(X). If X = ∂H then we simply write dSinstead of dS,∂H .

We use Lemma 2.4 to improve Lemma 2.2 as follows.

Proposition 2.5. Let X ⊂ ∂H be a thick subsurface and let D,E ⊂ H be essential

discs with boundary in X. If there is an essential simple closed curve α ⊂ X which

intersects ∂D, ∂E in at most k ≥ 1 points then dS,X(D,E) ≤ 2k + 4.

Proof. Let D,E be essential discs in normal position as in the proposition whichare not disjoint.

Let α be an essential simple closed curve in X which intersects both ∂D and ∂Ein at most k ≥ 1 points. We may assume that these intersection points are disjointfrom ∂D ∩ ∂E.

Let p ≥ 2 (or q ≥ 2) be the number of outer components ofD−E (or of E−D). Ifp = 2, q = 2 then Lemma 2.4 shows that either dE,X(D,E) ≤ 4 or ∂D, ∂E intersectsome I-bundle generator γ in precisely two points, and we have dS,X(D,E) = 1.

Let j ≤ k, j′ ≤ k be the number of intersection points of D,E with α. Ifmin{j, j′} ≤ 1 then dE,X(D,E) ≤ log2 k+3 by Lemma 2.2. Thus it suffices to showthe following. If max{p, q} ≥ 3 and min{j, j′} ≥ 2 then there is a simple surgerytransforming the pair (D,E) to a pair (D′, E′) with the following properties.

(1) D′ is disjoint from D, E′ is disjoint from E.(2) Either D = D′ or E = E′.(3) The total number of intersections of α with D′ ∪E′ is strictly smaller than

j + j′.


To this end assume without loss of generality that q ≥ 3. If j/2 > j′/3 thenchoose an outer component E1 of E − D with at most j′/3 intersections with α.This is possible because E −D has at least three outer components. Let D1 be acomponent of D −E1 which intersects α in at most j/2 points. Then D1 ∪E1 is adisc which is disjoint from D and has at most j/2 + j′/3 < j intersections with α.

On the other hand, if j/2 ≤ j′/3 then choose an outer component D1 of D − Ewith at most j/2 intersections with α. Let E1 be a component of E −D1 with atmost j′/2 intersections with α and replace E by the disc E1 ∪D1 which is disjointfrom E and intersects α in at most j/2 + j′/2 < j′ points.

This is what we wanted to show. �

For easy reference we note

Corollary 2.6. Let H be a handlebody of genus g ≥ 2 without spots. Let D,E ⊂ Hbe discs and assume that there is a simple closed curve γ on ∂H which intersects

both ∂D, ∂E in at most k ≥ 1 points.

(1) If g = 2n is even then dS(D,E) ≤ 2k + 4.(2) If g = 2n+ 1 is odd then dE(D,E) ≤ 2k + 4.

Remark: The arguments in this section use the fact that every simple surgeryof a disc at an outer component of another disc yields an essential disc in H. Theyare not valid for handlebodies with spots.

3. Distance in the curve graph

The purpose of this section is to establish an estimate for the distance in the curvegraph of the boundary of the handlebody H which will be essential for a geometricdescription of the superconducting disc graph. The results in this section are validfor an arbitrary oriented surface S of genus g ≥ 0 with m ≥ 0 punctures and3g − 3 +m ≥ 2.

The idea is to use train tracks on S. We refer to [PH92] for all basic notions andconstructions regarding train tracks.

A train track η (which may just be a simple closed curve) is carried by a traintrack τ if there is a map F : S → S of class C1 which is homotopic to the identity,with F (η) ⊂ τ and such that the restriction of the differential dF of F to thetangent line of η vanishes nowhere. Write η ≺ τ if η is carried by τ . If η ≺ τ thenthe image of η under a carrying map is a subtrack of τ which does not depend onthe choice of the carrying map. Such a subtrack is a subgraph of τ which is itselfa train track. Write η < τ if η is a subtrack of τ .

A train track τ is called large [MM99] if each complementary component ofτ is either simply connected or a once punctured disc. A simple closed curve ηcarried by τ fills τ if the image of η under a carrying map is all of τ . A diagonal

extension of a large train track τ is a train track ξ which can be obtained fromτ by subdividing some complementary components which are not trigons or oncepunctured monogons.

A trainpath on τ is an immersion ρ : [k, ℓ] → τ which maps every interval[m,m + 1] diffeomorphically onto a branch of τ . We say that ρ is periodic ifρ(k) = ρ(ℓ) and if the inward pointing tangent of ρ at ρ(k) equals the outwardpointing tangent of ρ at ρ(ℓ). Any simple closed curve carried by a train track τdefines a periodic trainpath and a transverse measure on τ . The space of transversemeasures on τ is a cone in a finite dimensional real vector space. Each of its extreme


rays is spanned by a vertex cycle which is a simple closed curve carried by τ . Avertex cycle defines a periodic trainpath which passes through every branch at mosttwice, in opposite direction [H06, Mo03].

Let η be a large train track. If η ≺ τ then τ is large as well. In particular, ifη′ < η is a large subtrack of η and if ξ is a diagonal extension of η′, then a carryingmap F : η → τ induces a carrying map of η′ onto a large subtrack τ ′ of τ , and itinduces a carrying map of ξ onto a diagonal extension of τ ′.

Definition 3.1. A pair η ≺ τ of large train tracks is called wide if every simpleclosed curve which is carried by a diagonal extension of a large subtrack of η fills adiagonal extension of a large subtrack of τ .

We have

Lemma 3.2. If σ ≺ η ≺ τ and if the pair η ≺ τ is wide then σ ≺ τ is wide.

Proof. Let σ′ be a large subtrack of σ and let ξ be a diagonal extension of σ′. Thenthe carrying map σ → η maps σ′ onto a large subtrack η′ of η, and it maps ξ to adiagonal extension ζ of η′. Similarly, η′ is mapped to a large subtrack τ ′ of τ , andζ is mapped to a diagonal extension ρ of τ ′.

A simple closed curve α carried by ξ is carried by ζ. In particular, since η ≺ τis wide, α fills a large subtrack of ρ. From this the lemma follows. �

A splitting and shifting sequence is a finite sequence (τi)0≤i≤n of train tracks sothat for each i, τi+1 can be obtained from τi by a sequence of shifts followed by asingle split. We allow the split to be a collision, i.e. to reduce the number of edges.Note that τi+1 is carried by τi for all i and the pair τi+1 ≺ τi is not wide.

Recall from the introduction the definition of the curve graph CG of S. For anessential simple closed curve c on S let i(c) ∈ {0, . . . , n} be the largest number withthe following property. There is a large subtrack η of τi(c) so that c is carried by adiagonal extension ξ of η and fills ξ. If no such number exists then put i(c) = 0.

Define a projection P : CG → (τi)0≤i≤n by

P (c) = τi(c).

Extend the map P to the edges of CG by mapping an edge to the image of one ofits endpoints.

Lemma 3.3. Let c, d be disjoint simple closed curves on S. Assume that P (c) = τi.If τi ≺ τj is wide then P (d) = τs for some s ≥ j.

Proof. Assume that P (c) = τi ≺ τj is wide. By the definition of the map P , thereis a large subtrack η of τi so that c fills a diagonal extension ξ of η. By Lemma 4.4of [MM99], since d is disjoint from c, d is carried by a diagonal extension ζ of ξ.Then ζ is a diagonal extension of η.

Since τi ≺ τj is wide, d fills a diagonal extension of a large subtrack of τj . Thisimplies that P (d) = τs for some s ≥ j. �

Define a distance function dg on (τi)0≤i≤n as follows. For i < j, the gap distance

dg(τi, τj) between τi and τj is the smallest number k > 0 so that there is a sequencei0 = i < i1 < · · · < ik = j with the property that for each p < k, the pairτip+1

≺ τip is not wide. Note that this defines indeed a distance since for each ℓthe pair τℓ+1 ≺ τℓ is not wide and hence dg(τi, τj) ≤ j − i. Moreover, the triangleinequality is immediate from Lemma 3.2.


The following is a consequence of Lemma 3.3. For its formulation, define a mapP from a metric space X to a metric space Y to be coarsely L-Lipschitz for someL > 1 if d(Px, Py) ≤ Ld(x, y) + L for all x, y ∈ X.

Corollary 3.4. The map P : CG → ((τi), dg) is coarsely 2-Lipschitz.

Define a map Υ : (τi)0≤i≤n → CG by associating to the train track τi one of itsvertex cycles. We have

Lemma 3.5. There is a number p > 1 not depending on (τi)0≤i≤n so that the map

Υ : ((τi), dg) → CG is coarsely p-Lipschitz.

Proof. It suffices to show the following. If τ ≺ η is not wide then the distance inCG between a vertex cycle α of τ and a vertex cycle β of η is uniformly bounded.

To this end note that if α is a simple closed curve which is carried by a large traintrack ξ then the image of α under a carrying map is a subtrack of ξ. If this subtrackis not large then α is disjoint from an essential simple closed curve α′ which canbe represented by an edge-path in ξ (possibly with corners) which passes througha branch of ξ at most a uniformly small number of times. Since a vertex cycle ofξ passes through each branch of ξ at most twice, this implies that α′ intersects avertex cycle of ξ in uniformly few points (see [H06] for details). In particular, thedistance in CG between α and a vertex cycle of ξ is uniformly bounded [MM99].

On the other hand, if τ is another large train track and if ξ is a diagonal extensionof a large subtrack τ ′ of τ then a vertex cycle of ξ intersects a vertex cycle of τ inuniformly few points. Hence the distance in CG between a vertex cycle of τ and avertex cycle of ξ is uniformly bounded. Together we deduce that the distance inCG between α and a vertex cycle of τ is uniformly bounded.

Now by definition, if τ ≺ η is not wide then there is a curve α which is carriedby a diagonal extension ξ of a large subtrack τ ′ of τ and such that the followingholds true. A carrying map τ → η induces a carrying map of ξ onto a diagonalextension ζ of a large subtrack of η. The train track ζ carries α and so that α doesnot fill a large subtrack of ζ. Since a carrying map ξ → ζ maps a large subtrack ofξ onto a large subtrack of ζ, the curve α does not fill a large subtrack of ξ.

By the above discussion, the distance in CG between α and any vertex cycle ofboth τ and η is uniformly bounded. This shows the lemma. �

Call a map Φ of a metric space (X, d) into a subset A of X a coarse Lipschitz

retraction if there is a number L > 1 with the following properties.

(1) d(Φ(x),Φ(y)) ≤ Ld(x, y) + L.(2) d(x,Φ(x)) ≤ L whenever x ∈ A.

We are now ready to show

Corollary 3.6. For any splitting and shifting sequence (τi)0≤i≤n the map Υ ◦P is

a coarse L-Lipschitz retraction of CG for a number L > 1 not depending on (τi).

Proof. Let dCG be the distance in the curve graph of S. By Corollary 3.4 andLemma 3.5 it suffices to show that dCG(α,Υ ◦ P (α)) ≤ L for a universal constantL > 1 and every vertex cycle α of a train track τi from the sequence.

To this end observe that since α is a vertex cycle of τi, α is carried by each ofthe train tracks τj for j ≤ i, moreover α does not fill a diagonal extension of a largesubtrack of τi. On the other hand, by definition of a wide pair, if τi ≺ τj is widethen α fills a large subtrack of τj . This means that P (α) = τs for some s ≥ j sothat the pair τi ≺ τs+1 is not wide. The corollary now follows from Lemma 3.5. �


Remark: The above discussion immediately implies that the image under Υ ofa splitting and shifting sequence of train tracks is an unparametrized quasi-geodesicin CG. This was earlier established in [MM04] (see also [H06]).

4. Quasi-geodesics in the superconducting disc graph

In this section we prove the third part of Theorem 2, Theorem 3 and Corollary1 from the introduction.

For a thick subsurface X of ∂H recall from Section 2 the definition of the graphSDG(X). Our goal is to show that the natural map which associates to a discits boundary defines a quasi-isometric embedding of SDG(X) into the curve graphCG(X) of X. To simplify the notation we identify in the sequel a disc in H with itsboundary circle. Thus we view the vertex set of SDG(X) as a subset of the curvegraph CG(X) of X.

The argument is based on the results in Section 2-3 and a construction from[MM04]. This construction uses a specific type of surgery sequences of discs whichcan be related to train tracks which is defined as follows.

Let H be a handlebody of genus g ≥ 2 without spot. Let D,E ⊂ H be two discsin normal position. Let E′ be an outer component of E −D and let D1 be a discobtained from D by simple surgery at E′.

Let α be the intersection of ∂E′ with ∂H. Then up to isotopy, the boundary∂D1 of the disc D1 contains α as an embedded subarc. Moreover, α is disjoint fromE. In particular, given an outer component E′′ of E −D1, there is a distinguishedchoice for a disc D2 obtained from D1 by simple surgery at E′′. The disc D2 isdetermined by the requirement that α is not a subarc of ∂D2. Then for an outercomponent of E−D2 there is a distinguished choice for a disc D3 obtained from D2

by simple surgery at an outer component of E−D2 etc. We call a surgery sequence(Di) of this form a nested surgery path in direction of E. Note that the boundaryof each disc Di is composed of a single subarc of ∂D and a single subarc of ∂E.

Recall from Section 2 the definition of a thick subsurface of ∂H. Recall that athick subsurface is different from a four-holed sphere and a one-holed torus. Thefollowing result is due to Masur and Minsky (this is Lemma 4.2 of [MM04] whichis based on Lemma 4.1 and the proof of Theorem 1.2 in that paper).

Proposition 4.1. Let X ⊂ ∂H be a thick subsurface and let D,E ⊂ ∂H be any

discs with boundary in X. Let D = D0, . . . , Dn be a nested surgery path in direction

of E which connects D to a disc Dn disjoint from E. Then for each i ≤ n there is

a train track τi on X with a single switch such that the following holds true.

(1) τi carries ∂E and ∂E fills up τi.(2) τi+1 ≺ τi.(3) The disc Di intersects τi only at the switch.

The train tracks τi in the proposition are constructed as follows.Let α = ∂D, β = ∂E. Assume that the curves α, β are smooth (for a smooth

structure on ∂H) and fill up X. This means that the complementary componentsof α ∪ β are all polygons or once holed polygons where in our setting, a hole is aboundary component of X. Let P be a complementary polygon which has at least6 sides. Such a polygon exists since the Euler characteristic of X is negative. Itsedges are subsegments of α and β. Let I be a boundary edge of P contained inα. Collapse α− I to a single point with a homotopy F of X. This can be done in


such a way that the restriction of F to β is nonsingular everywhere. The resultinggraph has a single vertex. Collapsing the bigons in the graph to single arcs yieldsa train track τ with a single switch [MM04].

Let b ⊂ β be an outer arc for E −D and let a ⊂ α− I be the subarc of α whichis bounded by the endpoints of b and which does not intersect the interval I. Thena ∪ b is the boundary of a disc D1 obtained from D by nested surgery at b. Thenew train track τ1 obtained from the above construction is obtained from β ∪ a bycollapsing the arc a to a single point (we refer to [MM04] for details).

In the remainder of this section we identify a disc in H with its boundary circleon ∂H. Thus we view the vertex set of the superconducting disc graph SDG(X)as a subset of the vertex set of the curve graph CG(X) of X. We are now ready toshow

Theorem 4.2. Let X ⊂ ∂H be a thick subsurface. There is a number L > 1only depending on H but not on X such that the vertex inclusion defines an L-quasi-isometric embedding SDG(X) → CG(X).

Proof. As before, let dS,X be the distance in SDG(X) and let dCG,X be the distancein CG(X). We have to show the existence of a number c > 0 with the followingproperty. If D,E are discs with boundary in X then

dS,X(D,E) ≤ cdCG,X(∂D, ∂E).

By Proposition 4.1, there is a nested surgery path D = D0, . . . , Dn connectingthe disc D0 = D to a disc Dn which is disjoint from E, and there is a sequence(τi)0≤i≤n of train tracks onX such that τi+1 ≺ τi for all i < n and thatDi intersectsτi only at the switch.

By Theorem 2.3.1 of [PH92], there is a splitting and shifting sequence τ0 = η0 ≺· · · ≺ ηs = τn connecting τ0 to τn and a sequence 0 = u0 < · · · < un = s sothat ηuq

= τq for 0 ≤ q ≤ n. Since the disc Di intersects τi only at the switch,the distance in the curve graph CG(X) between ∂Di and a vertex cycle of τi isuniformly bounded, independent of i and X [MM99, H06]. Now the discs Di andDi+1 are disjoint and consequently the distance in CG(X) between a vertex cycleof τi and a vertex cycle of τi+1 is uniformly bounded. Thus by Corollary 3.6 andthe definition of the no-gap distance, it suffices to show the existence of a numberb > 0 with the following property. Let k < i be such that the pair τi ≺ τk is not

wide; then dS,X(Di, Dk) ≤ b.Since τi ≺ τk is not wide there is a large subtrack τ ′i of τi, a diagonal extension

ζi of τ′i and a simple closed curve α carried by ζi with the following property. Let

τ ′k be the image of τ ′i under a carrying map τi → τk and let ζk be the diagonalextension of τ ′k which is the image of ζi under a carrying map induced by a carryingmap τ ′i → τ ′k. Then α does not fill a large subtrack of ζk.

Since ζi is a diagonal extension of the large subtrack τ ′i of τi and since Di

intersects τi only at the switch, the intersection number between ∂Di and ζi isbounded from above by a constant κ ≥ 2 only depending on the topological typeof ∂H.

For each p ∈ [k, i], the image of τ ′i under a carrying map τi → τp is a largesubtrack τ ′p of τp, and there is a diagonal extension ζp of τ ′p which carries α. Wemay assume that ζu ≺ ζp for u ≥ p. The disc Dp intersects ζp in at most κ points.

For each p ∈ [k, i] let βp ≺ ζp be the subtrack of ζp filled by α. Then βp isconnected and not large. The union Yp of a thickening of βp with the components


ofX−βp which are either simply connected or once holed discs is a proper connectedsubsurface of X for all p. The boundary of Yp can be realized as a union of simpleclosed curves which are embedded in βp (but with cusps). The carrying map βp+1 →βp maps Yp+1 into Yp. In particular, either the boundary of Yp+1 coincides up tohomotopy with the boundary of Yp or Yp+1 is a proper subsurface of Yp. This meansthat there is a boundary circle of Yp+1 which is essential in Yp and hence in X.In other words, the subsurfaces Yp are nested, and hence their number is boundedfrom above by a universal constant h > 0.

Since ∂Dp intersects ζp in at most κ points, the number of intersections between∂Dp and ∂Yp is bounded from above by a universal constant χ > 0 (κ times themaximal number of branches of a train track on X will do). As a consequence,there are h essential simple closed curves c1, . . . , ch in X so that for every p ∈ [k, i]there is some r(p) ∈ {1, . . . , h} with

ι(∂Dp, cr(p)) ≤ χ.

Each of the curves cj is a fixed boundary component of one of the subsurfaces Yp.By reordering, assume that r(i) = 1. Let v1 be the minimum of all numbers

p ∈ [k, i] such that r(v1) = 1. Proposition 2.5 shows that

dS,X(Di, Dv1) ≤ 2χ+ 8.

On the other hand, we have dS,X(Dv1, Dv1−1) = 1. Again by reordering, assume

that r(v1 − 1) = 2 and repeat this construction with the discs Dv1−1, . . . , Dk andthe curve c2. In a ≤ h steps we construct in this way a decreasing sequencei ≥ v1 > · · · > va = k such that dS,X(Dvu

, Dvu−1) ≤ 2χ + 9 for all u ≤ a. This

implies that

dS,X(Di, Dk) ≤ h(2χ+ 9)

which is what we wanted to show. �

By a result of Bell and Fujiwara [BF08], the asymptotic dimension of the curvegraph of a subsurface Y of ∂H is bounded from above independent of Y . Thus asan immediate consequence of Theorem 4.2 we observe

Corollary 4.3. For a thick subsurface X of ∂H the graph SDG(X) is hyperbolic,

and its asymptotic dimension is bounded independent of X.

Corollary 4.4. Let H be a handlebody of genus g ≥ 2 without spots.

(1) If g = 2n is even then the vertex inclusion SDG → CG is a quasi-isometric

embedding.

(2) If g = 2n+1 is odd then the vertex inclusion EDG → CG is a quasi-isometric

embedding.

The remainder of this section is devoted to the proof of Corollary 1 from theintroduction. To this end recall that a hyperbolic geodesic metric space Y admitsa Gromov boundary. This boundary is a topological space on which the isometrygroup of Y acts as a group of homeomorphisms.

Let L be the space of all geodesic laminations on ∂H (for some fixed hyperbolicmetric) equipped with the coarse Hausdorff topology. In this topology, a sequence(µi) converges to a lamination µ if every accumulation point of (µi) in the usualHausdorff topology contains µ as a sublamination. Note that the coarse Hausdorfftopology on L is not T0, but its restriction to the subspace ∂CG ⊂ L of all minimal


geodesic laminations which fill up ∂H (i.e. which intersect every simple closed geo-desic transversely) is Hausdorff. The space ∂CG equipped with the coarse Hausdorfftopology can naturally be identified with the Gromov boundary of CG [K99, H06].

Let∂H ⊂ ∂CG

be the closed subset of all geodesic laminations which are limits in the coarse Haus-dorff topology of boundaries of discs in H. The handlebody group Map(H) acts on∂CG as a group of transformations preserving the subset ∂H. The Gromov bound-ary of SDG can now fairly easily be determined from Corollary 4.4. To this end wefirst observe

Lemma 4.5. The Gromov boundary of SDG is a closed Map(H)-invariant subsetof ∂H.

Proof. Since the vertex inclusion SDG → CG defines a quasi-isometric embedding,the Gromov boundary of SDG is the subset of the Gromov boundary of CG of allendpoints of quasi-geodesic rays in CG which are contained in SDG.

By the main result of [H06] (see [K99] for an earlier account of a similar state-ment), a simplicial quasi-geodesic ray γ : [0,∞) → CG defines the endpoint lam-ination ν ∈ ∂CG if and only if the curves γ(i) converge as i → ∞ in the coarseHausdorff topology to ν. As a consequence, the Gromov boundary of SDG is asubset of ∂H, and this subset is clearly Map(H)-invariant.

We are left with showing that the Gromov boundary of SDG is a closed subsetof ∂CG. To this end note that by Corollary 4.4, there is a number p > 1 suchthat for every L > 1, any L-quasi-geodesic in SDG is an Lp-quasi-geodesic in CG.Moreover, for a suitable choice of p, any vertex in SDG can be connected to anypoint in the Gromov boundary of SDG by a p-quasi-geodesic.

Now let (νi) be a sequence in the Gromov boundary of SDG which converges in∂CG to a lamination ν. Let ∂D be the boundary of a disc and let γ : [0,∞) → CGbe a quasi-geodesic ray issuing from γ(0) = ∂D with endpoint ν. By hyperbolicityof CG and by the discussion in the previous paragraph, there is a number R > 0 andfor every k ≥ 0 there is some i(k) > 0 such that a p-quasi-geodesic in SDG ⊂ CGconnecting γ(0) to νi(k) passes through the R-neighborhood of γ(k) in CG. Sincek > 0 was arbitrary, this implies that the entire quasi-geodesic ray γ is containedin the R-neighborhood of the subset SDG of CG. Using once more hyperbolicity,we conclude that there is a quasi-geodesic ray in CG connecting γ(0) to ν which isentirely contained in SDG. But this just means that ν is contained in the Gromovboundary of SDG. �

The handlebody group Map(H) naturally acts on the Gromov boundary of SDGas a group of homeomorphisms. Since Map(H) is a subgroup of the mapping classgroup Mod(∂H), by naturality this action is compatible with the action of themapping class group on the Gromov boundary of the curve graph. From Lemma4.5 and the following observation (which is essentially contained in Theorem 1.2 of[M86]), we conclude that ∂H is indeed the Gromov boundary of SDG.

Lemma 4.6. The action of the handlebody group Map(H) on ∂SDG is minimal.

Proof. Let (∂Di) be a sequence of boundaries of discs Di converging in the coarseHausdorff topology to a geodesic lamination µ ∈ ∂H. For each i let Ei be a discwhich is disjoint fromDi. Since the space of geodesic laminations equipped with the


usual Hausdorff topology is compact, up to passing to a subsequence the sequence(∂Ei) converges in the Hausdorff topology to a geodesic lamination ν which does notintersect µ (we refer to [K99, H06] for details of this argument). Now µ is minimaland fills up ∂H and therefore the lamination ν contains µ as a sublamination. Thisjust means that (∂Ei) converges in the coarse Hausdorff topology to µ.

Since the genus of H is at least two, for every separating disc in H we canfind a disjoint non-separating disc. Thus the discussion in the previous paragraphshows that every µ ∈ ∂H is a limit in the coarse Hausdorff topology of a sequenceof non-separating discs. However, the handlebody group acts transitively on non-separating discs. Minimality of the action of Map(H) on ∂SDG follows. �

As an immediate consequence of Lemma 4.5 and Lemma 4.6 we obtain

Corollary 4.7. (1) If g = 2n ≥ 2 is even then ∂H is the Gromov boundary of

SDG.(2) If g = 2n+ 1 is odd then ∂H is the Gromov boundary of EDG.

5. Hyperbolic thinnings of hyperbolic graphs

In this section we consider a graph G which is obtained from a hyperbolic graphEG by deleting some of the edges. We establish a sufficient condition for G to behyperbolic. This condition is related to the work of Farb [F98] and will be used inthe later sections to deduce from hyperbolicity of the superconducting disc graphhyperbolicity of the electrified disc graph, and from hyperbolicity of the electrifieddisc graph hyperbolicity of the disc graph.

To begin with, let (G, d) be any (not necessarily locally finite) metric graph (i.e.edges have length one). For a given family H = {Hc | c ∈ C} of complete connectedsubgraphs of G define the H-electrification of G to be the metric graph (EG, dE)which is obtained from G by adding vertices and edges as follows. For each c ∈ Cthere is a unique vertex vc ∈ EG − G. This vertex is connected with each of thevertices of Hc by a single edge of length one, and it is not connected with any othervertex.

We say that the family H is r-bounded for a number r > 0 if diam(Hc ∩Hd) ≤ rfor c 6= d ∈ C where the diameter is taken with respect to the intrinsic path metricon Hc and Hd. A family which is r-bounded for some r > 0 is simply calledbounded.

In the sequel all parametrized paths γ in G or EG are supposed to be simplicial.This means that the image of every integer is a vertex, and the image of an integralinterval [k, k + 1] is an edge or a single vertex.

Call a simplicial path γ in EG efficient if for every c ∈ C we have γ(k) = vc forat most one k. Note that if γ is an efficient simplicial path in EG which passesthrough γ(k) = vc for some c ∈ C then γ(k − 1) ∈ Hc, γ(k + 1) ∈ Hc.

Definition 5.1. The family H has the bounded penetration property if it is r-bounded for some r > 0 and if for every L > 0 there is a number p(L) > 2r withthe following property. Let γ be an efficient L-quasi-geodesic in EG, let c ∈ C andlet k ∈ Z be such that γ(k) = vc. If the distance in Hc between γ(k − 1) andγ(k + 1) is at least p(L) then every efficient L-quasi-geodesic γ′ in EG with thesame endpoints as γ passes through vc. Moreover, if k′ ∈ Z is such that γ′(k′) = vcthen the distance in Hc between γ(k−1), γ′(k′−1) and between γ(k+1), γ′(k′+1)is at most p(L).


Let H be as in Definition 5.1. Define an enlargement γ of an efficient simplicialL-quasi-geodesic γ : [0, n] → EG with endpoints γ(0), γ(n) ∈ G as follows. Let0 < k1 < · · · < ks < n be those points such that γ(ki) = vci for some ci ∈ C. Thenγ(ki − 1), γ(ki + 1) ∈ Hci . For each i ≤ s replace γ[ki − 1, ki + 1] by a simplicialgeodesic in Hci with the same endpoints.

For a number k > 0 define a subset Z of the metric graph G to be k-quasi-convexif any geodesic with both endpoints in Z is contained in the k-neighborhood of Z.In particular, up to perhaps increasing the number k, any two points in Z can beconnected in Z by a (not necessarily continuous) path which is a k-quasi-geodesicin G. The goal of this section is to show

Theorem 5.2. Let G be a metric graph and let H = {Hc | c} be an r-boundedfamily of complete connected subgraphs of G. Assume that the following conditions

are satisfied.

(1) There is a number δ > 0 such that each of the graphs Hc is δ-hyperbolic.(2) The H-electrification EG of G is hyperbolic.

(3) H has the bounded penetration property.

Then G is hyperbolic. Enlargements of geodesics in EG are uniform quasi-geodesics

in G. The subgraphs Hc are uniformly quasi-convex.

For the remainder of this section we assume that G is a graph with a family Hof subgraphs which has the properties stated in Theorem 5.2.

For a number R > 2r call c ∈ C R-wide for an efficient L-quasi-geodesic γ inEG if the following holds true. There is some k ∈ Z such that γ(k) = vc, andthe distance between γ(k − 1), γ(k + 1) in Hc is at least R. Note that since H isr-bounded, c is uniquely determined by γ(k − 1), γ(k + 1). If R = p(L) is as inDefinition 5.1 then we simply say that c is wide.

Lemma 5.3. Let L ≥ 1 and let γ1, γ2 be two efficient L-quasi-geodesics in EC with

the same endpoints. If c ∈ C is 3p(L)-wide for γ1 then c is wide for γ2.

Proof. By definition, if c is 3p(L)-wide for γ1 then there is some k so that γ1(k) = vcand that the distance in Hc between γ1(k−1) and γ1(k+1) is at least 3p(L). Sinceγ2 is an efficient L-quasi-geodesic with the same endpoints as γ1, by the boundedpenetration property there is some k′ so that γ2(k

′) = vc, moreover the distancein Hc between γ1(k − 1) and γ2(k

′ − 1) and between γ1(k + 1) and γ2(k′ + 1) is at

most p(L). Thus by the triangle inequality, the distance in Hc between γ2(k′ − 1)

and γ2(k′ + 1) is at least p(L) which is what we wanted to show. �

Define the Hausdorff distance between two closed subsets A,B of a metricspace to be the infimum of the numbers b > 0 such that A is contained in theb-neighborhood of B and B is contained in the b-neighborhood of A.

Lemma 5.4. For every L > 0 there is a number κ(L) > 0 with the following

property. Let γ1, γ2 be two efficient simplicial L-quasi-geodesics in EG connecting

the same points in G, with enlargements γ1, γ2. Then the Hausdorff distance in Gbetween the images of γ1, γ2 is at most κ(L).

Proof. Let γ : [0, n] → EG be an efficient simplicial L-quasi-geodesic with endpointsγ(0), γ(n) ∈ G. Let R > p(L) and assume that c ∈ C is not R-wide for γ. If there issome u ∈ {1, . . . , n−1} such that γ(u) = vc then γ(u−1), γ(u+1) ∈ Hc. Since c isnot R-wide for γ, γ(u− 1) can be connected to γ(u+ 1) by an arc in Hc of length


at most R. In particular, if no c ∈ C is R-wide for γ then an enlargement γ of γ isan L-quasi-geodesic in EG for a universal constant L = L(L,R) > 0. Then γ is a

L-quasi-geodesic in G as well (note that the inclusion G → EG is 1-Lipschitz).Let γi : [0, ni] → EG be efficient L-quasi-geodesics (i = 1, 2) with the same

endpoints in G. Assume that no c ∈ C is wide for γ1. By Lemma 5.3, no c ∈ C isR = 3p(L)-wide for γ2. Let γi be an enlargement of γi. By the above discussion, the

arcs γi are L = L(L, 3p(L))-quasi-geodesics in EG. In particular, by hyperbolicityof EG, the Hausdorff distance in EG between the images of γi is bounded from aboveby a constant b− 1 > 0 only depending on L and R.

We have to show that the Hausdorff distance in G between these images is alsouniformly bounded. For this let x = γ1(u) be any vertex on γ1 and let y = γ2(w) bea vertex on γ2 of minimal distance in EG to x. Then dE(x, y) ≤ b (here as before, dEis the distance in EG, and we let d be the distance in G). Let ζ be a geodesic in EGconnecting x to y. Since y is a vertex on γ2 of minimal distance to x, ζ intersectsγ2 only at its endpoints.

We claim that there is a universal constant χ > 0 such that no c ∈ C is χ-widefor ζ. Namely, since γ1 does not pass through any of the special vertices in EG, theconcatenation ξ = ζ ◦ γ1[0, u] is efficient. Thus ξ is an efficient L′-quasi-geodesic

in EG with the same endpoints as γ2[0, w] where L′ > L > L only depends on L.

Hence by the bounded penetration property, if c ∈ C is p(L′)-wide for ζ then the

L-quasi-geodesic γ2[0, w] passes through the vertex vc which is a contradiction.As a consequence of the above discussion, the length of an enlargement of ζ

is bounded from above by a fixed multiple of dE(γ1(u), γ2(w)), i.e. it is uniformlybounded. This shows that d(γ1(u), γ2(w)) is uniformly bounded. As a consequence,the image of γ1 is contained in a neighborhood of uniformly bounded diameter inG of the image of γ2. Exchanging γ1 and γ2 we conclude that indeed the Hausdorffdistance in G between the images of the enlargements γ1, γ2 is bounded by a numberonly depending on L.

Now let γj : [0, nj ] → EG be arbitrary efficient L-quasi-geodesics (j = 1, 2)connecting the same points in G. Then there are numbers 0 < u1 < · · · < uk < n1such that for every i ≤ k, γ1(ui) = vci where ci ∈ C is wide for γ1, and there areno other wide points for γ1. Put u0 = −1 and uk+1 = n1 + 1.

By the bounded penetration property, there are numbers wi ∈ {1, . . . , n2 − 1}such that γ2(wi) = γ1(ui) = vci for all i. Moreover, the distance in Hci betweenγ1(ui−1) and γ2(wi−1) and between γ1(ui+1) and γ2(wi+1) is at most p(L). Sinceγ1, γ2 are L-quasi-geodesics by assumption, we may assume that the special verticesvci appear along γ2 in the same order as along γ1, i.e. that 0 < w1 < · · · < wk < n2.is immediate from the Put w0 = −1 and wk+1 = n2 + 1.

For each i ≤ k, define a simplicial edge path ζi : [ai, ai+1] → EG connectingζi(ai) = γ1(ui + 1) ∈ Hci to ζi(ai+1) = γ1(ui+1 − 1) ∈ Hci+1

as the concatentationof the following three arcs. A geodesic in Hci connecting γ1(ui + 1) to γ2(wi + 1)(whose length is at most p(L)), the arc γ2[wi +1, wi+1 − 1] and a geodesic in Hci+1

connecting γ2(wi+1 − 1) to γ2(ui+1 − 1). Let moreover ηi = γ1|[ui + 1, ui+1 − 1](i ≥ 0). Then ηi, ζi are efficient uniform quasi-geodesics in EG with the sameendpoints, and ηi does not have wide points.

For each i let νi be an enlargement of the arc νi = γ2[wi + 1, wi+1 − 1]. By

construction, there is an enlargement ζi of the efficient quasi-geodesic ζi which con-tains νi as a subarc and whose Hausdorff distance in G to νi is uniformly bounded.


Let ηi be an enlargement of ηi. Then ζi, ηi are enlargements of the efficient uni-form quasi-geodesics ζi, ηi in EG with the same endpoints, and ηi does not havewide points. Therefore by the first part of this proof, the Hausdorff distance in Gbetween ηi and ζi is uniformly bounded. Hence the Hausdorff distance between ηiand νi is uniformly bounded as well.

There is an enlargement γ1 of γ1 which can be represented as

γ1 = ηk ◦ σk ◦ · · · ◦ σ1 ◦ η0

where for each i, σi is a geodesic inHci connecting γ1(ui−1) to γ1(ui+1). Similarly,there is an enlargement γ2 of γ2 which can be represented as

γ2 = νk ◦ τk ◦ · · · ◦ τ1 ◦ ν0

where for each i, τi is a geodesic in Hci connecting γ2(wi − 1) to γ2(wi + 1).For each i the distance in Hci between γ1(ui − 1) and γ2(wi − 1) is at most

p(L), and the same holds true for the distance between γ1(ui + 1) and γ2(wi + 1).Since Hci is δ-hyperbolic for a constant δ > 0 not depending on ci, the Hausdorffdistance in Hci between any two geodesics connecting γ1(ui − 1) to γ1(ui + 1) andconnecting γ2(wi−1) to γ2(wi+1) is uniformly bounded. Together with the abovediscussion, this shows the lemma. �

Let for the moment X be an arbitrary geodesic metric space. Assume that forevery pair of points x, y ∈ X there is a fixed choice of a path ρx,y connecting x toy. The thin triangle property for this family of paths states that there is a universalnumber C > 0 so that for any triple x, y, z of points in X, the image of ρx,y iscontained in the C-neighborhood of the union of the images of ρy,z, ρz,x.

For two vertices x, y ∈ G let ρx,y be an enlargement of a geodesic in EG connectingx to y. We have

Proposition 5.5. The thin triangle inequality property holds true for the paths

ρx,y.

Proof. Let x1, x2, x3 be three vertices in G and for i = 1, 2, 3 let γi : [0, ni] → EGbe a geodesic connecting xi to xi+1.

By hyperbolicity of EG there is a number L > 0 not depending on the pointsxi, and there is a vertex y ∈ EG with the following property. For i = 1, 2, 3 letβi : [0, pi] → EG be a geodesic in EG connecting xi to y. Then for all i, αi = β−1

i+1◦βiis an L-quasi-geodesic connecting xi to xi+1.

We claim that without loss of generality we may assume that the quasi-geodesicsαi are efficient. Namely, since the arcs βi are geodesics, they do not backtrack.Thus if α1 is not efficient then there is a common point y on β1 and β2. Lets1 < p1 be the smallest number so that β1(s1) = β2(s2) for some s2 ∈ β2[0, p2].Then the distance between y and βi(si) (i = 1, 2) is uniformly bounded, and α1 =(β2[0, s2])

−1 ◦β1[0, s1] is an efficient L-quasi-geodesic connecting x1 to x2. Replace

y by β1(s1), replace βi by βi = βi[0, si] (i = 1, 2) and and replace β3 by a geodesic

β3 : [0, p3] → EG connecting x3 to β1(s1). Thus up to increasing the number L by auniformly bounded amount we may assume that the quasi-geodesic α1 is efficient.

Assume from now on that β1, β2, β3 are such that the quasi-geodesic α1 = β−12 ◦β1

is efficient. Using the notation from the second paragraph of this proof, if there issome s < p3 such that β3(s) is contained in α1 then let s3 be the smallest numberwith this property. Replace the point y = βi(pi) by β3(s3), replace β3 by β3[0, s3]


and for i = 1, 2 replace βi by the subarc of α1 connecting xi to β3(s3). With thisconstruction, up to increasing the number L by a uniformly bounded amount andperhaps replacing β1, β2 by uniform quasi-geodesics rather than geodesics we mayassume that all three quasi-geodesics αi = β−1

i+1 ◦ βi (i = 1, 2, 3) are efficient.Resuming notation, assume from now on that the quasi-geodesics αi are efficient.

By Lemma 5.4, the Hausdorff distance between an enlargement of the geodesic γiand any choice of an enlargement of the efficient uniform quasi-geodesic αi withthe same endpoints is uniformly bounded. Thus it suffices to show the thin triangleproperty for enlargements of the quasi-geodesics αi.

If y ∈ G then an enlargement of the quasi-geodesic αi is the concatenation of anenlargement of the quasi-geodesic βi and an enlargement of the quasi-geodesic β−1

i+1

which have endpoints in G. Hence in this case the thin triangle property followsonce more from Lemma 5.4.

If y = vc for some c ∈ C then we distinguish two cases.Case 1: c ∈ C is wide for each of the quasi-geodesics αi.Recall that y = βi(pi). By hyperbolicity of Hc, there is a number R > 0 not

depending on c such that for all i ∈ {1, 2, 3} the image of any geodesic in Hc

connecting βi(pi − 1) to βi+1(pi+1 − 1) is contained in the R-neighborhood of theunion of the images of any two geodesics connecting βj(pj−1) to βj+1(pj+1−1) forj 6= i and where indices are taken modulo three. In other words, the thin triangleproperty holds true for such geodesics.

Now let αi be an enlargement of αi and let ζi be the subarc of αi which connectsβi(pi − 1) to βi+1(pi+1 − 1). By the definition of an enlargement, ζi is a geodesicin Hc. Thus by the discussion in the previous paragraph and by the fact that wemay use the same enlargement of the arc βi+1[0, pi+1−1] for the construction of anenlargement of αi and αi+1, the thin triangle property holds true for some suitablechoice and hence any choice of an enlargement of the quasi-geodesics αi which iswhat we wanted to show.

Case 2: For at least one i, c ∈ C is not wide for the quasi-geodesic αi.Assume that this holds true for the quasi-geodesic α1. Then the distance in Hc

between β1(p1 − 1) and β2(p2 − 1) is uniformly bounded (depending on the quasi-geodesic constant for α1). Replace the point y by β1(p1 − 1), replace the quasi-

geodesic β1 by β1 = β1[0, p1−1], replace the quasi-geodesic β2 by the concatentation

β2 of β2[0, p2 − 1] with a geodesic in Hc connecting β2(p2 − 1) to β1(p1 − 1), and

replace the geodesic β3 by the concatentation β3 of β3 with the edge connecting vcto β1(p1−1). The resulting arcs βi are efficient uniform quasi-geodesics in EG, and

they connect the points xi to y ∈ G. Moreover, the quasi-geodesics β−1i+1 ◦ βi are

effficient as well and hence we are done by the above proof for the case y ∈ G. �

Now we are ready to show

Corollary 5.6. G is hyperbolic. Enlargements of geodesics in EG are uniform

quasi-geodesics in G.

Proof. For any pair (x, y) of vertices in G let ηx,y be a reparametrization on [0, 1]of the path ρx,y. By Proposition 3.5 of [H07], it suffices to show that there is somen > 0 such that the paths ηx,y have the following properties (where d is the distancein G).

(1) If d(x, y) ≤ 1 then the diameter of ηx,y[0, 1] is at most n.


(2) For x, y and 0 ≤ s < t < 1 the Hausdorff distance between ηx,y[s, t] andηηx,y(s),ηx,y(t)[0, 1] is at most n.

(3) For all vertices x, y, z the set ηx,y[0, 1] is contained in the n-neighborhoodof ηx,y[0, 1] ∪ ηy,z[0, 1].

Properties 1) and 2) above are immediate from Lemma 5.4. The thin triangleproperty 3) follows from Proposition 5.5. �

The following corollary is an immediate consequence of Corollary 5.6.

Corollary 5.7. There is a number k > 0 such that each of the subgraphs Hc (c ∈ C)is k-quasi-convex.

6. Decreasing conductivity: The electrified disc graph for m = 0

The goal of this section is to show hyperbolicity of the electrified disc graphfor handlebodies H without spot on the boundary. We also determine its Gromovboundary and show more generally hyperbolicity of the electrified disc graph of athick subsurface of ∂H.

Thus let X ⊂ ∂H be a thick subsurface. Recall that X is connected, and bythe remark after Lemma 2.1, X is distinct from a sphere with at most four holesand from a torus with a single hole. As before, we denote by dCG,X the distance inthe curve graph CG of X, by dS,X the distance in the superconducting disc graphSDG(X) of X and by dE,X the distance in the electrified disc graph EDG(X) of X.By Theorem 4.2, the graph SDG(X) is quasi-isometrically embedded in the curvegraph of X. In particular, it is δ-hyperbolic for a number δ > 0 only depending onthe genus of H.

If X does not contain any I-bundle generator then EDG(X) = SDG(X) andthere is nothing to show. Thus assume that there are I-bundle generators c ⊂ X.Let E(c) ⊂ EDG(X) be the complete subgraph of EDG(X) whose vertices are discsintersecting c in precisely two points and let E = {E(c) | c}. By definition, SDG(X)is 2-quasi-isometric to the E-electrification of EDG(X). Thus to show hyperbolicityof EDG(X) it suffices to show that the family E is r-bounded and satisfies properties(1) and (3) in Theorem 5.2.

We begin with establishing property (1). To this end define for a compact(not necessarily orientable) surface F with boundary ∂F and for a fixed boundarycomponent ρ of F the electrified arc graph C ′(F, ρ) as follows. Vertices of C ′(F, ρ)are essential embedded arcs in F with both endpoints in ρ. Two such arcs areconnected by an edge of length one if either they are disjoint or if they are disjointfrom a common essential simple closed curve. If F is non-orientable, then we requirethat an essential simple closed curve does not bound a Moebius band.

The following statement is well known but hard to find in the literature. Wegive a proof for completeness.

Lemma 6.1. Let F be a compact surface with boundary which is not a sphere with

at most three holes, a projective plane with at most three holes or a one-holed Klein

bottle. Let ρ be a boundary circle of F . Then C ′(F, ρ) is 4-quasi-isometric to the

curve graph of F .

Proof. Define the arc and curve graph A(F, ρ) of F to be the graph whose verticesare arcs with endpoints on ρ or essential simple closed curves in F . Two such arcsor curves are connected by an edge of length one if they can be realized disjointly.


Consider first the case that F is a one-holed torus, a four holed sphere a four-holed projective plane. In this case two simple closed curves in F are connected byan edge in the curve graph of F if they intersect in the minimal number of points(one or two). Let α be an essential simple closed curve in F . Cutting F open alongα yields a three-holed sphere, the disjoint union of two three holed spheres or thedisjoint union of a three holed sphere and a three holed projective plane. Thusthere is a unique essential arc Λ(α) with endpoints on ρ which is disjoint from α.The distance between α, β in the curve graph of F equals one if and only if the arcsΛ(α),Λ(β) are disjoint. This means that the map Λ which associates to a simpleclosed curve α in F the unique arc Λ(α) with endpoints on ρ which is disjoint fromα defines an isometry of the curve graph of F onto the arc graph of (F, ρ). This arcgraph is the complete subgraph of A(F, ρ) whose vertex set consists of arcs withendpoints on ρ. Moreover, in the special case at hand the arc graph is just thegraph C ′(F, ρ). This yields the statement of the lemma.

Now assume that the surface F is such that two vertices in the curve graph of Fare connected by an edge if they can be realized disjointly. Then for any two disjointessential simple closed curves α, β in F there is an essential arc with endpoints on ρwhich is disjoint from both α, β. In particular, for every simplicial path γ in A(F, ρ)connecting two vertices in A(F, ρ) which are arcs there is a path of at most doublelength in C ′(F, ρ) connecting the same endpoints. This path can be obtained fromγ as follows. If γ(i), γ(i+1) are both simple closed curves then replace γ[i, i+1] bya simplicial path in A(F, ρ) of length 2 with the same endpoints whose midpoint isan arc disjoint from γ(i), γ(i+ 1). In the resulting path, a simple closed curve α isadjacent to two arcs disjoint from α and hence we can view this path as a path inC ′(F, ρ). Thus the vertex inclusion C ′(F, ρ) → A(F, ρ) is a quasi-isometry.

We are left with showing that A(F, ρ) is quasi-isometric to the curve graph ofF . However, this is well known, and the proof will be omitted. �

A thick subsurface X of ∂H is not a four-holed sphere. Thus if c is a separatingI-bundle generator for X then the base of the I-bundle either has positive genusor is a sphere with at least four holes. Similarly, if c is a non-separating I-bundlegenerator for X then we may assume that the base F of the I-bundle is not aprojective plane with three holes. Namely, if F is a projective plane with threeholes and if α is a distinguished boundary component of F then there is up tohomotopy a unique essential arc β in F with boundary on α. The I-bundle over βis the unique disc in the oriented I-bundle over F which intersects the curve α inprecisely two points.

If the boundary of the base of the I-bundle is connected then X = ∂H, and weobserved in Section 2 that in this case the base surface of any I-bundle generatorin X is orientable. In particular, the base of the I-bundle defined by an I-bundlegenerator c ⊂ X is not a one-holed Klein bottle.

We use Lemma 6.1 to verify the first condition in Theorem 5.2 for the family E .

Lemma 6.2. Let X ⊂ ∂H be a thick subsurface and let c be an I-bundle generator

in X, with base surface F . Then the map which associates to a disc D ∈ E(c) the

projection of ∂D to F extends to a 2-quasi-isometry of E(c) onto the electrified arc

graph C′(F, c) of F .

Proof. Let c be an I-bundle generator in X and let F be the base surface of theI-bundle generated by c. Let V be the oriented I-bundle over F as in the definition


of an I-bundle generator and let Φ : V → H be a corresponding embedding..Then Φ(∂V ) ∩ ∂H = X (up to isotopy). There is an orientation reversing bundleinvolution ϕ of V which exchanges the endpoints of the fibres. The involutionpreserves X ⊂ ∂V and the curve c. The quotient of X under this involutionequals the base surface F of the I-bundle. The projection of c is the distinguishedboundary component of F , again denoted by c.

Up to isotopy, if the boundary ∂D of a disc D in H is contained in X and inter-sects the curve c in precisely two points then ∂D is invariant under the involutionϕ. Thus the map Θ : V(C ′(F, c)) → V(E(c)) which associates to an arc α in F withendpoints on c the I-bundle over α is a bijection. Here V(C ′(F, c)) (or V(E(c))) isthe set of vertices of C ′(F, c) (or E(c)).

If α, β ∈ V(C ′(F, c)) are connected by an edge then either α, β are disjoint andso are Θ(α),Θ(β), or α, β are disjoint from a simple closed curve and the sameholds true for Θ(α),Θ(β). Thus Θ extends to a 1-Lipschitz map C ′(F, c) → E(c).

We are left with showing that Θ−1 : V(E(c)) → V(C ′(F, c)) is 2-Lipschitz. Tothis end let α, β ∈ V(C ′(F, c)) be such that Θ(α),Θ(β) are connected by an edge inE(c). If Θ(α),Θ(β) are disjoint then the same holds true for α, β and hence α, β areconnected by an edge in C ′(F, c). Otherwise Θ(α),Θ(β) are disjoint from a simpleclosed curve γ.

Since ∂Θ(α) ∪ ∂Θ(β) is invariant under the involution ϕ, ∂Θ(α) ∪ ∂Θ(β) isdisjoint from γ ∪ ϕ(γ). As a consequence, the projection of γ to F is a union ofessential arcs with boundary on c and closed curves (not necessarily simple) whichare disjoint from α ∪ β. Then there either is a simple arc in F with endpoints onc or there is an essential simple closed curve in F which is disjoint from α ∪ β. Asa consequence, the distance in C′(F, c) between α ∪ β is at most two. The lemmafollows. �

From Lemma 6.2, Lemma 6.1 and hyperbolicity of the curve graph of X [MM99]and see [BF07] for the curve graph of a non-orientable surface) we immediatelyobtain

Corollary 6.3. There is a number δ > 0 such that each of the graphs E(c) is

δ-hyperbolic.

Note that the number δ > 0 in the statement of the corollary only depends onH but not on X. In fact, the main result of [HPW13] together with Lemma 6.2shows that it can even be chosen independent of H.

Next we check that the family E = {E(c) | c} where c runs through all I-bundlegenerators in X is r-bounded for some r > 0.

Lemma 6.4. Let X ⊂ ∂H be a thick subsurface. There is a number r > 0 such

that if c, d are distinct I-bundle generators in X then diam(E(c) ∩ E(d)) ≤ r.

Proof. Let c ⊂ X be an I-bundle generator and let F be the base surface of theI-bundle defined by c.

Assume first that the curve c is separating. Then the base F of the I-bundlecan naturally be identified with a component of X − c. A simple closed curve bdisjoint from c is contained in a component X1 of X−c, and it is disjoint from someessential simple arc β in X1 with endpoints on c. Now β = X1 ∩ ∂D for some discD ∈ E(c) and hence b is not discbusting. Thus any I-bundle generator d distinctfrom c intersects c.


Let α be a component of d−c. A disc D ∈ E(c)∩E(d) intersects α in at most twopoints. The distance between ∂D ∩X1 and α in the electrified arc graph C ′(F, c)of F ∼ X1 is uniformly bounded. By Lemma 6.2, this implies that the diameter inE(c) of E(c) ∩ E(d) is uniformly bounded.

The same argument applies if c is non-separating. A disc in E(c) is the I-bundleover a simple arc in F with boundary on c, and every essential simple arc in F withendpoints on c arises in this way. Let d be another I-bundle generator in X andlet d0 be the projection of d to F . Then d0 either is a (perhaps empty) collectionof arcs with boundary on c, or it is a closed curve, possibly with self-intersections.The projection to F of the boundary ∂D of a disc D ∈ E(c)∩ E(d) is an embeddedarc with endpoints on c which intersects d0 in at most two points. Now the setof arcs in F with both endpoints on c which intersect d0 in at most 2 points hasuniformly bounded diameter in the electrified arc graph C ′(F, c). �

We are left with the verification of the bounded penetration property. To thisend recall from [MM00] the definition of a subsurface projection. Namely, let againX ⊂ ∂H be a thick subsurface and let Y ⊂ X be an incompressible, open connectedsubsurface which is distinct from X, a three-holed sphere and an annulus. The arcand curve graph AC(Y ) of Y (here we do not specify a boundary component) is thegraph whose vertices are isotopy classes of arcs with endpoints on ∂Y or essentialsimple closed curves in Y , and two such vertices are connected by an edge of lengthone if they can be realized disjointly. The vertex inclusion of the curve graph of Yinto the arc and curve graph is a quasi-isometry [MM00].

There is a projection πY of the curve graph CG(X) of X into the space of subsetsof AC(Y ) which associates to a simple closed curve in X the homotopy classes ofits intersection components with Y . For every simple closed multicurve c, thediameter of πY (c) in AC(Y ) is at most one. If c can be realized disjointly from Ythen πY (c) = ∅.

In the sequel we call a subsurface Y of X proper if Y is non-peripheral, incom-pressible, open and connected and different from an annulus, a three-holed sphereor X.

As before, call a path γ in a metric graph G simplicial if γ maps each interval[k, k + 1] (where k ∈ Z) isometrically onto an edge of G. The following lemma is aversion of Theorem 3.1 of [MM00].

Lemma 6.5. For every number L > 1 there is a number ξ(L) > 0 with the following

property. Let Y be a proper subsurface of X and let γ be a simplicial path in CG(X)which is an L-quasi-geodesic. If πY (v) 6= ∅ for every vertex v on γ then

diamπY (γ) < ξ(L).

Proof. By hyperbolicity, for every L > 1 there is a number n(L) > 0 so that forevery L-quasi-geodesic γ in CG(X) of finite length, the Hausdorff distance betweenthe image of γ and the image of a geodesic γ′ with the same endpoints does notexceed n(L).

Now let Y ⊂ X be a proper subsurface. By Theorem 3.1 of [MM00], there is anumberM > 0 with the following property. If ζ is any simplicial geodesic in CG(X)and if πY (ζ(s)) 6= ∅ for all s ∈ Z in the domain of ζ then

diam(πY (ζ)) ≤M.


Let L > 1, let γ : [0, k] → CG(X) be a simplicial path which is an L-quasi-geodesic and assume that

diam(πY (γ(0) ∪ γ(k))) ≥ 2M + L(2n(L) + 4).

Our goal is to show that γ passes through the set A ⊂ CG(X) of all essential simpleclosed curves in X − Y . The diameter of A in CG(X) is at most two.

To this end let γ′ be a simplicial geodesic in CG(X) with the same endpoints asγ. Theorem 3.1 of [MM00] shows that there is some u ∈ Z such that γ′(u) ∈ A.Then γ passes through the n(L)-neighborhood of A.

Let s + 1 ≤ t − 1 be the smallest and the biggest number, respectively, so thatγ(s + 1), γ(t − 1) are contained in the n(L)-neighborhood of A. Then γ[0, s] (orγ[t, k]) is contained in the complement of the n(L)-neighborhood of A. Since γ isan L-quasi-geodesic, a geodesic connecting γ(0) to γ(s) (or connecting γ(t) to γ(k))is contained in the n(L)-neighborhood of γ[0, s] (or of γ[t, k]) and hence it does notpass through A. In particular,

diam(πY (γ(0) ∪ γ(s))) ≤M and diam(πY (γ(t) ∪ γ(k))) ≤M.

As a consequence, we have

(2) diam(πY (γ(s) ∪ γ(t))) ≥ L(2n(L) + 4).

Since dCG,X(γ(s + 1), A) ≤ n(L) and dCG,X(γ(t − 1), A) ≤ n(L) and since thediameter of A is at most 2, we obtain dCG,X(γ(s), γ(t)) ≤ 2n(L) + 2. Now γ isa simplicial L-quasi-geodesic in CG(X) and hence the length t − s of γ[s, t] is atmost L(2n(L) + 2) + L = L(2n(L) + 3). For all ℓ ∈ Z the curves γ(ℓ), γ(ℓ + 1)are disjoint and therefore if γ(ℓ), γ(ℓ + 1) both intersect Y then the diameter ofπY (γ(ℓ) ∪ γ(ℓ+ 1)) is at most one. Thus if γ(ℓ) intersects Y for all ℓ then

diam(πY (γ(s) ∪ γ(t))) ≤ L(2n(L) + 3).

This contradicts inequality (2) and completes the proof of the lemma. �

For simplicity of notation, in the remainder of this section we identify discs inH with their boundaries. In other words, for a thick subsurface X of ∂H we viewthe vertex sets of the graphs SDG(X), EDG(X) as subsets of the vertex set of thecurve graph CG(X) of X.

Let SDG0(X) be the E-electrification of E(X). For each I-bundle generator c inX, the graph SDG0(X) contains a special vertex vc,. The vertex set of SDG0(X)is the union of the set of all discbounding simple closed curves in X and the set{vc | c}. In particular, there is a natural vertex inclusion V(SDG0(X)) → CG(X)which maps the special vertex vc to the simple closed curve c. Since SDG(X) isquasi-isometric to the E-electrification of E(X), Theorem 4.2 shows that this vertexinclusion extends to a quasi-isometric embedding SDG0(X) → CG(X). Moreover,the images of any two vertices in SDG0(X) which are connected by an edge can beconnected in CG(X) by an edge-path of length at most two. This is a consequenceof the fact that the distance in CG(X) between any two simple closed curves whichintersect in at most two points is at most two.

Now we are ready to show

Lemma 6.6. For every thick subsurface X of ∂H the family E has the bounded

penetration property.


Proof. Let L ≥ 1 and let γ : [0, n] → SDG0(X) be an efficient simplicial L-quasi-geodesic. Let γ0 be a simplicial arc in CG(X) which is obtained from γ by viewingthe vertices of SDG0(X) as vertices in CG(X) and connecting two adjacent verticesin γ by a geodesic in CG(X) whose length is at most two. The arc γ0 is a uniformquasi-geodesic in CG(X) which passes through an I-bundle generator c at mostonce, and it passes through c if and only if it passes through a simple closed curvewhich is disjoint from c.

Let now c be a separating I-bundle generator in X and let πc : CG(X) →AC(X−c) be the subsurface projection. By Lemma 6.5, there is a numberM(L) > 0with the following property. Denote by dAC,X−c the distance in the arc and curvegraph of the surface X − c. If dAC,X−c(π

c(γ(0)), πc(γ(n))) ≥ M(L) then there issome k0 ∈ Z such that γ(k0) = c. Equivalently, there is some k < n such thatγ(k) = vc. Moreover, the distance in AC(X− c) between πc(γ(0)) and πc(γ(k− 1))and between πc(γ(k + 1)) and πc(γ(n)) is at most M(L).

As a consequence, if γ′ : [0, n′] → SDG0(X) is another efficient quasi-geodesicwith the same endpoints, then there is some k′ such that γ′(k′) = vc, and

dAC,X−c(πc(γ(k − 1)), πc(γ′(k′ − 1))) ≤ 2M(L),

dAC,X−c(πc(γ(k + 1)), πc(γ′(k′ + 1))) ≤ 2M(L).

Lemma 6.2 and Lemma 6.1 now show that the distance in E(c) between γ(k −1), γ′(k′ − 1) and between γ(k + 1), γ′(k′ + 1) is uniformly bounded as well. Inparticular, the bounded penetration property holds true for the subgraph E(c) andfor quasi-geodesics connecting two discs whose boundaries have projections of largediameter into X − c.

On the other hand, if γ : [0, n] → SDG0(X) is any efficient L-quasi-geodesic andif γ(k) = vc for some I-bundle generator c then using once more Lemma 6.5, weconclude that the distance in AC(X − c) between πc(γ(0)) and πc(γ(k − 1)) andbetween πc(γ(k + 1)) and πc(γ(n)) is at most M(L). Therefore the reasoning inthe previous paragraph shows that whenever the distance in E(c) between γ(k −1), γ(k + 1) is sufficiently large then the distance in AC(X − c) between πc(γ(0))and πc(γ(n)) is bigger than M(L). In other words, the conclusion in the previousparagraph applies, and the bounded penetration property for separating I-bundlegenerators follows.

Now assume that c is non-separating. Let πc be the subsurface projection asbefore. The above argument shows that if the distance in AC(X − c) betweenπc(γ(0)) and πc(γ(n)) is at least M(L) then there is some k so that γ(k) = vc.Moreover, we have γ(k − 1) ∈ E(c), γ(k + 1) ∈ E(c). As a consequence, the curvesγ(k − 1), γ(k + 1) are invariant under the orientation reversing involution ϕ of Xwhich preserves c and extends to an involution of the I-bundle defined by c.

Let F be the base of the I-bundle and let α, β ∈ C ′(F, c) be the projections ofγ(k − 1), γ(k + 1). By Lemma 6.2, the distance in E(c) between γ(k − 1), γ(k + 1)is comparable to the distance in C ′(F, c) between α, β. Since γ(k − 1), γ(k + 1)are invariant under the involution ϕ, the main result of [RS09] shows that thisdistance is also comparable to the distance between πc(γ(k − 1)) and πc(γ(k + 1))in AC(X−c). In particular, if γ′ is any other efficient L-quasi-geodesic in SDG0(X)with the same endpoints then there is some k′ with γ(k′) = vc, and the distance inE(c) between γ(k − 1), γ′(k′ − 1) and between γ(k + 1) and γ′(k′ + 1) is uniformlybounded. The bounded penetration property follows in this case.


Finally we can invert this argument as in the case of a separating I-bundlegenerator and complete the proof of the lemma. �

We can now apply Theorem 5.2 to conclude

Corollary 6.7. For every thick subsurface X of ∂H, the graph EDG(X) is δ-hyperbolic for a number δ > 0 not depending on X. There is a number k > 0 such

that for every I-bundle generator c the subgraph E(c) of EDG(X) is k-quasi-convex.

Next we show Theorem 4 from the introduction. Thus let as before H be ahandlebody of even genus g = 2n ≥ 2 without spots. Let dE be the distance inEDG. If c is an I-bundle generator in ∂H then let πc be the subsurface projectionof a simple closed curve in ∂H into the arc and curve-graph of ∂H − c.

For a subset A of a metric space Y and a number C > 0 define diam(A)C to bethe diameter of A if this diameter is at least C and let diam(A)C = 0 otherwise.The notation ≍ means equality up to a universal multiplicative constant.

Corollary 6.8. Let H is a handlebody of genus g = 2n ≥ 2 without spot. Then

there is a number C > 0 such that

dE(D,E) ≍ dCG(∂D, ∂E) +∑

c

diam(πc(∂D ∪ ∂E))C

where c runs through all I-bundle generators on ∂H.

Proof. Let SDG0 be the E-electrification of EDG. For an I-bundle generator c in∂H denote by vc the special vertex in SDG0 defined by c.

Let γ : [0, k] → SDG0 be a geodesic. By Corollary 5.6, an enlargement γ of γ isa uniform quasi-geodesic in EDG. Thus it suffices to show that the length of γ isuniformly comparable to the right hand side of the formula in the corollary.

Now by Theorem 4.2, there is a number L > 1 such that a simplicial arc inCG constructed from γ as in the paragraph preceding Lemma 6.6 is an L-quasi-geodesic in the curve graph CG of ∂H. Lemma 6.6 shows that if γ is an enlargementof γ then the diameter of the intersection of γ with E(c) equals the diameter ofπ∂H−c(γ(0) ∪ γ(k)) up to a universal multiplicative and additive constant. This iswhat we wanted to show. �

We complete this section with the proof of Corollary 2 from the introduction.Thus let H be a handlebody of even genus g = 2n ≥ 2 without spot on theboundary. Let L be the space of all geodesic laminations on ∂H equipped with thecoarse Hausdorff topology.

Recall that an I-bundle generator in H is a separating simple closed curve in ∂Hwhich decomposes ∂H into two surfaces of genus n = g/2 with connected boundary.For such an I-bundle generator c let ∂E(c) ⊂ L be the set of all geodesic laminationswhich consist of two minimal components filling up ∂H − c and which are limitsin the coarse Hausdorff topology of boundaries of discs contained in E(c). Eachlamination µ ∈ E(c) is invariant under the orientation reversing involution τc of ∂Hwhich fixes c pointwise and preserves the boundaries of the discs in E(c). Using thenotations from Section 4, define

∂EDG = ∂H ∪⋃

c

∂E(c) ⊂ L

where the union is over all I-bundle generators c ⊂ ∂H. Then ∂EDG is a Map(H)-space.


Proposition 6.9. The Gromov boundary of EDG can naturally be identified with

∂EDG. A quasi-geodesic ray γ : [0,∞) → EDG represents the boundary point

µ ∈ EDG if and only if the sequence (∂γ(i)) converges in the coarse Hausdorff

topology to µ. The action of Map(H) on ∂EDG is minimal.

Proof. We show first that the subspace ∂EDG of L is Hausdorff.A point λ ∈ ∂EDG either is a minimal geodesic lamination which fills up ∂H, or

it is a geodesic lamination with two minimal components which fill up ∂H − c forsome I-bundle generator c. Let ν 6= λ be another such lamination. We claim thatν and λ intersect. This means that for some fixed hyperbolic metric on ∂H, thegeodesic representatives of ν, λ intersect transversely.

If either ν or λ fills up ∂H then this is obvious. Otherwise ν fills up the com-plement of an I-bundle generator c, and λ fills up the complement of an I-bundlegenerator c′. Now the simple closed curve c is the only minimal geodesic laminationwhich is distinct from a component of ν and which does not intersect ν. The lam-ination λ consists of two minimal components which are not simple closed curvesand therefore the geodesic laminations ν, λ indeed intersect.

Since ν, λ ∈ ∂EDG intersect, by the definition of the coarse Hausdorff topologythere are neighborhoods U of λ, V of ν in L so that any two laminations λ′ ∈U, ν′ ∈ V intersect. In particular, the neighborhoods U, V are disjoint. This showsthat ∂EDG is Hausdorff.

Next we show that for every simplicial quasi-geodesic ray γ : [0,∞) → EDG thesequence (∂γ(i)) of boundary curves converges as i → ∞ in the coarse Hausdorfftopology to a lamination µ ∈ ∂EDG.

To this end let SDG0 be the E-electrification of EDG. For a number L > 1 definean unparametrized L-quasi-geodesic in the graph SDG0 to be a path η : [0,∞) →SDG0 with the following property. There is some n ∈ (0,∞], and there is anincreasing homeomorphism ρ : [0, n) → [0,∞) such that η ◦ρ is an L-quasi-geodesicin SDG0.

Let p > 1 be sufficiently large that every point in the Gromov boundary of EDGis can be represented by a p-quasi-geodesic ray. Let γ : [0,∞) → EDG be such asimplicial p-quasi-geodesic ray. We claim that there is a number p′ > 1 such thatγ viewed as a path in SDG0 is an unparametrized p′-quasi-geodesic in SDG0.

Namely, for each i > 0 let ζi be an enlargement of a geodesic in SDG0 withendpoints γ(0), γ(i). Then there is a number L > 1 such that ζi is an L-quasi-geodesic in EDG. By hyperbolicity, established in Corollary 6.7, the Hausdorffdistance in EDG between γ[0, i] and the image of ζi is uniformly bounded, andhence the same holds true if this Hausdorff distance is measured with respect tothe distance in SDG0 ⊃ EDG. Thus the Hausdorff distance in SDG0 between γ[0, i]and a geodesic with the same endpoints is uniformly bounded. Since i > 0 wasarbitrary, this implies that γ is an unparametrized p′-quasi-geodesic in SDG0 for anumber p′ > 0 only depending on p.

As a consequence, if the diameter of γ[0,∞) in SDG0 is infinite then up toparametrization, γ[0,∞) is a p′-quasi-geodesic ray in SDG0 and hence by Corollary4.4, the sequence (∂γ(i))is an unparametrized quasi-geodesic in CG. By the mainresult of [K99, H06], this implies that the simple closed curves ∂γ(i) converge asi → ∞ in the coarse Hausdorff topology to a lamination µ ∈ ∂H ⊂ ∂CG which iswhat we wanted to show.


Now assume that the diameter of γ[0,∞) in SDG0 is finite. By Corollary 6.7,the subgraphs E(c) of EDG where c runs through all I-bundle generators in ∂H areuniformly quasi-convex. Moreover, there is a number M > 0 not depending on cwith the following properties.

(1) If D,E ∈ EDG are any two vertices and if c is an I-bundle generator so thatthe distance in E(c) of some shortest distance projection of D,E into E(c)is at least M then a geodesic connecting D to E in SDG0 passes throughthe special vertex vc defined by c.

(2) Let γ : [0, ℓ] → EDG be a p-quasi-geodesic. If there is some k ≤ ℓ andsome I-bundle generator c such that the distance in E(c) of some shortestdistance projection of γ(0), γ(k) into E(c) is at least 2M then for each ℓ ≥ kthe distance in E(c) of a shortest distance projection of γ(0), γ(ℓ) into E(c)is at least M .

For k > 0 let C1(k) (or C2(k)) be the set of all I-bundle generators c so that thedistance in E(c) between a shortest distance projection of γ(0), γ(k) into E(c) is atleast M (or 2M). By property (2) above, for ℓ ≥ k we have C2(ℓ) ⊂ C1(k).

The diameter of the image of any simplicial geodesic in SDG0 equals the length ofthe geodesic and hence it is is bounded from below by the number of special verticesit passes through. Since the diameter of γ[0,∞) in SDG0 is finite by assumption,by property (1) the cardinality of C1(k) is bounded independent of k.

By property (2) above, we deduce that there is some k0 > 0 so that C2(ℓ) ⊂C1(k0) for all ℓ ≥ k0. Since the diameter of γ[k0,∞) in SDG0 is finite, it followsthat there is some I-bundle generator c so that γ[k0,∞) is contained in a uni-formly bounded neighborhood of E(c). Now γ is a p-quasi-geodesic in EDG andthe embedding E(c) → EDG is a quasi-isometry. Thus by hyperbolicity, there is aquasi-geodesic ζ in E(c) whose Hausdorff distance to γ[k0,∞) is bounded. FromLemma 6.2, Lemma 6.1 and the main result of [K99, H06] we conclude that the se-quence (∂ζ(j)) converges as j → ∞ in the coarse Hausdorff topology to a laminationµ ∈ ∂E(c).

As observed in the beginning of this proof, a minimal geodesic lamination whichdoes not intersect µ either is a component of µ, or it is the simple closed curve c.A limit in the coarse Hausdorff topology of a subsequence of (∂γ(j)) is a geodesiclamination which does not intersect µ, moreover its support does not contain c.Thus the sequence (∂γ(j)) converges in the coarse Hausdorff topology to µ (werefer to [K99, H06] for a detailed discussion).

As a consequence, there is a map Λ from the Gromov boundary of EDG into∂EDG which associates to the equivalence class of a simplicial p-quasi-geodesic rayγ the limit in the coarse Hausdorff topology of the sequence (∂γ(i)). Since theenlargement of any quasi-geodesic ray in SDG0 is a quasi-geodesic ray in EDG,Corollary 4.7 implies that the image of Λ contains ∂H. Since for every I-bundlegenerator c the subgraph E(c) of EDG is uniformly quasi-convex, Lemma 6.2 impliesthat ∂E(c) is contained in the image as well. Therefore the map Λ is surjective,and injectivity follows from the description of the boundary of the curve graph in[K99, H06].

Equivariance under the action of Map(H) is immediate from the construction.The reasoning in the proof of Lemma 4.5 shows that Λ is continuous and open. Inother words, Λ is an equivariant homeomorphism of the Gromov boundary of EDGonto ∂EDG.


Since Map(H) acts transitively on I-bundle generators and on non-separatingdiscs, we conclude as in the proof of Lemma 4.6 that the action of Map(H) on∂EDG is minimal. �

Remark: 1) It should be possible to relate in a more systematic way the Gromovboundary of a hyperbolic graph G as in Theorem 5.2 to the Gromov boundary ofits E-electrification and the Gromov boundaries of the family E of subgraphs.

2) Proposition 6.9 and its proof are equally valid for the electrified disc graph of athick subsurface X of ∂H and give an explicit description of the Gromov boundaryas a union of the Gromov boundary of the superconducting disc graph and theboundaries of the quasi-convex subgraphs defined by I-bundle generators.

7. Decreasing conductivity: The disc graph for m = 0

In this section we give an alternative proof of the following result of Masur andSchleimer [MS13].

Theorem 7.1. The disc graph for a handlebody without spots is hyperbolic.

The idea of the proof is to successively thin out the electrified disc graph usingthe construction in Section 5 as presented in Section 6. The main building block forthe construction is Corollary 6.7 which states that for every thick subsurface X ofthe boundary ∂H of a handlebody H of genus g ≥ 2 without spots, the electrifieddisc graph EDG(X) of X is hyperbolic.

For technical reason we slightly weaken the definition of a thick subsurface of∂H as follows. Define a connected properly embedded subsurface X of ∂H to bevisible if ∂H−X does not contain the boundary of any disc, if none of the boundarycomponents of X is discbounding and if moreover X contains the boundary of atleast one disc. Thus a thick subsurface is visible, but a visible subsurface may notbe filled by boundaries of discs. Note that if X is visible but not thick then thediameter of the electrified disc graph EDG(X) of X equals one.

Let DG(X) be the disc graph of the visible subsurface X. The next lemma showsthat if X is a visible five holed sphere or two holed torus then DG(X) is hyperbolic.

Lemma 7.2. Let X ⊂ ∂H be a visible subsurface which is a five-holed sphere or a

two-holed torus. Then the vertex inclusion DG(X) → EDG(X) is a quasi-isometry.

Proof. Let X ⊂ ∂H be a visible subsurface. Let γ : [0, k] → EDG(X) be a geodesic.By modifying γ while increasing its length by at most a factor of two we may assumethat for each i, either γ(i) is disjoint from γ(i + 1), or there is an essential simpleclosed curve which is not discbounding and which is disjoint from both γ(i), γ(i+1)but there is no discbounding curve disjoint from both γ(i), γ(i+ 1).

If X is a five-holed sphere then every simple closed curve c in X is separating,and X − c is the disjoint union of a four holed sphere X1 and a three holed sphere.Any two essential simple closed curves α, β in X which are disjoint from c arecontained in X1. If c is not discbounding and if c is disjoint from a discboundingsimple closed curve then X1 is a visible four holed sphere. A visible four holedsphere contains the boundary of precisely one disc. This implies that for all i thedisc γ(i) is disjoint from γ(i + 1) and therefore γ is in fact a simplicial path inDG(X).

Similarly, ifX is a two-holed torus then a simple closed curve c inX either is non-separating and X − c is a four-holed sphere, or c is separating and c decomposes X


into a three-holed sphere and a one-holed torus. A one-holed visible torus containsthe boundary of precisely one disc. Thus the argument in the previous paragraphis valid in this situation as well and shows the lemma. �

From now on let X be a visible subsurface of ∂H which is not a sphere withat most five holes or a torus with at most two holes. Our next goal is to showhyperbolicity of a graph EDG(2, X) which lies between DG(X) and EDG(X). Itsvertices are isotopy classes of essential discs with boundary in X, and two suchdiscs D,E are connected by an edge of length one if either D,E are disjoint or if∂D, ∂E are disjoint from a multicurve β ⊂ ∂X consisting of two components whichare not freely homotopic.

Call a simple closed curve c in X admissible if c has the following properties.

(1) c is neither discbounding nor discbusting in X.(2) Either c is non-separating or c decomposes X into a three-holed sphere X1

and a second component X2 which contains the boundary of at least onedisc.

Note that if c is an admissible simple closed curve and if d is any other simpleclosed curve then a tubular neighborhood of c∪d contains an essential simple closedcurve which is disjoint from c. This is obvious if c, d are disjoint, so assume thatc∩d 6= ∅. Let d0 be a component of d−c chosen in such a way that if c is separatingthen d0 is not contained in the three-holed sphere component of X − c. Then d0 iscontained in a component of X − c which neither is a three holed sphere nor a one-holed torus and hence one of the boundary components of a tubular neighborhoodof c ∪ d0 is an essential simple closed curve in X distinct from c.

For an admissible simple closed curve c in X define H(c) to be the completesubgraph of EDG(2, X) whose vertex set consists of all discs which are disjointfrom c.

Lemma 7.3. H(c) is isometric to EDG(X − c).

Proof. A disc D ∈ H(c) is disjoint from c and hence it defines a vertex in EDG(X−c). Any two such discs D,E are connected by an edge in EDG(2, X) if and only ifeither they are disjoint or if there is a pair β1, β2 of disjoint essential simple closedcurves which are disjoint from both D,E.

If one of these curves, say the curve β1, is disjoint from c then this means thatD,E viewed as vertices in EDG(X− c) are connected by an edge. Otherwise by theremark preceding the lemma, there is an essential simple closed curve containedin a tubular neighborhood of c ∪ β1 which is disjoint from c,D,E and once again,D,E are connected by an edge in EDG(X − c). �

Lemma 7.3 and Corollary 6.7 imply that there is a number δ > 0 so that eachof the graphs H(c) is δ-hyperbolic.

Let H = {H(c) | c} be the family of all these subgraphs of EDG(2, X) where cruns through all admissible curves in X. Our goal is to apply Theorem 5.2 to thefamily H of complete subgraphs of EDG(2, X). As in Section 6, we check that theconditions in this theorem are satisfied.

Lemma 7.4. The family H is bounded.

Proof. Let c, d ⊂ X be distinct admissible simple closed curves. If c, d are disjointthen by definition, any two discs in H(c)∩H(d) are connected in EDG(2, X) by anedge of length one. Thus the diameter of H(c) ∩H(d) equals one.


If c, d intersect then by the remark preceding Lemma 7.3, there is a simple closedcurve d′ disjoint from c which is disjoint from any disc in H(c) ∩ H(d) and henceas before, any two discs H(c) ∩H(d) are connected in EDG(2, X) by an edge. �

Lemma 7.5. EDG(X) is quasi-isometric to the H-electrification of EDG(2, X).

Proof. Let G be the H-electrification of EDG(2, X). We show first that the vertexinclusion EDG(X) → G is coarsely Lipschitz.

Namely, if D,E are any two vertices in EDG(X) which are connected by an edgeand if D,E are disjoint then they are connected by an edge in EDG(2, X) as well.

If D,E are disjoint from a common simple closed curve c and if c either is admis-sible or discbounding, then by the definition of the H-electrification of EDG(2, X),their distance in G is at most two.

On the other hand, if c is not admissible and not discbounding then c is aseparating simple closed curve in X which is neither discbounding nor discbustingand which decomposes X into two components X1, X2 which are not three-holedspheres. Two discs D,E with ∂D ⊂ X1, ∂E ⊂ X2 are disjoint and hence connectedby an edge in EDG(2, X). If ∂D, ∂E are contained in the same component of X−c,say in X1, then the second component X2 contains an essential simple closed curved. Then ∂D, ∂E are disjoint from the multi-curve c ∪ d with two components andhence once more, D,E are connected in EDG(2, X) by an edge. As a consequence,the vertex inclusion of EDG(X) into G is indeed coarsely Lipschitz.

That this inclusion is in fact a quasi-isometry is immediate from the definitions.Namely, if c is admissible then any two vertices in H(c) are connected in EDG(X)by an edge. �

We use Lemma 7.5 and Theorem 5.2 to show hyperbolicity of EDG(2, X).

Corollary 7.6. EDG(2, X) is hyperbolic. Enlargements of geodesics in EDG(X)are uniform quasi-geodesics in EDG(2, X).

Proof. it suffices to show that the family H satisfy the conditions in Theorem 5.2.This family is bounded by Lemma 7.4.

For an admissible simple closed curve c, δ-hyperbolicity of H(c) for a numberδ > 0 not depending on c follows from Corollary 7.3 and Corollary 6.7.

The second property in Theorem 5.2 is a consequence of Lemma 7.5 and Corollary6.7,

To show the bounded penetration property, recall that enlargements of geodesicsin SDG(X) are uniform quasi-geodesics in EDG(X). Let c be an admissible simpleclosed curve and let X1 be the component of X − c which is not a three-holedsphere. By Lemma 6.5, Lemma 6.6 and the proof of Corollary 6.7, an enlargementof a geodesic in SDG(X) passes through two points of large distance in H(c) =EDG(X − c) if and only if one of the following two possibilities holds true.

(1) The diameter of the subsurface projection of the endpoints into the arc andcurve graph of X1 is large.

(2) There is an I-bundle generator β ⊂ X1 such that the diameter of thesubsurface projection of the endpoints into the arc and curve graph ofX1 − β is large.

From this the bounded penetration property follows as in the proof of Lemma6.6. �


Proof of Theorem 7.1: For k ≥ 1 define EDG(k) to be the graph whose vertexset is the set of all discs in H and where two such discs are connected by an edgeof length one if and only if either they are disjoint or they are both disjoint from amulticurve in ∂H with a least k components. Note that if k equals the cardinalityof a pants decomposition for ∂H then EDG(k) equals the disc graph of H.

We show by induction on k that the graph EDG(k) is hyperbolic and that more-over enlargements of quasi-geodesics in EDG(k − 1) are uniform quasi-geodesics inEDG(k).

The case k = 1 is just Corollary 6.7, and the case k = 2 was shown in Corollary7.6. Thus assume that the claim holds true for k− 1 ∈ [2, 3g− 3). Let D,E be anytwo discs in H. Let γ be a geodesic in EDG(k − 1) connecting D to E.

Let i ≥ 0 be such that the discs γ(i), γ(i + 1) are not disjoint. Then they aredisjoint from a multicurve α in ∂H with at least k−1 components. We may assumethat none of the components of α is discbounding. Since ∂γ(i), ∂γ(i+ 1) intersectthey are contained in the same component X of ∂H − α. Then X is a visiblesubsurface of ∂H.

If either ∂H − X contains a multicurve with k components or if X − (∂γ(i) ∪∂γ(i+1)) contains an essential simple closed curve then γ(i) is connected to γ(i+1)by an edge in EDG(k). Otherwise replace the edge γ[i, i + 1] in EDG(k − 1) by ageodesic γik in EDG(X) with the same endpoints. The concatenation of these arcsis a curve γk which is an enlargement of γ. For all j, the discs γk(j), γk(j+1) eitherare disjoint or disjoint from a multicurve containing at least k components whichare not discbounding.

This process stops in the moment the component X of ∂H − α is a five-holedsphere or a two-holed torus since by Lemma 7.2, for such a surface the vertexinclusion DG(X) → EDG(X) is a quasi-isometry. �

For a thick subsurface Y of ∂H denote as before by πY the subsurface projectionof simple closed curves into the arc and curve graph of Y . If c is an I-bundlegenerator in a thick subsurface Y then let πc be the subsurface projection intoY − c.

The following corollary is now immediate from our construction. It was earlierobtained by Masur and Schleimer (Theorem 19.9 of [MS13]). The formulation wegive is however different from the formulation in [MS13] since we subsume thecontributions arising from I-bundles in the distance functions of the electrified discgraphs of thick subsurfaces. Denote as before by dD the distance in the disc graphof H.

Corollary 7.7. There is a number C > 0 such that

dD(D,E) ≍∑

Y

diam(πY (E ∪D))C +∑

c

diam(πc(E ∪D))C

where Y runs through all thick subsurfaces of ∂H, where c runs through all I-bundle generators in thick subsurfaces of ∂H, and the diameter is taken in the arc

and curve graph.

For a thick subsurface Y of ∂H let ∂EDG(Y ) be the Gromov boundary ofEDG(Y ). Define

∂DG = ∪Y ∂EDG(Y ) ⊂ L


where the union is over all thick subsurfaces of ∂H and where this union is viewedas a subspace of L, i.e. it is equipped with the coarse Hausdorff topology. Theproof of the following statement is completely analogous to the proof of Corollary6.9 and will be omitted.

Corollary 7.8. ∂DG is the Gromov boundary of DG.

8. The electrified disc graph for a handlebody with a single spot

A handlebody with spots is a handlebody H of genus g ≥ 1 with m ≥ 1 markedpoints, called spots, on the boundary. We view the boundary ∂H of such a handle-body as a surface with m punctures.

A disc in a handlebody with spots is a disc D in H whose boundary ∂D isdisjoint from the spots. We require that ∂D is an essential simple closed curve in∂H, i.e. it is neither contractible nor homotopic into a spot. Two such discs areisotopic if there is an isotopy between them which is disjoint from the spots. Weuse the terminology introduced in Section 2 also for handlebodies with spots.

For the remainder of this section we consider a handlebody H of genus g ≥ 2with a single spot p. Our goal is to prove the third part of Theorem 2 from theintroduction.

Recall from the introduction the definition of the disc graph DG and of theelectrified disc graph EDG of H. Let moreover CG be the curve graph of ∂H.

Let H0 be the handlebody obtained from H by forgetting the spot and let

Φ : H → H0

be the natural forgetful map. Let CG(∂H0) be the curve graph of ∂H0, and letDG(H0) or EDG(H0), respectively, be the disc graph or the electrified disc graphof H0.

Lemma 8.1. The map Φ induces a simplicial surjection

Π : CG → CG(∂H0)

which restricts to simplicial surjections EDG → EDG(H0) and DG → DG(H0).

Proof. Since H has a single spot, the image under the map Φ of an essential simpleclosed curve γ on ∂H is an essential simple closed curve Φ(γ) on ∂H0. The curveγ is discbounding if and only if this is the case for Φ(γ). Moreover, if γ, δ aredisjoint then this holds true for Φ(γ),Φ(δ) as well. This immediately implies thelemma. �

Theorem 7.1 of [KLS09] shows that for every simple closed curve c in ∂H0 thepreimage Π−1(c) of c in the curve graph CG of ∂H is a tree Tc which can be describedas follows.

Let H2 be the universal covering of ∂H0 and let S(c) be the preimage of c inH2. Let T be the tree whose vertex set is the set of components of H2 −S(c), andwhere two components are connected by an edge if their closures intersect. Viewthe spot p as the basepoint for the fundamental group of ∂H0. Then π1(∂H0, p)acts transitively on T as a group of simplicial isometries. The tree Tc is π1(∂H0, p)-equivariantly isomorphic to T (see Section 7 of [KLS09] for details).

A section for the projection Π : CG → CG(∂H0) is a map Λ : CG(∂H0) → CG sothat Π ◦ Λ = Id. The next observation is essentially due to Harer (see [KLS09]).


Lemma 8.2. For each γ ∈ CG there is an isometric section Λ : CG(H0) → CG for

the projection Π : CG → CG(H0) passing through γ.

Proof. Fix a hyperbolic metric on ∂H0. Every simple closed curve on ∂H0 can berepresented by a unique simple closed geodesic. The union of these geodesics hasarea zero and hence there is a point x ∈ ∂H0 which is not contained in any suchgeodesic. View x as the marked point in ∂H.

Define a map Λ : CG(∂H0) → CG as follows. To a vertex of CG(∂H0), i.e. asimple closed curve on ∂H0, associate the isotopy class Λ(γ) of γ viewed as a curvein ∂H = (∂H0, x). Clearly Π ◦Λ is the identity on vertices of CG(∂H0). If α, β aredisjoint simple closed curves on ∂H0 then their geodesic representatives are disjointas well, and Λ(α),Λ(β) are disjoint simple closed curves in ∂H. Thus Λ extends to asimplicial section. Since Π is distance non-increasing, Λ is an isometric embeddingof CG(∂H0) into CG.

Consider the Birman exact sequence

(3) 0 → π1(∂H0, x) → Mod(∂H) → Mod(∂H0) → 0.

The action of π1(∂H0, x) on the curve graph of ∂H is fibre preserving. Indeed,the restriction of this action to a fixed fibre is just the action of π1(∂H0, x) on thegeometric realization of the fibre as described in the text preceding this proof. Inparticular, this action is transitive on vertices in the fibres (see [KLS09] for details).Equivalently, for every simple closed curve α on ∂H and any curve β ∈ Π−1(Π(α))there is a mapping class ψ ∈ π1(∂H0, x) < Mod(∂H) so that ψ(α) = β.

Since the map ψ acts as a fibre preserving simplicial isometry on the curve graphof ∂H, the composition ψ◦Λ is a new isometric section for the map Π. This impliesthat for every vertex of CG there is an isometric section for the projection Π whoseimage contains that point. This is what we wanted to show. �

Remark: 1) The proof of Lemma 8.2 contains some additional information.Namely, two simple closed curves γ, δ on ∂H are contained in the image of anisometric section as constructed by Harer [KLS09] and used in the proof of Lemma8.2 if the spot is not contained in a component of ∂H− (γ∪δ) which is a puncturedbigon, i.e. a component whose boundary consists of a single subarc of γ and a singlesubarc of δ.

2) In [MjS12] (see also [H05]), the authors define a metric graph bundle to consistof graphs V,B and a surjective simplicial map p : V → B with the followingproperties.

(1) For each b ∈ V(B), Fb = p−1(b) is a connected subgraph of V , and theinclusion maps i : V(Fb) → V are uniformly metrically proper for the pathmetric db induced on Fb as measured by a non-decreasing surjective functionf : N → N.

(2) For any two adjacent vertices b1, b2 of V(B), each vertex x1 of Fb1 is con-nected by an edge with a vertex in Fb2 .

If the fibres are trees then we speak about a metric tree bundle. One consequenceof the main result in [MjS12] can be formulated as follows. Let p : V → B be ahyperbolic metric tree bundle over a hyperbolic base B. Let C ⊂ B be a quasi-convex subspace and let W = p−1(C) be the preimage of C. Then W is hyperbolic.

By Lemma 8.2, the graph projection Π : CG → CG(H0) satisfies the secondproperty in the definition of a metric tree bundle. However, the first property


states that for every b ∈ B the distance in V between any two points x, y ∈ Fb

is bounded from below by f(db(x, y)), and this property is violated for map Π.As an example, let c1, c2 be two non-separating simple closed curves in ∂H0, letx ∈ ∂H0− (c1∪c2) and let ζ be a loop in π1(∂H0−c1, x) which fills ∂H0−c1. Thenthe point pushing map β defined by ζ via the Birman exact sequence (3) acts asa hyperbolic isometry and hence with positive translation length on the fibre overc2 although the distance in CG(H) between any two points on the orbit is at mosttwo.

Thus we can not use Theorem 4.2 and the results in [MjS12] to deduce fromhyperbolicity of the curve graph of ∂H0 and ∂H that the superconducting disc graphof H (and hence the electrified disc graph if the genus g of H is odd) is hyperbolic.We expect that this is nevertheless the case, however we did not attempt to answerthis question.

It turns out that unlike in the case of handlebodies without spots, boundaries ofdiscs inH are not quasi-convex in the curve graph of ∂H, so we can not hope to con-struct a version of a superconducting disc graph for H which is quasi-isometricallyembedded in the curve graph of ∂H. Moreover, the bounded penetration propertyused in Section 6 to prove hyperbolicity of the electrified disc graph was deducedfrom properties of subsurface projections for curve graphs. Therefore hyperbolicityof the superconducting disc graph for a handlebody with a single spot would notcontradict the main result of this section.

For the remainder of this section assume that the genus g = 2n of H is even.Define an I-bundle generator in H to be a simple closed curve c on the boundary∂H of H whose image under the forgetful map Φ is an I-bundle generator for thehandlebody H0 = Φ(H).

By the second paragraph in the proof of Lemma 6.4, any two distinct I-bundlegenerators in H0 intersect. Thus any two disjoint I-bundle generators in ∂H arecontained in the same fibre for the projection Π, and they bound a three-holedsphere, with the spot as one of the holes. For such a pair (c1, c2) of disjoint I-bundle generators in ∂H define

RE(c1, c2)

to be the complete subgraph of EDG whose vertex set is the set of all discs whoseboundaries intersect c1, c2 in precisely two points. The image of RE(c1, c2) underthe projection Π is contained in E(Π(c1)).

For an I-bundle generator c in ∂H0 let as before E(c) ⊂ EDG(H0) be the completesubgraph of EDG(H0) which consists of all discs whose boundaries intersect c inprecisely two points. Let X be a component of ∂H0 − c. By Lemma 6.2, the mapwhich associates to a disc D ∈ E(c) the intersection arc ∂D∩X is a quasi-isometryof E(c) onto the electrified arc graph of X.

Corollary 8.3. For every pair (c1, c2) of disjoint I-bundle generators in ∂H there

is a quasi-isometric section Λc1,c2 : E(Π(c1)) → RE(c1, c2) for the projection Π.

Proof. Let c be an I-bundle generator in ∂H0. Choose a hyperbolic metric on∂H0 and let c be the geodesic representative of c. Choose a point x ∈ c whichis not contained in any simple closed geodesic which intersects c transversely. Asdescribed in the proof of Lemma 8.2, use x as a basepoint in ∂H to define anisometric section E(c) → EDG.


The preimage of c in ∂H = (∂H0, x) is an embedded arc c0 in ∂H with endpointson the marked point. This arc determines two simple closed curves c1, c2 in ∂Hwhich bound a once punctured annulus and are disjoint from c0. Then (c1, c2)is a pair of disjoint I-bundle generators in ∂H, and the image of E(c) under theembedding E(c) → EDG is contained in RE(c1, c2).

The corollary now follows from the fact that the subgroup π1(∂H0, x) of Map(H)acts transitively on pairs of disjoint I-bundle generators which project to c. �

For an I-bundle generator c1 in ∂H let E(c1) be the complete subgraph of EDGof all discs whose boundaries intersect c1 in precisely two points. Let X1 be thecomponent of ∂H − c1 which does not contain the spot. Then X1 is a compactsurface of genus n = g/2 ≥ 1 with connected boundary.

Every disc D ∈ E(c1) intersects X1 in a single arc Ψ(D) with endpoints on c1.If we denote by C ′(X1, c1) the electrified arc graph of X1 then this defines a map

Ψ : E(c1) → C ′(X1, c1).

Lemma 6.2 and Corollary 8.3 show that whenever c2 is an I-bundle generator in∂H which is disjoint from c1 then the restriction of the map Ψ to RE(c1, c2) issurjective.

Lemma 8.4. Let (c1, c2) be a pair of disjoint I-bundle generators in ∂H and let

X1 ⊂ ∂H − c1 be the component of ∂H − c1 not containing c2.

(1) The map Ψ : E(c1) → C ′(X1, c1) is a quasi-isometry.

(2) The inclusion RE(c1, c2) → E(c1) is a quasi-isometry.

Proof. We follow the argument in the proof of Lemma 6.2 and show first that themap Ψ : E(c1) → C ′(X1, c1) is coarsely Lipschitz of Lipschitz constant at most two.

Namely, the map Φ restricts to a simplicial map E(c1) → E(Φ(c1)), moreoverthe restriction of Φ to X1 is a homeomorphism onto Φ(X1) ⊂ ∂H0. Thus the claimfollows from Lemma 6.2.

Next we show that for each D ∈ E(c1) the diameter of Ψ−1(Ψ(D)) ⊂ E(c1) is atmost one. Namely, the genus of X1 is positive and hence for every disc D ∈ E(c1)there is a simple closed curve γ ⊂ X1 which is disjoint from D. Then γ is disjointfrom any disc in Ψ−1(Ψ(D)) and therefore by the definition of the electrified discgraph, the diameter of Ψ−1(Ψ(D)) is at most one.

To complete the proof of the lemma it now suffices to show that for any twoarcs α, β ⊂ X1 of distance one in C ′(X1, c1) there are discs D,E ∈ RE(c1, c2) withΨ(D) = α,Ψ(E) = β and such that the distance in RE(c1, c2) between D,E equalsone. But this just follows from surjectivity of the restriction of Ψ to RE(c1, c2). �

We use this discussion to show the second part of Theorem 2 from the introduc-tion.

Proposition 8.5. Let H be a handlebody of even genus g = 2n ≥ 2 with one spot.

The electrified disc graph EDG of H contains quasi-isometrically embedded R2.

Proof. Let (c1, c2) be a pair of disjoint I-bundle generators in ∂H and let Xi bethe component of ∂H − ci not containing ci+1 (indices are taken modulo two). ByLemma 8.4, the map Ψ : D → Ψ(D) = ∂D ∩ X1 restricts to a surjective quasi-isometry of the subgraphRE(c1, c2) of EDG onto the electrified arc graph C ′(X1, c1)of X1. The graph C ′(X1, c1) in turn is quasi-isometric to the curve graph of X1.


Let σ1 be a pseudo-Anosov element of the mapping class group Mod(X1) of X1.Then σ1 acts as a hyperbolic isometry on the curve graph of X1. In particular, thetranslation length of σ1 on the curve graph of X1 is positive.

Choose an essential arc α in X1; then the map j ∈ Z → σj1(α) ∈ C ′(X1, c1) is

a quasi-geodesic. By Corollary 8.3, for each j there is a disc ρ(j) ∈ RE(c1, c2) so

that Ψ(ρ(j)) = σj1(α). By Lemma 8.4 and Corollary 6.7, the map j → Φ(ρ(j)) ∈

EDG(H0) is a quasi-geodesic. Since the projection Π : EDG → EDG(H0) is distancenon-increasing, Corollary 8.3 implies that ρ is a quasi-geodesic in EDG.

Let p be the spot in ∂H, contained in the three-holed sphere B in ∂H withboundary c1 ∪ c2. For i = 1, 2 let ζi be a loop based at p which is disjoint fromXi+1, which intersects B in two embedded arcs connecting p to a point on ci andwhich fills Xi. This means that ζi decomposes Xi into discs (in particular, ζi hasself-intersections). The concatenation ζ = ζ2 ◦ ζ1 is a loop based at p which fills∂H0 (indices are taken modulo two).

By a result of Kra ([Kr81], see Theorem 4.2 [KLS09] for an alternative proof),the point pushing homeomorphism defined by ζ via the Birman exact sequence (3)is a pseudo-Anosov element σ2 in the mapping class group Mod(∂H) of ∂H. Thusthere is a number χ ∈ (0, 12 ] with the following property. For every simple closedcurve γ on ∂H and every n ∈ Z, the distance in the curve graph CG between γ andσn2 (γ) is at least 2χ|n|.Since the boundaries of the discs ρ(i) (i ∈ Z) intersect c1 in 2 points, the distance

in CG between the boundaries of any two discs ρ(i), ρ(j) ∈ RE(c1, c2) is at mostfour. Thus denoting again by dE the distance in EDG we conclude that

(4) dE(ρ(i), σn2 (ρ(j))) ≥ χ|n| − 8 ∀i, j, n.

We claim that

(5) dE(ρ(j), σn2 (ρ(j))) ≤ 2|n| ∀j, n.

Assume for simplicity of notation that n is positive. Let E = ρ(j) ∈ RE(c1, c2).The loop ζ1 does not intersect X2 and hence the point pushing homeomorphismβ1 defined by ζ1 via the sequence (3) can be realized by a diffeomorphism whichpreserves X2 pointwise. In particular, β1 preserves the set E(c2), and if Ψ denotesthe map which associates to a discD the intersection of ∂D withX2 then β1◦Ψ = Ψ.

As in the proof of Lemma 8.4, this implies that the distance between E and β1(E)in E(c2) and hence in EDG equals one. Moreover, β1 maps c1 to a simple closedcurve c3 ⊂ ∂H −X2 which is disjoint from c2, and we have β1(E) ∈ RE(c3, c2).

Similarly, the point pushing homeomorphism β2 defined by the loop ζ2 preservesthe simple closed curve c3 and hence we conclude as before that the distance in E(c3)between β2(β1(E)) = σ2(E) and β1(E) equals one. To summarize, the distancebetween E and σ2(E) is at most two. Now σ2 acts as an isometry on EDG andtherefore

dE(σn2 (E), σ2(E)) ≤

∑

i

dE(σi+12 (E), σi

2(E)) ≤ 2|n|

which is what we wanted to show.As a consequence,

(6) dE(ρ(j), σn2 (ρ(j))) ∈ [χ|n| − 8, 2|n|] ∀j, n.


Since the map σ2 acts as an isometry on EDG and since ρ is a quasi-geodesic,there is a number b > 0 such that

(7) dE(σn2 (ρ(i)), σ

n2 (ρ(j))) ∈ [|i− j|/b− b, b|i− j|+ b] ∀i, j, n.

The estimates (6) and (7) together imply that the map (i, k) ∈ Z2 → σk

2 (ρ(i)) iscoarsely Lipschitz.

We are left with showing that there is a number u > 0 so that

dE(σn2 (ρ(i)), σ

k2 (ρ(j))) ≥ umax{|k − n|, |i− j|} − u−1.

Namely, choose u = χ/4b. If |k − n| ≥ |i− j|/4b then inquality (4) shows

dE(σn2 (ρ(i)), σ

k2 (ρ(j))) = dE(ρ(i), σ

k−n2 (ρ(j)))

≥ χ|k − n| − 8 ≥χ

4bmax{|k − n|, |i− j|} − 8.

Otherwise by inequality (5), σk2 (ρ(i)) can be connected to σn

2 (ρ(i)) by an arc oflength at most 2|k−n| ≤ |i− j|/2b. But the distance between σn

2 (ρ(i)), σn2 (ρ(j)) is

at least |i− j|/b− b and hence the claim follows from the triangle inequality.As a consequence, the map (j, n) → σn

2 (ρ(j)) is a quasi-isometric embedding.The proposition follows. �

Remark: 1) The above proposition is equally valid for the electrified disc graphEDG(X) of a thick subsurface X with a single spot in the boundary ∂H of Hprovided that X contains a separating I-bundle generator.

2) The proof of the proposition above depends in an essential way on the existenceof a quasi-geodesic in the electrified disc graph of the handleobody H0 withoutspot consisting of discs whose boundaries intersect a fixed simple closed curve inuniformly few points. It is not valid for handlebodies of odd genus. As indicatedin the remark after Lemma 8.2, we conjecture that the electrified disc graph of ahandlebody of odd genus with a single spot is hyperbolic.

9. The disc graph of a handlebody with a single spot

The goal of this section is to construct quasi-isometrically embedded copies ofR

2 in the disc graph of handlebodies with a single spot. The case of even genusdoes not follow from Proposition 8.5 since the quasi-isometrically embedded R

2

constructed in its proof is not quasi-isometrically embedded in the disc graph.The arc graph A(S) of a compact surface S of genus n ≥ 1 with connected

boundary ∂S is the graph whose vertices are isotopy classes of embedded arcs in Swith endpoints on the boundary and where two arcs are connected by an edge oflength one if and only if they can be realized disjointly. The arc graph A(S) of Sis hyperbolic, however the inclusion of A(S) into the arc and curve graph of S is aquasi-isometry only if the genus of S equals one [MS13] (see also [HPW13]).

We begin with the case of a handle body H of even genus 2n = g ≥ 2 with asingle spot. For a pair (c1, c2) of disjoint I-bundle generators in ∂H let RD(c1, c2)be the complete subgraph of the disc graph DG of H of all discs which intersectboth curves c1, c2 in precisely two points. We have

Lemma 9.1. Let (c1, c2) be a pair of disjoint I-bundle generators in ∂H.

(1) The graph RD(c1, c2) is quasi-isometric to the product of the arc graph

A(X) of a compact surface X of genus g/2 with connected boundary and

the real line.


(2) There is a coarse one-Lipschitz retraction DG → RD(c1, c2). In particular,

RD(c1, c2) is a quasi-convex subset of DG.

Proof. Let X be the subsurface of ∂H which is bounded by c1 and does not containc2. Then X is a compact surface of genus g/2 with connected boundary. Definea map Ψ1 : DG → A(X) by associating to a disc D a component of ∂D ∩X. ByLemma 8.4, the map Ψ1 is coarsely well defined, coarsely Lipschitz and surjective.Moreover, its restriction to RD(c1, c2) is simplicial.

Let B ⊂ ∂H be the once punctured annulus bounded by c1, c2. The boundary ofa discD ∈ RD(c1, c2) intersects B in two disjoint arcs with endpoints on the distinctboundary components of B. Let Λ be a diffeomorphism of ∂H which preserves thecomplement of B (and hence the boundary of B) pointwise and which pushes thespot in B one full turn around a central loop in the annulus. The isotopy class of Λis contained in the kernel of the homomorphism Mod(∂H) → Mod(∂H0) induced bythe spot closing map. The map Λ extends to a diffeomorphism of the handlebodyH, and it generates an infinite cyclic group of simplicial isometries of RD(c1, c2).

Figure B

Choose a complete finite area hyperbolic metric on ∂H and realize c1, c2 bygeodesics for this metric. For any pair of points a1 ∈ c1, a2 ∈ c2 choose a geodesicarc α(a1, a2) ⊂ B with endpoints a1, a2 such that any two of these arcs connectingdistinct pairs of points on c1, c2 intersect in at most one point.

An essential simple closed curve on ∂H can be represented by a unique geodesicβ. If β bounds a disc then β crosses through the annulus B. This means that thereis at least one subsegment b of β ∩B connecting c1 to c2. Any other such segmentis disjoint from b.

Let x1, x2 be the endpoints of b on c1, c2. There is a number ℓ ∈ Z such thatup to homotopy with fixed endpoints, b = Λℓ(α(x1, x2)). Define Ψ2(β) = ℓ. If b′

is another component of β ∩ B with endpoints y1, y2 on c1, c2 then b′ is disjointfrom b and hence b′ = Λℓ′(α(y1, y2)) for some ℓ′ with |ℓ − ℓ′| ≤ 1. Thus the mapΨ2 is coarsely well defined, and it maps DG into Z. Since disjoint discs intersectB in disjoint arcs, the above discussion also shows that Ψ2 is coarsely 2-Lipschitz,moreover it is coarsely surjective.

Define a distance d0 on A(X) × Z by d0((α, a), (β, b)) = dA(X)(α, β) + |a − b|where dA(X) denotes the distance in A(X) and let

Ψ = (Ψ1,Ψ2) : DG → (A(X)× Z, d0).

By the above consideration, the map Ψ is coarsely Lipschitz, and its restrictionto RD(c1, c2) is coarsely injective and coarsely surjective. Namely, a disc D ∈RD(c1, c2) is determined by the homotopy class relative to c1 of its intersection


with X and by the intersection arc with B of the geodesic representative of itsboundary.

Now a coarse inverse of Ψ|RD(c1, c2) is coarsely Lipschitz as well. This showsthe first part of the lemma, and the second follows by composing Ψ with a coarseinverse of Ψ|RD(c1, c2). �

As a consequence we obtain

Corollary 9.2. The disc graph of a handlebody of even genus g = 2n ≥ 2 with one

spot contains quasi-isometrically embedded copies of R2.

Proof. By the second part of Lemma 9.1, for every pair (c1, c2) of disjoint I-bundlegenerators there is a coarse Lipschitz retraction of DG onto RD(c1, c2). Thus thesubspace RD(c1, c2) is quasi-isometrically embedded in DG. The corollary nowfollows from the first part of Lemma 9.1. �

In the remainder of this section we extend the construction used to show Corol-lary 9.2 to handlebodies of odd genus g = 2n+ 1 ≥ 3 with a single spot.

As before, let H0 be the handlebody with the spot removed and let Φ : H → H0

be the spot forgetful map. Let (c1, c2) be a pair of non-separating simple closedcurves in ∂H which form a bounding pair with the property that H0 is an I-bundleover a component of ∂H0 − Φ(c1 ∪ c2). Choose a simple closed curve c′ in ∂H sothat the three curves c1, c2, c

′ bound a sphere B with four holes, one hole being thespot. The image of B under the map Φ is a three-holed sphere Φ(B) ⊂ ∂H0.

Let RD(c1, c2, c′) be the complete subgraph of DG whose vertex set is the set

of all discs which intersect c1 ∪ c2 in precisely two points and such that there is nocomponent b of ∂D ∩ B whose image under Φ is homotopic into the boundary ofΦ(B). The following observation corresponds to the second part of Lemma 9.1 andis established with a similar proof.

Lemma 9.3. There is coarse Lipschitz retraction DG → RD(c1, c2, c′). In partic-

ular, RD(c1, c2, c′) is a quasi-convex subset of DG.

Proof. Let X1 be the component of ∂H− (c1∪c2) not containing c′. The boundary

∂D of every disc D in H intersects X1.Let τ be the orientation reversing involution of ∂H0 which preserves c1 and c2

pointwise and extends to the fibre preserving involution of the I-bundle structurefor H0 defined by c1 ∪ c2. Let β be a component of ∂D ∩X1; then Φ(β) ∪ τ(Φ(β))is the boundary of a disc E(β) in H0. If β

′ is another component of ∂D ∩X1 thenE(β), E(β′) are disjoint.

Let p be the spot in B. The fundamental group of a three holed sphere is the freegroup F2 with two generators. Thus point pushing of p along loops in the three-holed sphere Φ(B) ⊂ ∂H0 defines a subgroup of the handlebody group isomorphicto F2. For every disc D ⊂ H0 which intersects Φ(c1 ∪ c2) in precisely two points,the group F2 acts transitively on the preimages of D in RD(c1, c2, c

′).The boundary ∂D of a disc D in H intersects the four holed sphere B in a

collection of disjoint arcs. As in the proof of Lemma 9.1, these arcs can be usedto determine a preimage Θ(D) ∈ RD(c1, c2, c

′) of the disc Φ(D). The lemmafollows. �

Now we can argue as for handlebodies of even genus and conclude


Proposition 9.4. The disc graph of a handlebody of odd genus with one spot

contains quasi-isometrically embedded copies of R2.

Proof. Choose a section of the projection RD(c1, c2, c′) → RD(Φ(c1),Φ(c2)) where

the latter is the set of all discs in H0 whose boundaries intersect Φ(c1 ∪ c2) inprecisely two points. Let U0 be the image of this section. Using the notationsfrom the proof of Lemma 9.3, let Λ be a diffeomorphism of ∂H which preserves thecomplement of the four holed sphere B pointwise and is the point-pushing map ofa loop based at p which fills B. For example, a closed curve which goes aroundeach of the three boundary circles of B once will do.

Let U = ∪jΛj(U0). The restriction to U of the projection Π : DG → DG(H0) is a

graph projection with fibre Z. Moreover, for any disc D0 ∈ RD(Φ(c1),Φ(c2)), andfor all i.j, the distance between any two preimages of D0 Ui,Uj equals |i−j| up to afixed multiplicative constant. This implies that U is quasi-isometric to the productof RD(Φ(c1),Φ(c2)) with the real line. As in the proof of Lemma 9.1 we concludethat RD(c1, c2, c

′) contains quasi-isometrically embedded R2. The proposition now

follows from Lemma 9.3. �

10. Distance and intersection II: The case m ≥ 2

In this section we begin to investigate graph of discs in handlebodies with atleast two spots on the boundary. Our main goal is the proof of the fourth part ofTheorem 2.

Reall that an essential connected subsurface X of ∂H is thick if it satisfies thefollowing properties.

(1) No boundary component of X is discbounding.(2) Every disc intersects X.(3) X is filled by boundaries of discs.

As before, we investigate more generally the electrified disc graph of a thick sub-surface X ⊂ ∂H which contains at least two spots. Let CG (or CG(X)) be the curvegraph of ∂H (or of the thick subsurface X). We continue to use the assumptionand notations from Section 2-9.

As mentioned at the end of Section 2, the fact that surgery of discs in handlebod-ies with spots may produce peripheral discs causes substantial difficulty. Indeed,for handlebodies H with a single spot and genus g ≥ 2, boundaries of discs do notform a quasi-convex subset of the curve graph of ∂H (see the remark in Section 8).Instead our strategy is to establish first a weaker analog of Lemma 2.1 and thenanalyze in detail discs which become peripheral after a single surgery.

As a warm-up, observe that if X is a thick subsurface of ∂H with a single spotthen Lemma 2.1 is valid without modification. Namely, if X contains a single spot,then for any two discs D,E with boundary in X which are not disjoint and for anyouter component E′ of E−D, at least one of the discs obtained from D by surgeryat E′ is not peripheral. Thus in this case the proof of Lemma 2.1 is valid and shows

Lemma 10.1. Let X ⊂ ∂H be a thick subsurface containing a single spot and let

D,E be discs in H with boundary in X. Then

dD,X(D,E) ≤ ι(∂D, ∂E)/2 + 1.

As a consequence, a thick subsurface X of ∂H can not be a four-holed spherewhere a single hole is a spot.


Example: If X contains 2 spots then Lemma 10.1 is false in general. Namely,let H be a handlebody of genus g ≥ 2 with precisely two spots and let α ⊂ ∂H be asimple closed curve which bounds a twice punctured disc D0 ⊂ ∂H. Then α boundsa properly embedded disc in H. Choose an embedded arc β ⊂ ∂H which meetsD0 exactly at its endpoints and with the property that β intersects every disc in Hwhich is disjoint from D0. This implies that none of the two simple closed curvesβ1, β2 obtained by connecting the endpoints of β by a subarc of α is discbounding.Let X ⊂ ∂H be a tubular neighborhood of D0 ∪ β. Then X is a thick subsurfaceof ∂H which is a four-holed sphere (here a hole may be a puncture or a boundarycircle). A simple closed curve γ ⊂ X is discbounding if and only if it separates thetwo punctures of X from the two boundary components. Thus the graph EDG(X)consists of countably many points, and it does not have edges. More precisely, anytwo discbounding simple closed curves in X intersect in at least four points, andno two such curves are disjoint from a common essential simple closed curve in X.

For a thick subsurface X ⊂ ∂H which is not a four-holed sphere there is a(weaker) version of Lemma 2.1 which is formulated in the next lemma. For con-venience of notation in the proof, we define the intersection between a peripheralcurve α in ∂H and any other curve β in ∂H as ι(α, β) = 0.

Lemma 10.2. Let X ⊂ ∂H be a thick subsurface containing n ≥ 2 spots which is

different from a four-holed sphere. Then for any two discs D,E with boundary in

X we have

dE,X(D,E) ≤ ι(∂D, ∂E)/2 + 1.

Proof. As in the proof of Lemma 2.1, we proceed by induction on ι(∂D, ∂E). Thecase ι(∂D, ∂E) = 0 is immediate from the definitions. Thus assume that the claimof the lemma holds true whenever ι(∂D, ∂E) ≤ k − 1 for some k ≥ 1.

Let α = ∂D, β = ∂E ⊂ X be discbounding simple closed curves of intersectionnumber k. If there is an essential simple closed curve γ ⊂ X disjoint from α∪β thendE,X(D,E) ≤ 1 by the definition of the electrified disc graph and there is nothingto show. Thus assume that α, β decompose X into discs and one-holed discs (herea hole may either be a spot of ∂H or a boundary component of X). Then sucha complementary region is a polygon or one-holed polygon with sides alternatingbetween subarcs of α and subarcs of β. A complementary polygon has at least foursides. A punctured bigon is a complementary component which is a one-holed discbounded by a single subarc of α and a single subarc of β.

Let E′ be an outer component of E − D (see Section 2 for the terminology).Surgery of D at E′ yields two discs B1, B2 in H with boundary in X. The bound-aries of these discs are simple closed curves α1, α2 in X which are disjoint from α,moreover ι(αi, β) ≤ k−2 = ι(α, β)−2. If at least one of the discs B1, B2 is essential,say if this holds true for B1, then B1 is a disc with dD,X(D,B1) = 1 = dE,X(D,B1)and ι(∂B1, ∂E) ≤ k − 2. Thus the induction hypothesis can be applied to B1 andE and shows the lemma.

If both discs B1, B2 are non-essential then α = ∂D bounds a twice punctureddisc D0 which is embedded in X ⊂ ∂H. The curve β = ∂E decomposes D0 intotwo once punctured discs A1, A2 and a set of rectangles.

The once punctured disc Ai (i = 1, 2) is bounded by a subarc ai of α and asubarc bi of β. Let pi be the spot contained in Ai. The arc ai is an outer arc forthe disc E. For i = 1, 2 surger E at the outer arc ai to a disc whose boundary βi is


A1 A2

α

β

a1

Figure C

obtained from β by replacing the arc bi with the arc ai. Then ι(α, βi) ≤ ι(α, β)− 2and hence if either β1 or β2 is essential then the claim follows as before from theinduction hypothesis.

Thus we are left with the case that both curves β1, β2 are peripheral. Then βbounds a disc E0 ⊂ X punctured at p1, p2. The intersection D0 ∩ E0 is a unionof the once punctured bigons A1, A2 and a collection of rectangles. More precisely,if R = E0 − (A1 ∪ A2) then the intersections of R with ∂D0 decompose R into achain of discs R1, . . . , Rs such that for even i, the rectangle Ri contained in thetwice punctured disc D0, and for odd i the rectangle Ri is contained in X − D0.The number s of rectangles is odd, and ι(∂D, ∂E) = ι(α, β) = 2s+ 2.

Assume for the moment that s = 1, i.e. that the rectangle R does not intersectD0. Then R is homotopic relative to α to an embedded arc in X − D0. Sinceby assumption X is not a four-holed sphere, X contains an essential simple closedcurve γ which is disjoint from D0 ∪ R. Then γ is disjoint from both D0, E0 andhence from D and E. Thus dE,X(D,E) ≤ 1 which is what we wanted to show.

If s ≥ 3 then assume without loss of generality that the arc a1 ⊂ A1 is containedin the boundary of the rectangle R1. Homotope R1 relative to α to a rectangle R1

such that the side ρ of R1 which is opposite to a1 is a subarc of a2 ⊂ α. Attachto ρ a disc G ⊂ A2 punctured at p2. The union A1 ∪ R1 ∪ G is a disc B0 ⊂ Xpunctured at p1 and p2. The boundary of B0 bounds a properly embedded disc Bin H. By the case s = 1 discussed above, we have dE,X(D,B) ≤ 1.

On the other hand, ι(∂B, β) ≤ 2s− 2 = ι(α, β)− 4 and hence the claim followsas before from the induction hypothesis. �

Example: Lemma 10.2 is false if we replace the electrified disc graph by thedisc graph. Namely, let H be a handlebody of genus g ≥ 2 with two spots. Let


X ⊂ ∂H be a twice punctured thick subsurface different from a four holed spherewith the following property. If β ⊂ X is any discbounding simple closed curve thenβ bounds a twice punctured disc in X. Such a thick subsurface can be constructedas follows.

Choose a twice punctured disc D0 ⊂ ∂H and let β ⊂ ∂H be a simple arc withendpoints on D0 whose interior is disjoint from D0. Let β1, β2 be the two disjointsimple closed curves which are concatenations of β with one of the two subarcs of∂D0 = α with the same endpoints. Let H0 be the handlebody obtained from H byclosing the spots. Assume that β is chosen in such a way that the projection of βito H0 has large distance in the curve graph of ∂H0 from any discbounding simpleclosed curve. Choose simple closed curves γ1, γ2 ⊂ ∂H −D0 which bound togetherwith β1 a three-holed sphere. The projections to H0 of the curves γ1, γ2 havelarge distance to any discbounding curve. Let X ⊂ ∂H be the subsurface boundedby γ1, γ2 and β2. Then X is thick, and X is a thick five holed sphere. Everydiscbounding curve in X bounds a twice punctured disc in X. In particular, anytwo discbounding curves in X intersect in at least four points. As a consequence,the disc graph of X is a countable collection of points and does not have edges.

We next establish a version of Proposition 2.5 for discs D,E in handlebodieswith at least two spots on the boundary which become peripheral after closing oneof the spots. To ease terminology we say that a disc D ⊂ H encloses two spots

p1, p2 in ∂H if D is homotopic with fixed boundary to a disc D0 ⊂ ∂H puncturedat the points p1, p2. Note that the distance in the following lemma is taken inthe electrified disc graph. This is the first indication that for handlebodies withat least two spots on the boundary, the electrified disc graph is quasi-isometricallyembedded in the curve graph. The above example shows that we can not replacethe electrified disc graph by the disc graph.

Lemma 10.3. Let X ⊂ ∂H be a thick subsurface which is different from a four-

holed sphere and let D,E be two discs with boundary in X which enclose the same

spots p1 6= p2 ∈ X. If there is a simple closed curve γ ⊂ X which intersects ∂D, ∂Ein at most k ≥ 1 points then dE,X(D,E) ≤ 2k + 12.

Proof. Let D,E be as in the lemma, with boundaries ∂D = α, ∂E = β. Then α, βbound discs D0, E0 ⊂ X ⊂ ∂H punctured at the points p1, p2. Up to isotopy, theintersection D0 ∩E0 is a union of two once punctured bigons A1, A2 and a disjointunion of rectangles. The punctured bigon Ai (i = 1, 2) is bounded by a subarc ofα and a subarc of β. Assume that Ai contains the spot pi (i = 1, 2).

Let γ ⊂ X be a simple closed curve which intersects both α, β in at most kpoints. If γ is not disjoint from α then γ ∩D0 is a union of at most k/2 pairwisedisjoint arcs. By modifying γ with an isotopy we may assume that these arcs aredisjoint from the punctured discs A1, A2. For i = 1, 2 choose a compact arc ci ⊂ D0

which connects the punctured bigon Ai to γ and whose interior is disjoint from γ.Let moreover γ0 be one of the two subarcs of γ which connect the two endpoints ofc1, c2 on γ. The concatenation c1 ◦ γ0 ◦ c

−12 of c1, γ0, c

−12 is an embedded arc in X

connecting A1 to A2.Let C0 ⊂ X be an embedded rectangle with two opposite sides contained in the

interior of ∂A1 ∩D0, ∂A2 ∩D0 which is a thickening of the arc c1 ◦ γ0 ◦ c−12 . The

union of C0 with A1 ∪ A2 is a disc B0 punctured at p1, p2 whose boundary ∂B0


intersects γ in at most two points, one intersection point each near the endpointsof c1, c2, and it intersects ∂D0 = α in at most 2k points.

Let B ⊂ H be a properly embedded disc with boundary ∂B = ∂B0. The discB encloses the spots p1, p2. By Lemma 10.2, we have dE,X(D,B) ≤ k + 1. Thusvia replacing D by B we may assume that γ intersects D in at most two points.Repeating this construction with the disc E implies that it suffices to show thefollowing. If the simple closed curve γ ⊂ X intersects each of the curves ∂D, ∂E inat most two points then dE,X(D,E) ≤ 10. Note that since ∂D, ∂E are separatingsimple closed curves, the curve γ intersects ∂D0 = ∂D, ∂E0 = ∂E in either two orzero points.

In the case that γ is disjoint from both D0, E0 we are done, so assume (viapossibly exchanging D0 and E0) that γ intersects ∂E0 in precisely two points. Ifγ is not disjoint from D0 then we may assume that γ ∩ D0 is disjoint from E0.Moreover, in this case we may assume that each of the two curves obtained fromγ by replacing γ ∩ D0 by a subarc of α = ∂D with the same endpoints is notperipheral. Namely, otherwise γ bounds a disc B in H and the claim follows fromLemma 10.2 applied to D,B and to B,E.

Let R be the component of E0−D0 containing γ∩E0. Let E1 be the componentof E0−R which contains the once punctured disc A1. Note that E1 is disjoint fromγ. The intersection E1∩D0 is a finite union of disjoint rectangles. Let R ⊂ E1∩D0

be the component which is closest to the once punctured disc A2 in D0. Thismeans that there is an embedded arc c3 ⊂ D0 with one endpoint in R and thesecond endpoint in A2 which intersects E1 only at one endpoint. Let E2 ⊂ E1

be the union of the component of E1 − R containing A1 with R. Note that E2 isdisjoint from γ. The union of the once punctured disc E2, a thickening of the arcc3 and the once punctured disc A2 is a twice punctured disc V embedded in X. LetD1 ⊂ H be a properly embedded disc with boundary ∂D1 = ∂V . Then D1 enclosesthe spots p1, p2. Moreover, if the rectangle component R of E0−D0 containing theintersection of E0 with γ is the component of E0 −D0 which contains ∂A2 ∩ ∂D0

in its boundary then ι(∂D1, ∂E) = 4.We distinguish three cases.

Case 1: γ ∩D0 6= ∅ and R is contained in the component of D0 − γ containingA1, or, equivalently, γ ∩D0 lies between R and A2.

Then the punctured disc V and hence D1 is disjoint from the essential simpleclosed curve γ′ which is obtained from γ by replacing the γ ∩ D0 by the subarcof α with the same endpoints which contains A2 ∩ α. Since both D and D1 aredisjoint from γ′, we have dE,X(D,D1) = 1. Moreover, D1 intersects γ in preciselytwo points. Replace D by D1.

Case 2: γ ∩D0 = ∅.Then D1 is disjoint from γ, and dE,X(D,D1) = 1. Replace D by D1.

Case 3: γ ∩D0 6= ∅ and R is contained in the component of D0 − γ containingA2.

Then D1 is disjoint from γ. Thus the pairs of discs D,D1 and D1, E satisfy thehypothesis in Case 2. Therefore this case follows from two applications of Case 2,applied to D,D1 and D1, E provided that we can show that under the assumptionof Case 2 above, we have dE,X(D,E) ≤ 5.


We now continue to investigate Case 1 and Case 2. Using the above notationsfor Case 1 and Case 2, let Q be the component of D0 − R containing A2. Thecomponents of V ∩ E0 which are different from once punctured discs are up toisotopy contained in Q∩E0. Since the subdisc E1 ⊂ E0 −R is disjoint from Q thisimplies that the rectangle component R1 of E0 − V which contains γ ∩E0 has oneside on the boundary of the disc component of V ∩ E0 which is punctured at p1.

Reapply the above construction with the discs D1 and E, but with the roles ofthe punctures p1, p2 exchanged. Let V ′ be the twice punctured disc obtained fromthis construction. Since the component of E0 − V containing the intersection withγ is the rectangle adjacent to the bigon component of V ∩ E0 punctured at p1,the remark before Case 1 above shows that ∂V ′ ∩ ∂E0 consists of four points. Inparticular, if D2 ⊂ H is the properly embedded disc with boundary ∂D2 = ∂V ′

then dE,X(D2, E) ≤ 3 by Lemma 10.2.An application of the above analysis to D1 and E yields the following. If either γ

is disjoint from D1 or if γ is not disjoint from D2 then dE,X(D1, D2) = 1, moreoverdE,X(D2, E) ≤ 3 and hence dE,X(D1, E) ≤ 4.

To summarize, we have.

i) In Case 2 above, dE,X(D,E) ≤ 5.ii) In Case 1, either the pairs of discs D1, E also satisfy Case 1 and then

dE,X(D,E) ≤ 5, or there is a disc D′ which is disjoint from γ.iii) In Case 3, there is a disc D′ which is disjoint from γ.

As a consequence, either dE,X(D,E) ≤ 5 or there is a disc D′ with dE,X(D′, D) ≤5, dE,X(D′, E) ≤ 5 which is what we wanted to show. �

We use Lemma 10.2 and Lemma 10.3 to show the fourth part of Theorem 2.

Proposition 10.4. Let X ⊂ ∂H be a thick subsurface with n ≥ 2 spots which is

not a four-holed sphere. Then the map EDG(X) → CG(X) which associates to a

disc its boundary is a 16-quasi-isometry.

Proof. Let γ ⊂ X be any simple closed curve. Let p1, p2 be two spots of ∂Hcontained in X. Then there is an embedded arc α ⊂ X connecting p1 to p2 whichintersects γ in at most one point. A thickening of α is a discbounding simple closedcurve inX which intersects γ in at most two points. Thus any simple closed curve inX is at distance two in CG(X) from a discbounding simple closed curve. Moreover,by Lemma 10.2, for any disc D in X there is a disc D′ with boundary in X whichencloses the spots p1, p2 and such that dE,X(D,D′) ≤ 2.

As a consequence, it suffices to show the following. If the discs D,E both enclosethe spots p1, p2 in X then

dE,X(D,E) ≤ 16dCG,X(∂D, ∂E)

where dCG,X denotes the distance in the curve graph of X.Let (γi)0≤i≤ℓ be a geodesic in CG(X) connecting ∂D = γ0 to ∂E = γℓ. The

curve γi is disjoint from γi+1.Since for each i < ℓ the simple closed curves γi, γi+1 are disjoint there is a simple

arc in X connecting p1 to p2 which intersects each of the curves γi and γi+1 in atmost one point. A thickening of such an arc is a curve βi,i+1 which bounds a discBi,i+1 enclosing p1 and p2. The curve βi,i+1 intersects both γi and γi+1 in at mosttwo points.


By Lemma 10.3,dE,X(Bi−1,i, Bi,i+1) ≤ 16 ∀i.

This means that D can be connected to E by a path in EDG(X) whose length doesnot exceed 16dCG,X(∂D, ∂E). This shows the proposition. �

11. The disc graph for handlebodies with at least two spots

In this final section we investigate handlebodies H with at least two spots onthe boundary. Our goal is to show parts 3,4,5 of Theorem 1.

A handlebody of genus one with at most one spot on the boundary contains asingle disc. This is used to establish

Proposition 11.1. The disc graph of a solid torus with two spots is a tree.

Proof. Let H be a solid torus with two spots p1, p2 on the boundary. The handle-body H1 obtained from H by removing the spot p2 is a solid torus with one spoton the boundary. Let Φ1 : ∂H → ∂H1 be the spot closing map.

The handlebody H1 contains a single disc D1, and this disc is non-separating. IfD ⊂ H is any non-separating disc then Φ1(∂D) = ∂D1. Thus by the main result of[KLS09], the graph of non-separating discs in H is a tree T (compare the discussionin Section 8).

If D ⊂ H is a separating disc then Φ1(∂D) is peripheral. There is a single discin H which is disjoint from D, and this disc is non-separating. As a consequence,the disc graph of H is an extension of the tree T which attaches to each vertex inT a single edge whose second endpoint is univalent. Thus this graph is a tree aswell. The proposition follows. �

Next we investigate a handlebody H of genus g ≥ 2 with two spots p1, p2 on theboundary. Let H0 be the handlebody of genus g without spots and let Φ : H → H0

be the spot closing map. A disc D in H encloses the two spots p1, p2 if Φ(D) ⊂ H0

is contractible. A disc D which encloses the two spots defines the conjugacy classof a free splitting Fg+1 = Z ∗ Fg of the free group with g + 1 generators as follows.

Enlarge the two spots p1, p2 to two small compact disjoint discs B1, B2 in ∂Hwith pi ∈ ∂Bi. Construct a handlebody H ′ of genus g + 1 without spots on theboundary by identifying these two discs with an orientation reversing diffeomor-phism which identifies p1 and p2. The fundamental group of H ′ is the free groupFg+1 with g + 1 generators. We choose the image p of p2 as the basepoint for thefundamental group of H ′. Up to isotopy, we may assume that the disc D is disjointfrom B1 ∪ B2 and hence it determines a separating disc D′ in H ′. The choice ofthe basepoint p and the disc D′ define a free splitting

Fg+1 = Fg ∗ Z

which only depends on p2, D and not on any choices made. The free factor Fg

can naturally be identified with the fundamental group of the handlebody H. Agenerator α of the free factor Z is the image in H ′ of an embedded arc α0 in ∂Hwhose interior is disjoint from D ∪B1 ∪B2 and which connects the puncture p1 top2.

Now let E ⊂ H be another disc which encloses the two spots p1, p2. Then ∂Eintersects ∂D in at least four points. Assume for the moment that ∂E intersects∂D in precisely four points. Let V be the component of ∂H −D of positive genus.Then ∂E ∩ V consists of two parallel copies of a simple arc γ in V with endpoints


on ∂D. Connect the two endpoints of γ with the spot p2 by embedded arcs in∂H − D and denote by γ the resulting loop based at p2. For each of the twoorientations, the loop γ defines an element of π1(H, p2) = Fg which only dependson the free homotopy class of γ relative to ∂D and the choice of the orientation.Note that since we take the fundamental group of the handlebody H rather thanthe fundamental group of its boundary, the way the endpoints of γ are connectedto p2 in ∂H −D does not affect the result.

Let β0 be an embedded arc in ∂H which is disjoint from E and connects thepuncture p1 to p2. Up to homotopy, β0 ∩ V is an embedded arc in V which ishomotopic relative to ∂D to the arc γ equipped with one of the two orientations.In particular, the homotopy class with fixed endpoints of β0 is the concatenationof the homotopy class of the arc α0 and the homotopy class of the oriented loop γ.This conclusion also holds true if ∂E intersects ∂D in more than four points. Thusif β denotes the homotopy class of the closed curve in the handlebody H ′ definedby β0 then β equals the concatenation of α with the class of γ.

Choose a free basis A = {α1, . . . , αg} of Fg = π1(H, p2). As above let α be agenerator of the free factor Z of Fg+1 defined by the arc α0 and hence by the disc D

and the basepoint p2. Let A = {α1, . . . , αg, α} be the resulting free basis of Fg+1.

By the above discussion, a word in the letters A which represents the class β canbe obtained from α by concatenating the word defining γ (the reading here is fromleft to right). Thus replacing D by E determines an automorphism Ψ of Fg+1 byΨ(α) = αγ and Ψ(σ) = σ for each element σ ∈ Fg = π1(H, p2).

We now use a device from [SS10]. Namely, define the simple g + 1-length

|w|simpleg+1

of a word in the free basis A = {α1, . . . , αg} of Fg to be the greatest number t suchthat w is of the form w1w2 . . . wt where the Whitehead graph of wj with respect tothe basis A has no cut vertex for each j = 1, . . . , t. If the Whitehead graph of w hasa cut vertex then the simple g+1-length of w is defined to be zero. The terminologyhere is taken from [SS10] although it is not well adapted to the situation at hand.

The next lemma relates simple g + 1-length to the disc graph in H. To simplifynotation, in the sequel we call a sequence (Di) of discs in H a path in DG if forall i the disc Di is disjoint from Di+1. Thus such a sequence is the set of integralpoints on a simplicial path in DG connecting its endpoints.

Lemma 11.2. Let (Di)0≤i≤n be a path in DG which begins and ends with a disc

enclosing the two spots p1, p2. Let w ∈ π1(H, p2) be such that an arc connecting

p1 to p2 in ∂H − Dn defines the word αw in Fg+1 where α is defined by an arc

connecting p1 to p2 in ∂H −D0. Then

|w|simpleg+1 ≤ 2n.

Proof. Assume without loss of generality that the path (Di) connecting D0 to Dn

is of minimal length in DG. First we modify inductively the sequence (Di) withoutincreasing its length in such a way that each of the discs Di (1 ≤ i ≤ n− 1) eitheris non-separating or encloses the spots p1, p2.

In a first step, we modify the sequence keeping the discs D2i fixed in such a waythat for each i ≤ n/2 the disc D2i−1 with odd index either is non-separating orencloses the spots p1, p2.


Thus let ℓ ≤ n/2 and assume that the disc D2ℓ−1 is separating and does notenclose the spots; otherwise there is nothing to do. If D2ℓ−2, D2ℓ are contained indistinct components of H−D2ℓ−1 then they are disjoint. In this case we can removeD2ℓ−1 from the path (Di) and obtain a shorter path with the same endpoints. Sincethe path (Di) has minimal length this is impossible.

As a consequence, D2ℓ−2, D2ℓ are contained in the same component V of H −D2ℓ−1. Since D2ℓ−1 does not enclose the spots, the subsurface ∂(H − V ) of ∂H isnot a three holed sphere. Since H has precisely two spots, ∂(H − V ) has at most 3holes and hence the genus of ∂(H − V ) is positive. Thus there is a non-separating

disc D2ℓ−1 contained in H − V . Replace D2ℓ−1 by D2ℓ−1.Replace in this way any disc D2ℓ−1 with an odd index which is separating but

does not enclose p1, p2 by a non-separating disc without modifying the discs D2i

with even index. In a second step, we then replace the discs D2ℓ with even indexwhich are separating but do not enclose the spots in the same way.

We next modify the path (Di) with fixed endpoints and by increasing its lengthby a factor of at most 4 in such a way that in the resulting path (Ei)0≤i≤2u, for eachi the disc E2i encloses the spots p1, p2 and the disc E2i−1 is non-separating. Tothis end note that any two distinct discs Di which enclose the spots p1, p2 intersect.This means that if the disc Di from the above sequence encloses the spots then thediscs Di−1, Di+1 are non-separating. Thus it now suffices to replace any consecutivepair Di, Di+1 of disjoint non-separating discs by a path of length at most four withthe same endpoints whose vertices alternate between non-separating discs and discsenclosing the spots p1, p2.

Let i < n − 1 be such that the discs Di, Di+1 are both non-separating. IfH − (Di ∪Di+1) is connected then there is a disc B which encloses the spots p1, p2and which is disjoint from Di∪Di+1. Such a disc can be obtained by thickening anembedded arc in ∂H−(Di∪Di+1) which connects p1 to p2. Replace the consecutivepair Di, Di+1 by the path Di, B,Di+1 of length three.

If H − (Di ∪Di+1) is disconnected and if the spots p1, p2 are both contained inthe same component of ∂H − (Di ∪Di+1) then we can proceed as in the previousparagraph. Otherwise there is a component of ∂(H−(Di∪Di+1)) which is a surfaceof genus g ≥ 1 with three holes. One of the holes is a spot, the other two holes arethe boundary circles of Di, Di+1. Hence there is a nonseparating disc B which isdisjoint from Di ∪Di+1 and such that H − (Di ∪B) and H − (Di+1 ∪B) are bothconnected. Replace the consecutive pair Di, Di+1 by a path of length 5 of the formDi, A1, B,A2, Di+1 so that consecutive discs in this sequence are disjoint and thatthe discs A1, A2 both enclose the spots p1, p2. This completes the construction ofthe sequence (Ej).

For each i the disc E2i defines a free splitting of Fg+1 of the form Fg ∗ Z wherethe free factor Fg = π1(H, p2) is independent of i. The complementary free factoris generated by the homotopy class βi ∈ π1(H

′, p2) obtained by identifying theendpoints of an embedded arc in ∂H which is disjoint from E2i and connects thetwo spots p1, p2. For each i there is an element wi ∈ Fg so that up to conjugation,the element β2i+2 can be represented as β2iwi.

By construction, the disc E2i+1 in H is nonseparating. The set of all loops inH with basepoint p2 which do not intersect E2i+1 define a free factor Q of Fg ofcorank one. Since E2i+1 is disjoint from E2i and E2i+2, the word wi is containedin the free factor Q.


Since wi ∈ Q, by Theorem 2.4 of [S00] the Whitehead graph of wi has a cutvertex. But this just means that the simple g + 1-length of wi vanishes. Lemma4.9 of [SS10] now shows that the simple g + 1-length of the word w ∈ Fg so thatβ2u = β0w is at most u ≤ 2n. �

Now we are ready to show the third part of Theorem 1 from the introduction.

Proposition 11.3. For every n ≥ 1 the disc graph of H contains a quasiisometri-

cally embedded copy of Rn.

Proof. Let H1 be the handlebody with the spot p2 removed (so that H1 has asingle spot p1). Choose an embedded rose R in ∂H1 = ∂H ∪ {p2} with vertexp2 whose fundamental group π1(R, p2) maps isomorphically onto the fundamentalgroup of H via the inclusion R → H. The image of π1(R, p2) in π1(∂H1, p2) isa free subgroup Q of π1(∂H1, p2). The restriction to Q of the homomorphismπ1(∂H1, p2) → π1(H, p2) is an isomorphism. The g petals e1, . . . , eg of R determinea free basis a1, . . . , ag for Q and for π1(H, p2) = Fg. We may assume that for eachpetal ei there is a disc Ei in H which intersects the petal ei in a single point andhas no other intersections with R.

Consider the Birman exact sequence

(8) 0 → π1(∂H1, p2) → Map(H) → Map(H1) → 0

(we refer to [HH12] for a discussion). For the homotopy class [c] of a loop c in ∂H1

based at p2, the point pushing map ψ[c] of [c] is the element of the handlebody groupMap(H) which can be realized by a diffeomorphism fixing a small neighborhood of cin H pointwise and pushing the point p2 along the closed curve c. The identificationof the fibre of the Birman exact sequence with π1(H1, p2) is via point pushing maps.The map ψ[c] acts on π1(H1, p2) by conjugation with [c].

As in [SS10] we consider for t ≥ 1 the element

qt = at+11 . . . at+1

g at+11 at+1

2 at+11 ∈ Q.

Let D be a disc in H enclosing the spots p1, p2 which intersects the rose R in theboundary of a contractible neighborhood of the vertex p2 and which is disjoint fromthe discs Ei. We claim that for every t ≥ 1 the image of D under the point pushingmap by qt has distance at most 10 to D in the disc graph DG.

Consider first the case that the genus g of H is at least g ≥ 4. Then qt = uvwhere u = at+1

1 . . . at+1g−1 and v = at+1

g at+11 at+1

2 at+11 . The word u does not contain

the letter ag, and the word v does not contain the letter ag−1 since g−1 ≥ 3. Hencethe image of D under the point pushing map ψu defined by u is disjoint from thedisc Eg. Similarly, the image of the disc ψuD under the point pushing map ψuvu−1

defined by uvu−1 is disjoint from the disc ψuEg−1. Thus we have

dD(D,ψuv(D)) ≤ 4.

If g = 3 then write qt = uvw where u = at+11 at+1

2 , where v = at+13 at+1

1 andw = at+1

2 at+11 . The argument in the previous paragraph then shows that the

distance between D and ψqtD is at most 6. In the case g = 2 we write similarlyqt = uvwxz where u = at1 = w = z and v = at2 = x and obtain a distance bound ofat most 10.

Using the above notations, we follow [SS10] and define for each n ≥ 1 a mapΛ of the integral lattice Z

n into DG as the map which takes (k1, . . . , kn) ∈ Zn


to the disc which is the image of D under the point pushing map along the loopqk1

1 qk2

2 . . . qknn ∈ Q based at p2.

Let dD be the distance in the disc graph. We claim that

dD(Λ(k1, . . . , kn),Λ(ℓ1, . . . , ℓn)) ≤ 10

n∑

i=1

|ki − ℓi|.

We follow p.20 of [SS10]. Recall that the disc D defines a disc in the handlebodyH ′ corresponding to a free splitting Fg+1 = F ∗

g Z. The group Q acts by conjugationon such free splittings preserving the factor Fg. Now a separating disc in the handlebody H ′ corresponds to a conjugacy class of a free splitting of the fundamentalgroup Fg+1 of H ′. Therefore the action of Q ∼ π1(H) = Fg < Fg+1 on Fg+1 byconjugaction preserves the discs in H enclosing the spots p1, p2.

Let a be the generator of the complementary cyclic factor Z determined by thedisc D. Then the disc Λ(k1, . . . , kn) defines the splitting Fg∗ < aqk1

1 qk2

2 · · · qknn >.

Denoting by [F ∗g < b > the disc inH which encloses the spots p1, p2 and corresponds

to the free splitting where the complementary free factor is generated by b = aq forsome q ∈ Q we have

Λ(k1, k2, . . . , kn) = [F ∗g < aqk1

1 qk2

2 · · · qkn

n >]

= [F ∗g < qk1

1 · · · qkn

n a >]

→ [F ∗g < qk1

1 · · · qkn

n aqℓ1−k1

1 >]

= [F ∗g < qk2

2 · · · qkn

n aqℓ11 >]

→ [F ∗g < qk2

2 · · · qkn

n aqℓ11 qℓ2−k2

2 >]

= . . .

= [F ∗g < aqℓ11 q

ℓ22 · · · qℓnn >]

= Λ(ℓ1, . . . , ℓn).

At each arrow, we append the element qℓj−kj

j (j ∈ {1, . . . , n}), and by the above

discussion, this requires an edge path in DG of length at most 10|ℓj−kj | completingthe proof of the claim.

On the other hand, Lemma 4.15 of [SS10] and the estimate in the proof ofTheorem 5.4 of [SS10] show that there is a number c > 0 such that

n∑

i=1

|ki − ℓi| ≤ c|qk1

1 qk2

2 · · · qkn

n |simpleg+1 .

By Lemma 11.2 this implies that the distance in DG between D and its imageunder the point pushing by qk1

1 qk2

2 · · · qknn is bounded from above and below by a

fixed positive multiple of∑n

i=1 |ki − ℓi|. Together this shows the proposition. �

We now show the fifth part of Theorem 1 from the introduction.

Proposition 11.4. Let H be a handlebody of genus g ≥ 1 with m ≥ g + 3 spots.

Then the disc graph of H is hyperbolic.

Proof. Let H be as in the proposition. Let X ⊂ H be a thick subsurface. Weclaim that X contains at least three spots. Namely, let k ≥ 0 be the number ofcomponents of ∂H −X. Since the boundary of every disc intersects X, each suchcomponent contains at most one spot, and it has at least one boundary circle. Thus


successively attaching the components to X increases the genus of the resultingsurface by at least one in each step. This shows that the maximal number of suchcomponents equals g and hence X contains at least m− g ≥ 3 spots.

As a consequence, a thick subsurface X of ∂H is neither a four holed sphere nor aone-holed torus, moreover the electrified disc graph EDG(X) is quasi-isometricallyembedded in the curve graph of X. The construction in Section 7 now shows thatindeed the disc graph of H is hyperbolic. �

The proof of Proposition 11.4 is also valid for the disc graph of a handlebodyof genus 1 with three spots (whose only thick subsurface is a five holed spherecontaining three punctures). We expect that disc graphs for handlebodies of genusg with m ≥ 3 spots and where m is small compared to g are not hyperbolic. Weexpect that this can be deduced by a careful analysis of thick subsurfaces of ∂H.

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MATH. INSTITUT DER UNIVERSITAT BONNENDENICHER ALLEE 6053115 BONNGERMANY

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contents · the disc graph is hyperbolic for m = 0 and if m− g ≥ 3. if m= 1 and g≥ 2 then the...

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