Spotted disk and sphere graphs I

New York Journal of Mathematics

New York J. Math.27(2021) 881–902.

Spotted disk and sphere graphs I

Ursula Hamenstädt

Abstract. The disk graph of a handlebody𝐻of genus𝑔 ≥ 2with𝑚 ≥ 0 marked points on the boundary is the graph whose vertices are isotopy classes of disks disjoint from the marked points and where two vertices are connected by an edge of length one if they can be realized disjointly. We show that for 𝑚 = 1the disk graph contains quasi-isometrically embedded copies ofℝ². The same holds true for sphere graphs of the doubled handlebody with one marked points provided that𝑔is even.

Contents

1. Introduction 881

2. Once spotted handlebodies 883

3. Once spotted doubled handlebodies 896

References 901

1. Introduction

Thecurve graph𝒞𝒢 of an oriented surface 𝑆of genus𝑔 ≥ 0with𝑚 ≥ 0 punctures and3𝑔 − 3 + 𝑚 ≥ 2is the graph whose vertices are isotopy classes of essential (that is, non-contractible and not homotopic into a puncture) simple closed curves on𝑆. Two such curves are connected by an edge of length one if and only if they can be realized disjointly. The curve graph is a locally infinite hyperbolic geodesic metric space of infinite diameter [12].

A handlebody of genus𝑔 ≥ 1is a compact three-dimensional manifold𝐻 which can be realized as a closed regular neighborhood inℝ³of an embedded bouquet of 𝑔circles. Its boundary 𝜕𝐻is an oriented surface of genus𝑔. We allow that 𝜕𝐻is equipped with𝑚 ≥ 0marked points (punctures) which we call spots in the sequel. The groupMap(𝐻)of all isotopy classes of orienta- tion preserving homeomorphisms of𝐻which fix each of the spots is called the handlebody groupof𝐻. The restriction of an element ofMap(𝐻)to the bound- ary𝜕𝐻defines an embedding ofMap(𝐻)into the mapping class group of𝜕𝐻, viewed as a surface with punctures [16,17].

Received August 28, 2020.

2010Mathematics Subject Classification. 57M99.

Key words and phrases. Disk graphs, handlebodies with spots, embedded flats, sphere graphs.

Partially supported by ERC Grant “Moduli”.

ISSN 1076-9803/2021

881

Anessential diskin𝐻is a properly embedded disk(𝐷, 𝜕𝐷) ⊂ (𝐻, 𝜕𝐻)whose boundary𝜕𝐷is an essential simple closed curve in𝜕𝐻, viewed as a surface with punctures. An isotopy of such a disk is supposed to consist of such disks.

Thedisk graph 𝒟𝒢of 𝐻 is the graph whose vertices are isotopy classes of essential disks in𝐻. Two such disks are connected by an edge of length one if and only if they can be realized disjointly.

In [14,4,3] the following is shown.

Theorem 1. The disk graph of a handlebody of genus𝑔 ≥ 2without spots is hyperbolic.

The main goal of this article is to show that in contrast to the case of curve graphs, Theorem1is not true if we allow spots on the boundary.

Theorem 2. Let𝐻be a handlebody of genus𝑔 ≥ 2with one spot. Then the disk graph of𝐻contains quasi-isometrically embedded copies ofℝ². In particular, it is not hyperbolic.

Theorem2implies that disk graphs can not be used effectively to obtain a geometric understanding of the handlebody group Map(𝐻)of a handlebody 𝐻 of genus𝑔 ≥ 3 paralleling the program developed by Masur and Minsky for the mapping class group [13]. The analogue of the strategy of Masur and Minsky would consist of cutting a handlebody open along an embedded disk which yields a (perhaps disconnected) handlebody with one or two spots on the boundary and studying disk graphs in the cut open handlebody.

A systematic study of groups to which the strategy laid out by Masur and Minsky can be applied was recently initiated by Behrstock, Hagen and Sisto [1], and these groups are calledhierarchically hyperbolic. Such groups have quadratic Dehn functions, but for𝑔 ≥ 3the Dehn function ofMap(𝐻)is expo- nential [7]. HenceMap(𝐻)can not be hierarchically hyperbolic. However, the geometric mechanism behind an exponential Dehn function forMap(𝐻)is not detected by the failure of being hierarchically hyperbolic in an obvious way.

Theorem2 has an analogue for geometric graphs related to the outer au- tomorphism group Out(𝐹𝑔)of the free group on 𝑔 ≥ 2generators. Namely, doubling the handlebody𝐻yields a connected sum𝑀 = ♯𝑔𝑆²× 𝑆¹of𝑔copies of 𝑆² × 𝑆¹ with𝑚 marked points. A deep result of Laudenbach [11] shows thatOut(𝐹𝑛)is a cofinite quotient of the group of isotopy classes of homeomor- phisms of𝑀.

A doubled disk is an embedded essential sphere in 𝑀, which is a sphere which is not homotopically trivial or homotopic into a marked point. Thesphere graphof𝑀is the graph whose vertices are isotopy classes of embedded essential spheres in𝑀and where two such spheres are connected by an edge of length one if and only if they can be realized disjointly. As before, an isotopy of spheres is required to be disjoint from the marked points. The sphere graph of a doubled handlebody without marked points is hyperbolic [9].

Paralleling the result in Theorem2we have

Theorem 3. Let𝑔 ≥ 2and let𝑀be a doubled handlebody of genus𝑔with one marked point. If𝑔is even then the sphere graph of𝑀contains quasi-isometrically embedded copies ofℝ². In particular, it is not hyperbolic.

The argument in the proof of Theorem 3uses Theorem 2 and a result in [6] which relates the sphere graph in a connected sum♯_𝑔𝑆²× 𝑆¹for𝑔even to the arc graph of an oriented surface of genus𝑔∕2with connected non-empty boundary. A corresponding result for odd𝑔and a non-orientable surface with a single boundary component would yield Theorem3for odd𝑔 ≥ 3, but at the moment, such a result is not available.

As in the case of disk graphs, this indicates that sphere graphs are of limited use for obtaining an effective geometric understanding ofOut(𝐹𝑔). Note that as in the case of the handlebody group, for𝑔 ≥ 3the Dehn function ofOut(𝐹𝑔) is exponential [2,8].

In a sequel to this article [5], it is shown that the disk graph of a handle- body of genus𝑔 ≥ 2with two spots contains quasi-isometrically embeddedℝ², and the sphere graph of a doubled handlebody with two spots contains quasi- isometrically embeddedℝ^𝑛for every𝑛 ≥ 2. We conjecture that the disk graph of a handlebody𝐻 with𝑚 ≥ 3 spots is quasi-isometrically embedded in the curve graph of𝜕𝐻.

Acknowledgement:I am very grateful to the anonymous referee of this paper for numerous and detailed comments which helped to improved the exposition.

2. Once spotted handlebodies

The goal of this section is to construct quasi-isometrically embedded copies ofℝ²in the disk graph of a handlebody with a single spot.

Thus let𝐻be a handlebody of genus𝑔 ≥ 2with a single spot. Let𝐻0be the handlebody obtained from𝐻by removing the spot and let

Φ ∶ 𝐻 → 𝐻0

be the spot removal map. The image underΦof an essential (that is, not con- tractible or homotopic into the spot) diskbounding simple closed curve in𝜕𝐻 is an essential diskbounding simple closed curve in𝜕𝐻₀.

The handlebody 𝐻0 without spots can be realized as a fiber bundle over a surface𝐹 with non-empty connected boundary𝜕𝐹 whose fiber is the closed interval𝐼 = [0, 1]. Such a fiber bundle is called an𝐼-bundle. We summarize from Section 3 of [3] (p.381-383) some properties of such𝐼-bundles used in the sequel.

There are two different ways a handlebody𝐻0 of genus𝑔 can arise as an 𝐼-bundle over a surface𝐹 with connected boundary𝜕𝐹. In the first case, the surface𝐹 is orientable. Then the genus 𝑔 of 𝐻0 is even and the 𝐼-bundle is trivial. The genus of𝐹equals𝑔∕2, and the boundary𝜕𝐹of𝐹defines an isotopy class of a separating simple closed curve𝑐on𝜕𝐻0which decomposes𝜕𝐻0into two surfaces of genus𝑔∕2, with a single boundary component.

If the surface 𝐹 is non-orientable, then𝐹 is the orientable 𝐼-bundle over the connected sum of𝑔projective planes with a disk, and the𝐼-bundle is non- trivial. The boundary𝜕𝐹 defines the isotopy class of a non-separating simple closed curve𝑐in𝜕𝐻0. The complement of an open annulus about𝑐in𝜕𝐻0is the orientation cover of𝐹.

Following Definition 3.3 of [3], define an𝐼-bundle generatorfor𝐻₀to be an essential simple closed curve𝑐 ⊂ 𝜕𝐻0so that𝐻0can be realized as an𝐼-bundle over a compact surface𝐹with connected boundary𝜕𝐹and such that𝑐is freely homotopic to𝜕𝐹 ⊂ 𝜕𝐻0. The surface𝐹is then called thebaseof the𝐼-bundle.

An𝐼-bundle generator𝑐in𝜕𝐻0is diskbusting, which means that it has an essential intersection with every disk (see [14,4]). Namely, the base𝐹of the𝐼- bundle is a deformation retract of𝐻0. Thus if𝛾is any essential closed curve on

𝜕𝐻₀which does not intersect𝑐, then𝛾projects to an essential closed curve on𝐹.

Such a curve is not nullhomotopic in𝐻0and hence it can not be diskbounding.

As established in [14, 3,4], 𝐼-bundle generators play a special role for the geometry of the disk graph of𝐻0. Our goal is to take advantage of this fact for the understanding of the geometry of the handlebody with one spot. To this end define thearc graph𝒜(𝑋)of a compact surface 𝑋 of genus𝑛 ≥ 1with connected boundary𝜕𝑋 to be the graph whose vertices are isotopy classes of embedded essential arcs in𝑋with endpoints on the boundary, and isotopies are allowed to move the endpoints of an arc along𝜕𝑋. Two such arcs are connected by an edge of length one if and only if they can be realized disjointly. The arc graph𝒜(𝑋)of𝑋is hyperbolic [14].

For an 𝐼-bundle generator 𝑐in 𝐻0 let ℛ𝒟(𝑐)be the complete subgraph of the disk graph𝒟𝒢0of𝐻0consisting of disks which intersect𝑐in precisely two points. The boundary of each such disk is an𝐼-bundle over an arc in the base 𝐹of the𝐼-bundle corresponding to𝑐(see the discussion preceding Lemma 4.2 of [3]). Namely, the𝐼-bundle over an arc in𝐹with endpoints on𝜕𝐹is an em- bedded disk in𝐻0. On the other hand, the boundary of a disk in𝐻0defines the trivial element in the fundamental group of𝐻0. Thus if𝛽 is a diskbounding simple closed curve in𝜕𝐻0which intersects𝑐in precisely two points, then the homotopy classes relative to𝑐of the two components of𝛽−𝑐are exchanged un- der the orientation reversing involution of𝐻0which exchanges the endpoints of a fiber in the𝐼-bundle. As𝛽has two essential intersections with𝑐, this then implies that up to homotopy, the two components of𝛽 − 𝑐trace through the two different preimages of the same points in𝐹.

Now two disks intersecting𝑐in precisely two points are disjoint if and only if the corresponding arcs in𝐹are disjoint and hence we have

Lemma 2.1. The graphℛ𝒟(𝑐)is isometric to the arc graph𝒜(𝐹)of𝐹.

The arc graph of a surface𝐹 with non-empty boundary 𝜕𝐹 is a complete subgraph of another geometrically defined graph, the so-calledarc and curve graph. Its vertices are essential simple closed curves in𝐹or arcs with endpoints on𝜕𝐹, and two such arcs or curves are connected by an edge of length one if they can be realized disjointly. The arc and curve graph contains the curve

graph of𝐹as a complete subgraph, and the inclusion of the curve graph into the arc and curve graph is known to be a quasi-isometry unless𝐹 is a sphere with at most three holes or a projective plane with at most three holes (Lemma 4.1 of [3]). Recall that a map𝜑 ∶ 𝑋 → 𝑌be tween two metric spaces𝑋, 𝑌is an 𝐿-quasi-isometric embeddingif for all𝑥, 𝑦 ∈ 𝑋we have

𝑑(𝑥, 𝑦)∕𝐿 − 𝐿 ≤ 𝑑(𝜑(𝑥), 𝜑(𝑦)) ≤ 𝐿𝑑(𝑥, 𝑦) + 𝐿,

and it is called an𝐿-quasi-isometryif moreover its image is𝐿-dense, that is, for every𝑦 ∈ 𝑌there exists some𝑥 ∈ 𝑋such that𝑑(𝜑(𝑥), 𝑦) ≤ 𝐿.

The arc graph𝒜(𝐹)of𝐹is1-dense in the arc and curve graph of𝐹, but the inclusion of𝒜(𝐹)into the arc and curve graph of𝐹 is a quasi-isometry only if the genus of𝑋equals one [14] (see also [3]).

Acoarse 𝐿-Lipschitz retraction of a metric space (𝑋, 𝑑) onto a subspace𝑌 is a coarse 𝐿-Lipschitz mapΨ ∶ 𝑋 → 𝑌 (this means that 𝑑(Ψ(𝑥), Ψ(𝑦)) ≤ 𝐿𝑑(𝑥, 𝑦) + 𝐿 for some𝐿 ≥ 1 and all𝑥, 𝑦) with the additional property that there exists a number𝐶 > 0with𝑑(Ψ(𝑦), 𝑦) ≤ 𝐶for all𝑦 ∈ 𝑌. If𝑋 is a geo- desic metric space then the image𝑌of a coarse Lipschitz retraction is acoarsely quasi-convexsubspace of𝑋, that is, any two points in𝑌can be connected by a uniform quasi-geodesic (for the metric of𝑋) which is contained in a uniformly bounded neighborhood of𝑌.

Lemma 2.2. Let𝑐be an𝐼-bundle generator of the handlebody𝐻0. There exists a coarse Lipschitz retractionΘ0 ∶ 𝒟𝒢₀ → ℛ𝒟(𝑐)whose restriction toℛ𝒟(𝑐)is the identity.

Proof. If𝑐is aseparating𝐼-bundle generator, then the base of the𝐼-bundle can be identified with a component𝐹of𝜕𝐻0− 𝑐. Note that there two choices for the surface𝐹. One of these two choices will be fixed throughout this proof.

Since the boundary𝜕𝐷of a disk𝐷 is an embedded simple closed curve in

𝜕𝐻0and as𝑐 is diskbusting, the intersection𝜕𝐷 ∩ 𝐹 consists of a non-empty collection of pairwise disjoint simple arcs with endpoints on𝜕𝐹. The map

Υ₀∶ 𝒟𝒢₀→ 𝒜(𝐹)

which associates to a disk𝐷 a component of𝜕𝐷 ∩ 𝐹is coarsely well defined:

Although it depends on choices, any other choiceΥ₀^′ maps a disk𝐷 to an arc disjoint fromΥ0(𝐷). If we denote by𝑄 ∶ 𝒜(𝐹) → ℛ𝒟(𝑐)the map which asso- ciates to an arc𝛼in𝐹the𝐼-bundle over𝛼, then the disks𝑄(Υ0(𝐷)), 𝑄(Υ₀^′(𝐷)) are disjoint as well.

Furthermore, if𝐷, 𝐷^′are disjoint disks then the arcsΥ0(𝐷), Υ0(𝐷^′)are dis- joint and hence𝑑𝒟𝒢₀(𝑄Υ0(𝐷), 𝑄Υ0(𝐷^′)) ≤ 1. This shows that𝑄◦Υ0is coarsely one-Lipschitz. As a disk 𝐷 ∈ ℛ𝒟(𝑐) intersects 𝐹 in a single arc, we have 𝑄Υ0(𝐷) = 𝐷. Thus the map𝑄◦Υ0 is indeed a coarse one-Lipschitz retraction which completes the proof of the lemma in the case that𝑐is separating. Note however that the relation between the two Lipschitz retractions constructed in this way from the two distinct components of𝜕𝐻0− 𝑐is unclear.

The above argument does not extend to non-separating𝐼-bundle generators in any straightforward way. Namely, if𝑐is a non-separating𝐼-bundle generator, then although up to homotopy, a disk which intersects𝑐in precisely two points is invariant under the natural orientation reversing involutionΩof the corre- sponding𝐼-bundle which exchanges the two endpoints of a fiber, the projection to𝐹of the boundary of some other disk may have self-intersections, and hence there is no obvious projection of𝒟𝒢₀ontoℛ𝒟(𝑐)as in the case of a separating 𝐼-bundle generator.

Our strategy is to establish instead that the inclusion ℛ𝒟(𝑐) → 𝒟𝒢0 is a quasi-isometric embedding. Namely, if this holds true then as𝒟𝒢₀is hyper- bolic, the subspaceℛ𝒟(𝑐)isquasi-convex, that is, there exists a constant𝐶 > 0 such that any geodesic in𝒟𝒢₀connecting two points inℛ𝒟(𝑐)is contained in the𝐶-neighborhood ofℛ𝒟(𝑐). Then a (coarsely well defined) shortest distance projection𝒟𝒢₀ → ℛ𝒟(𝑐)is a coarsely Lipschitz retraction by hyperbolicity.

That the inclusionℛ𝒟(𝑐) → 𝒟𝒢₀ is indeed a quasi-isometric embedding follows from Theorem 10.1 of [14] (which can only be used indirectly as the

“holes” are not precisely specified) and, more specifically, Corollary 4.6 and Corollary 4.7 of [3]. These formulas establish that the distance in the disk graph between two disks𝐷, 𝐸which intersect a given𝐼-bundle generator𝑐with base 𝐹in precisely two points equals the distance in𝒜(𝐹)between the projections of 𝜕𝐷 and𝜕𝐸 to𝐹 up to a uniform constant not depending on𝑐. In view of Lemma2.1, this is what we want to show.

The details are as follows. Construct from the disk graph𝒟𝒢0of𝐻0another graph ℰ𝒟𝒢₀ with the same vertex set by adding additional edges as follows.

If𝐷, 𝐸 are two disks in𝐻0, and if up to homotopy,𝐷, 𝐸 are disjoint from an essentialsimple closed curve in𝜕𝐻0, that is, a simple closed curve which is not homotopic to zero, then we connect𝐷, 𝐸 by an edge inℰ𝒟𝒢₀. This graph is called theelectrified disk graphof𝐻0[3].

Let us denote byℰℛ𝒟(𝑐)the subgraph ofℰ𝒟𝒢₀whose vertex set consists of all disks which intersect the non-separating𝐼-bundle generator𝑐in precisely two points. Lemma 4.2 and Lemma 4.1 of [3] show that the map which asso- ciates to an arc in the non-orientable surface𝐹 the 𝐼-bundle over𝐹 is a uni- form quasi-isometry between the arc and curve graph of𝐹andℰℛ𝒟(𝑐). Fur- thermore, by Corollary 4.6 of [3], the inclusionℰ𝒟ℛ(𝑐) → ℰ𝒟𝒢₀is a uniform quasi-isometric embedding. Here uniform means with constants not depend- ing on𝑐.

Let𝜁 ∶ [0, 𝑚] → ℰℛ𝒟(𝑐)be a geodesic. Then𝜁is a uniform quasi-geodesic inℰ𝒟𝒢₀. Define theenlargement𝜁2of𝜁to be the edge path inℰℛ𝒟(𝑐)obtained from𝜁by replacing each edge𝜁[𝑘, 𝑘 + 1]by an edge path𝜁2[𝑖𝑘, 𝑖𝑘+1]with the same endpoints as follows.

If the disks𝜁(𝑘), 𝜁(𝑘 +1)are disjoint, then the edge path𝜁2[𝑖𝑘, 𝑖𝑘+1]just con- sists of the edge connecting these two points. Otherwise𝜁(𝑘), 𝜁(𝑘+1)intersect, but they are disjoint from an essential simple closed curve in𝜕𝐻0. As each disk 𝜁(𝑗)is an𝐼-bundles over an arc𝛼(𝑗)in the surface𝐹, this means that there is

an essential simple closed curve𝛽 ⊂ 𝐹 disjoint from both𝜁(𝑘), 𝜁(𝑘 + 1). We refer to Lemma 4.2 of [3] for a detailed explanation.

An essential subsurface of𝐹 containing𝜕𝐹is a component of𝐹 − 𝜉where 𝜉 is a collection of pairwise disjoint mutually not freely homotopic essential non-boundary parallel simple closed curves in𝐹. If𝜁(𝑘), 𝜁(𝑘 + 1)are disjoint from an essential simple closed curve in𝐹, then the subsurface ̂𝑋 of𝐹 filled by𝜁(𝑘), 𝜁(𝑘 + 1), defined to be the intersection of all essential subsurfaces of𝐹 which contain𝜁(𝑘),𝜁(𝑘 + 1), 𝜕𝐹, is not all of𝐹.

Let𝑋 ⊂ 𝜕𝐻0be the preimage of ̂𝑋in𝜕𝐻0. Then𝑋is an essential subsurface of𝜕𝐻0which contains the boundaries of the disks𝜁(𝑘), 𝜁(𝑘 + 1)and is invari- ant under the orientation reversing involutionΩ. No component of its bound- ary is diskbounding, and it contains𝑐as an𝐼-bundle generator. Furthermore, no essential simple closed curve in𝑋(here essential means non-peripheral) is disjoint from all disks with boundary in𝑋. This follows from the fact that no essential simple closed curve in𝑋 is disjoint from both𝜁(𝑘) and𝜁(𝑘 + 1) as 𝜁(𝑘), 𝜁(𝑘 + 1)are invariant underΩand their projection to𝐹fill the projection

̂𝑋of𝑋. A subsurface𝑋of𝜕𝐻₀with these properties is calledthickin [3].

The complete subgraphℰ𝒟𝒢(𝑋)of ℰ𝒟𝒢₀ whose vertex set is the set of all disks with boundary in𝑋 is an electrified disk graph for𝑋. By Corollary 4.6 of [3], its subgraphℰℛ𝒟(𝑐, 𝑋) of all disks which intersect𝑐in precisely two points is uniformly quasi-isometrically embedded in the electrified disk graph of𝑋. Note that Corollary 4.6 of [3] only states that this graph is uniformly quasi- convex, however Corollary 2.8 of [3] shows that indeed, the inclusion of each of these graphs into the electrified disk graph of𝑋is a uniform quasi-isometric embedding. Furthermore, by Lemma 4.2 of [3], the graphℰℛ𝒟(𝑐, 𝑋)is 4-quasi- isometric to the arc and curve graph of ̂𝑋where we require arcs to have end- points on the distinguished boundary component𝑐of ̂𝑋.

If ̂𝑋is the complement of an orientation reversing simple closed curve dis- joint from𝑐, then 𝑋 is the complement in𝜕𝐻0 of an essential simple closed curve. In this case we define𝜁₂[𝑖_𝑘, 𝑖_𝑘+1]to be the path inℰℛ𝒟(𝑐, 𝑋)connecting 𝜁(𝑘)to𝜁(𝑘 +1)which consists of𝐼-bundles over arcs in ̂𝑋defined by a geodesic in the arc and curve graph of ̂𝑋. That is, from a geodesic in the arc and curve graph of ̂𝑋we construct first an edge path of at most twice the length with the property that among two consecutive vertices, at least one is an arc, and then we view this edge path as an edge path in the graphℰℛ𝒟(𝑐, 𝑋). By Corollary 4.6 of [3],𝜁2[𝑖𝑘, 𝑖𝑘+1]is a uniform quasi-geodesic inℰ𝒟𝒢(𝑋). If the complement of ̂𝑋 contains an orientation preserving simple closed curve which does not bound a Möbius band, then the complement of𝑋in𝜕𝐻0contains at least two disjoint simple closed curves and we define𝜁2[𝑖𝑘, 𝑖𝑘+1]to be the edge between𝜁(𝑘)and 𝜁(𝑘 + 1).

The resulting edge path𝜁2inℰℛ𝒟(𝑐)has the property that two consecutive vertices, which are disks𝐷, 𝐸intersecting𝑐in two points, are either disjoint, or their boundaries lie in the same proper thickΩ-invariant subsurface𝑋of𝜕𝐻0

containing𝑐as an𝐼-bundle generator. Moreover,𝐷, 𝐸are connected by an edge

in the graphℰℛ𝒟(𝑐, 𝑋). In particular, the complement of the subsurface of𝜕𝐻0

filled by𝐷, 𝐸contains at least two disjoint essential simple closed curves.

Letℰ𝒟𝒢(2, 𝜕𝐻0)be the graph whose vertex set is the set of disks and where two disks are connected by an edge if either they are disjoint, or if they are dis- joint from a multicurve consisting of at least two non-homotopic components.

By Theorem 5.5 of [3], the graphℰ𝒟𝒢(2, 𝜕𝐻₀)is hyperbolic, and it is an electri- fication of the disk graph of𝐻0. This means that it has the same vertex set as the disk graph of𝐻0, and it is obtained from this disk graph by adding edges.

Theorem 5.5 of [3] also shows that the path𝜁2is a uniform quasi-geodesic in ℰ𝒟𝒢(2, 𝜕𝐻0). Namely, following Section 5 of [3], define a simple closed curve 𝛾 ⊂ 𝜕𝐻0 to beadmissibleif𝛾 is neither diskbounding nor diskbusting. Each such curve defines a thick subsurface of𝜕𝐻0. Writeℰ𝒟𝒢(𝜕𝐻0− 𝛾)to denote the electrified disk graph of𝜕𝐻₀− 𝛾and letℱ(𝛾)to be the complete subgraph ofℰ𝒟𝒢(2, 𝜕𝐻0)whose vertex set consists of all disks which are disjoint from𝛾.

A disk𝐷 ⊂ ℱ(𝛾)defines a vertex inℰ𝒟𝒢(𝜕𝐻0− 𝛾).

Following Section 2 of [3], define theenlargementof a uniform quasi-geodesic 𝜂 ∶ [0, 𝑛] → ℰ𝒟𝒢₀with no backtracking as follows. Assume that𝜂(𝑗), 𝜂(𝑗 + 1) ∈ ℰ𝒟𝒢(𝜕𝐻0− 𝛾)for some admissible simple closed curve𝛾and some𝑗 < 𝑛;

then replace the edge𝜂[𝑗, 𝑗 + 1]by a geodesic (or uniform quasi-geodesic) in ℰ𝒟𝒢(𝜕𝐻₀− 𝛾). Note that if𝜂(𝑗), 𝜂(𝑗 + 1) are disjoint from an essential sim- ple closed curve in𝜕𝐻0 − 𝛾, then there is an edge between𝜂(𝑗), 𝜂(𝑗 + 1) in ℰ𝒟𝒢(𝜕𝐻0− 𝛾). Theorem 5.5 of [3] states that enlargements of uniform quasi- geodesics inℰ𝒟𝒢0are uniform quasi-geodesics inℰ𝒟𝒢(2, 𝜕𝐻0).

Now the above construction takes as input a geodesic in ℛ𝒟(𝑐)and asso- ciates to it an enlargement, chosen in such a way that this enlargement con- sists of disks whose boundaries intersect𝑐in precisely two points. Using once more Theorem 5.5 of [3], this shows that inclusion defines a quasi-isometric embedding of the complete subgraph ofℰ𝒟𝒢(2, 𝜕𝐻0)of disks which intersect 𝑐in precisely two points into the graphℰ𝒟𝒢(2, 𝜕𝐻0).

This construction can be iterated. In the next step, we modify the path𝜁2to a path𝜁3 by replacing suitable edges by edge paths as follows. Consider two consecutive vertices𝜁2(𝑘), 𝜁2(𝑘 + 1)of 𝜁2. These are disks which intersect 𝑐 in precisely two points. If they are not disjoint, then there exists an essential simple closed curve𝛾 ⊂ 𝐹 which is disjoint from both 𝜁2(𝑘), 𝜁2(𝑘 + 1). If𝛾 is orientation preserving and does not bound a Möbious band, then𝛾has two disjoint preimages𝛾1, 𝛾2in𝜕𝐻0, and the complement of these preimages is an Ω-invariant thick subsurface𝑋 of𝜕𝐻0 containing𝑐as an𝐼-bundle generator.

Replace𝜁2[𝑘, 𝑘 + 1]by a geodesic inℰ𝒟𝒢(𝑋)with the same endpoints. This geodesic can be chosen to be the preimage of a geodesic in the arc and curve graph of𝐹 −𝛾. Proceed in the same way if the complement of the subsurface of 𝐹filled by𝜁(𝑘) ∩ 𝐹, 𝜁(𝑘 + 1) ∩ 𝐹only contains orientation reversing primitive simple closed curves.

In finitely many steps we construct in this way a path in the graphℛ𝒟(𝑐) connecting the endpoints of𝜁. Its length roughly equals the sum of the subsur- face projections of the projection of its endpoints to𝐹, where the sum is over all essential subsurfaces of𝐹containing the boundary𝜕𝐹. In particular, by the distance formula in Corollary 6.3 of [3], its length does not exceed a uniform multiple of the distance in𝒟𝒢₀between its endpoints. The statement also fol- lows as by the main result of [3], the so-called hierarchy paths, constructed from a geodesic inℰ𝒟𝒢₀in the above inductive fashion, are uniform quasi-geodesics in the disk graph.

As a consequence, taking the𝐼-bundle over an arc in𝐹 defines an isome- try between the arc graph of𝐹 and the graphℛ𝒟(𝑐), and this graph is quasi- isometrically embedded in𝒟𝒢₀. This is what we wanted to show.

Our goal is to use𝐼-bundle generators in𝜕𝐻0to construct quasi-isometrically embedded euclidean planes in the disk graph of𝐻. In analogy to [4], we define an𝐼-bundle generatorfor the spotted handlebody𝐻to be a simple closed curve in𝜕𝐻whose image under the spot forgetful mapΦis an𝐼-bundle generator in

𝜕𝐻0.

Let(𝑐1, 𝑐2) ⊂ 𝜕𝐻 be a pair of non-isotopic disjoint 𝐼-bundle generators so that𝜕𝐻 − {𝑐₁∪ 𝑐₂}has a connected component which is an annulus containing the spot in its interior. Then up to isotopy,Φ(𝑐1) = Φ(𝑐2) = 𝑐for an𝐼-bundle generator𝑐in𝐻0.

The following construction is due to Kra; we refer to [10] for details and for some applications. For its formulation, for a pair (𝑐1, 𝑐2)of disjoint 𝐼-bundle generators on𝜕𝐻as in the previous paragraph letℛ𝒟(𝑐1, 𝑐2)be the complete subgraph of the disk graph𝒟𝒢of𝐻whose vertex set consists of all disks which intersect each of the curves 𝑐₁, 𝑐₂ in precisely two points. Note that if 𝐷 ∈ ℛ𝒟(𝑐1, 𝑐2)then the image of𝐷 under the spot removing mapΦis contained inℛ𝒟(𝑐)where𝑐 = Φ(𝑐𝑖).

In the next lemma we denote by abuse of notation the map𝒟𝒢 → 𝒟𝒢0in- duced by the spot forgetful mapΦagain byΦ. Furthermore, for the remainder of this section we represent a disk by its boundary, that is, we view the disk graph as the complete subgraph of the curve graph of𝜕𝐻whose vertex set is the set of diskbounding curves.

Lemma 2.3. Let(𝑐1, 𝑐2)be a pair of I-bundle generators bounding a punctured annulus and let 𝑐 = Φ(𝑐1) = Φ(𝑐2). There exists a simplicial embedding𝜄 ∶ 𝒟𝒢0 → 𝒟𝒢with the following properties.

(1) Φ◦𝜄is the identity.

(2) 𝜄mapsℛ𝒟(𝑐)intoℛ𝒟(𝑐1, 𝑐2).

Proof. Note first that there is a natural orientation reversing involution𝜌0 of

𝜕𝐻0which exchanges the endpoints of the fibres of the interval bundle over the base𝐹. This involution fixes𝑐and preserves up to isotopy each diskbounding simple closed curve which intersects𝑐in precisely two points. We refer to the discussion before Lemma2.1for this fact.

Choose a hyperbolic metric𝑔0on𝜕𝐻0which is invariant under𝜌0and let̂𝑐be the geodesic representative of𝑐. This makes sense since the geodesic represen- tative of a simple closed curve is simple. Choose a point𝑝 ∈ ̂𝑐not contained in any diskbounding simple closed geodesic; this is possible since each diskbound- ing simple closed geodesic intersects ̂𝑐transversely in finitely many points and hence the set of all points of ̂𝑐contained in a diskbounding closed geodesic is countable. View𝑝as a marked point on𝜕𝐻0; then the geodesic representative of a diskbounding curve𝛼in𝜕𝐻0is a diskbounding curve𝜄(𝛼)in𝜕𝐻0−{𝑝}. Via identification of a disk with its boundary, this construction defines a simplicial embedding

𝜄 ∶ 𝒟𝒢₀ → 𝒟𝒢

with the property thatΦ◦𝜄equals the identity. Note that𝜄is simplicial and hence one-Lipschitz because the geodesic representatives of two disjoint simple closed curves are disjoint. Furthermore, we clearly have𝜄(ℛ𝒟(𝑐)) ⊂ ℛ𝒟(𝑐1, 𝑐2).

The situation in the following discussion is illustrated in Figure A. Let𝐵be the connected component of 𝜕𝐻 − {𝑐1, 𝑐2} containing the spot (this is a once spotted annulus). LetΛbe a diffeomorphism of𝜕𝐻which preserves the com- plement of𝐵(and hence the boundary of𝐵) pointwise and which pushes the spot in𝐵one full turn around a central loop in𝐵. The isotopy class ofΛis con- tained in the kernel of the homomorphismMod(𝜕𝐻) → Mod(𝜕𝐻0)induced by the spot removal mapΦ. The mapΛextends to a diffeomorphism of the handle- body𝐻. This can be seen as in the case of point-pushing in a surface: Identify the image of𝐵under the spot removal mapΦwith a closed annulus𝐴. Choose a neighborhood𝑁of the punctured annulus𝐵 in𝐻 which is homeomorphic to 𝐴 × [0, 1], with one interior point removed from𝐴 × {0}. Gradually undo the rotation of the marked point as one moves towards𝐴 × {1} ∪ 𝜕𝐴 × [0, 1].

Therefore the diffeomorphismΛgenerates an infinite cyclic group of simpli- cial isometries ofℛ𝒟(𝑐1, 𝑐2)which we denote again byΛ. With this notation, Φ◦Λ = Φ.

Figure A

LetΘ0∶ 𝒟𝒢₀→ ℛ𝒟(𝑐)be as in Lemma2.2. Define

Θ = Θ0◦Φ ∶ 𝒟𝒢 → ℛ𝒟(𝑐). (1)

Observe thatΘ(𝜄(𝐷)) = Θ0(𝐷)for all disks𝐷 ∈ 𝒟𝒢0. This then implies that Θ(𝜄(𝐷)) = 𝐷for all𝐷 ∈ ℛ𝒟(𝑐). Furthermore,Θis coarsely Lipschitz. Namely,

the puncture forgetful mapΦis simplicial and hence one-Lipschitz, and the mapΘ0is a coarse Lipschitz retraction by Lemma2.2. Moreover, we have

Θ(Λ(𝐷)) = Θ(𝐷) for all disks𝐷.

Recall from Lemma2.1thatℛ𝒟(𝑐)is isometric to the arc graph𝒜(𝐹)of𝐹.

Define a distance𝑑0onℛ𝒟(𝑐) × ℤby

𝑑0((𝛼, 𝑎), (𝛽, 𝑏)) = 𝑑ℛ𝒟(𝑐)(𝛼, 𝛽) + |𝑎 − 𝑏|

where𝑑ℛ𝒟(𝑐)denotes the distance inℛ𝒟(𝑐). Let moreover Ω = ∪𝑘Λ^𝑘𝜄(ℛ𝒟(𝑐)),

equipped with the restriction of the distance function of𝒟𝒢.

In the following lemma, the fact that the mapΨis well defined is part of the claim which is established in its proof.

Lemma 2.4. The mapΨ ∶ Ω → ℛ𝒟(𝑐) × ℤwhich maps𝐷 ∈ Λ^𝑘𝜄(ℛ𝒟(𝑐))to Ψ(𝐷) = (Θ(𝐷), 𝑘)is a bijective quasi-isometry.

Proof. Recall thatΘ(𝐷) = Θ(Λ^𝑘(𝐷))for all disks𝐷and all𝑘and that further- more the restriction ofΘto𝜄(ℛ𝒟(𝑐))is an isometry. In particular, if𝐷0, 𝐸0are distinct disks inℛ𝒟(𝑐)thenΘ(𝜄(𝐷0)) ≠ Θ(𝜄(𝐸0))and hence𝜄(𝐷0) ≠ Λ^𝑘(𝜄(𝐸0)) for all𝑘.

We claim that for every disk𝐷 ∈ Ωthe following hold true.

(1) 𝐷 ≠ Λ^𝑘(𝐷)for all𝑘 ≠ 0.

(2) If𝐷 ∈ 𝜄(ℛ𝒟(𝑐))thenΛ^𝑘𝐷 ∉ 𝜄(ℛ𝒟(𝑐))for all𝑘 ≠ 0.

(3) The disks𝐷andΛ(𝐷)can be realized disjointly.

(4) Two disks𝐷 ∈ Λ^𝑘𝜄(ℛ𝒟(𝑐)), 𝐸 ∈ Λ^𝓁𝜄(ℛ𝒟(𝑐))are disjoint only if|𝑘 − 𝓁| ≤ 1.

To show the claim let𝐷 ∈ Ωand for𝑘 ∈ ℤlet𝐷𝑘 = Λ^𝑘(𝐷). Figure A shows that for𝓁 ≥ 1, the disk𝐷𝑘+𝓁has precisely2𝓁−2essential intersections with𝐷𝑘, and these intersection points are up to isotopy contained in the annulus𝐵. This yields part (3) of the above claim, and part (4) follows from the same argument.

Furthermore, the twist parameter𝑘can be recovered from the geometric inter- section numbers betweenΛ^𝑘(𝐷)andΛ⁻¹(𝐷), 𝐷, Λ(𝐷). For example, if 𝑘 ≥ 2 then these intersection numbers equal2𝑘, 2𝑘 − 2, 2𝑘 − 4, respectively, and if 𝑘 ≤ −2then these intersection numbers are−2𝑘 − 4, −2𝑘 − 2, −2𝑘. This es- tablishes part (1) of the above claim, and part (2) follows from part (1) and the fact that the map𝜄is an embedding. In particular,Ω = ⊔𝑘Λ^𝑘𝜄(ℛ𝒟(𝑐))(disjoint union).

As a consequence, there exists a mapΨas claimed in the statement of the lemma, and this map is a bijection. NowΩ ⊂ ℛ𝒟(𝑐1, 𝑐2)and the restriction of the mapΘtoℛ𝒟(𝑐1, 𝑐2)is just the map induced by the spot forgetful map and hence it is one-Lipschitz. Part (4) of the above claim implies that the mapΨis two-Lipschitz.

AsΛ^𝑘𝜄(ℛ𝒟(𝑐))is isometric to𝒜(𝐹)for all𝑘, the inverse ofΨwhich associates to a pair(𝐷, 𝑘) ∈ ℛ𝒟(𝑐) × ℤthe diskΛ^𝑘(𝜄(𝐷))is coarsely one-Lipschitz. This

shows that indeed, the mapΨis a quasi-isometry.

The following proposition is the main remaining step towards a proof of The- orem2.

Proposition 2.5. There is a coarse Lipschitz retraction𝒟𝒢 → ∪𝑘Λ^𝑘𝜄(ℛ𝒟(𝑐)) = Ω. Moreover,Ωis a coarsely quasi-convex subset of𝒟𝒢.

Proof. For the construction of the Lipschitz retraction, we take advantage of the fact that any free homotopy class on a complete hyperbolic surface of finite area can be represented by a unique closed geodesic.

As in the proof of Lemma2.3, let𝜌₀be an orientation reversing involution of

𝜕𝐻0which fixes the𝐼-bundle generator𝑐pointwise. This involution determines an involution𝜌of the complement in𝜕𝐻of the interiorint(𝐵)of the annulus 𝐵which exchanges the curves𝑐1and𝑐2. Write as beforeΩ = ∪𝑘Λ^𝑘𝜄(ℛ𝒟(𝑐)).

Choose a complete finite area hyperbolic metric on𝜕𝐻(so that the marked point becomes a puncture) with the property that the involution𝜌of𝜕𝐻−int(𝐵) is an isometry for this metric which maps the geodesic representative ̂𝑐1of𝑐1to the geodesic representative ̂𝑐₂of𝑐₂. This metric restricts to a hyperbolic metric on the once punctured annulus𝐵with geodesic boundary. We use this hyper- bolic metric to determine for each pair of points𝑥𝑖 ∈ ̂𝑐𝑖 (𝑖 = 1, 2)a sample arc in𝐵connecting these two points as follows.

Choose a shortest geodesic arc𝛼connecting the two boundary components of 𝐵. By perhaps pulling back the hyperbolic metric with a diffeomorphism of 𝐵which preserves the boundary of 𝐵 pointwise, we may assume that𝛼is contained in the geodesic representative of one of the curves from 𝜄(ℛ𝒟(𝑐)).

Cutting𝐵open along𝛼yields a once punctured right angled rectangle𝑅with geodesic sides, where two distinguished sides come from the arc 𝛼. For any pair of points𝑥1, 𝑥2on the remaining two sides, choose a shortest geodesic arc 𝛼(𝑥1, 𝑥2)in𝑅connecting these two points. Such an arc is simple, but it may not be unique. By convexity, 𝛼(𝑥1, 𝑥2) is disjoint from𝛼if its endpoints are disjoint from the endpoints of𝛼. Note that as the spot of𝜕𝐻is a puncture for the hyperbolic metric, the geodesic arcs𝛼(𝑥₁, 𝑥₂)are disjoint from the spot, and 𝛼(𝑥1, 𝑥2)is not necessarily a shortest arc in𝐵with fixed endpoints.

This construction yields for any pair of points𝑥1 ∈ ̂𝑐1, 𝑥2 ∈ ̂𝑐2an oriented geodesic arc𝛼(𝑥1, 𝑥2) ⊂ 𝐵 with endpoints𝑥1, 𝑥2 such that any two of these arcs connecting distinct pairs of points on ̂𝑐1, ̂𝑐2intersect in at most two points.

Furthermore, each of these arcs intersects a geodesic representative of a curve in𝜄(ℛ𝒟(𝑐))in at most two points.

The geodesic arcs𝛼(𝑥1, 𝑥2)serve as a base marking to measure the twisting of a diskbounding simple closed curve relative to a simple closed curve in the set 𝜄(ℛ𝒟(𝑐)) ⊂ 𝒟𝒢. This is reminiscent to the definition of a twist parameter for a simple closed curve crossing through𝑐relative to a fixed marking of the surface

𝜕𝐻0. As we have to measure twisting about the puncture, we have to take care of

a pair of twist parameters about the simple closed curves𝑐1, 𝑐2. Our strategy to this end is to put the intersection of a simple closed diskbounding curve𝛽with

𝜕𝐻 − 𝐵into a normal form and use this normal form and the a priori chosen arcs𝛼(𝑥1, 𝑥2)to determine a twisting datum for𝛽. We next construct such a normal form for the intersection of𝛽with𝜕𝐻 − 𝐵using hyperbolic geometry.

Thus let𝛽be a diskbounding simple closed curve on𝜕𝐻. The intersection of 𝛽with𝜕𝐻−int(𝐵)consists of a non-empty collection𝜁of finitely many pairwise disjoint simple arcs with endpoints on ̂𝑐1, ̂𝑐2. Each such arc is freely homotopic relative to ̂𝑐1, ̂𝑐2to a unique geodesic arc which meets ̂𝑐1, ̂𝑐2orthogonally at its endpoints.

We claim that the components of the thus defined collection ̂𝜁of geodesic arcs are pairwise disjoint. However, some of these arcs may have nontrivial multiplicities as𝛽 ∩ (𝜕𝐻 − int(𝐵))may contain several components which are homotopic relative to the boundary. To verify the claim, double each compo- nent𝑋of the hyperbolic surface𝜕𝐻 − int(𝐵)along its boundary. The resulting, possibly disconnected, closed hyperbolic surface𝑆admits an isometric invo- lution𝜎preserving the components of𝑆whose fixed point set is precisely the image𝐶of the boundary of𝜕𝐻 −int(𝐵)in the doubled manifold. The double of the above collection𝜁of arcs is a collection of simple closed curves on𝑆which are invariant under𝜎.

The free homotopy classes of these closed curves are𝜎-invariant and hence the same holds true for their geodesic representatives: Namely, if𝛾is the ge- odesic representative of such a free homotopy class, then𝛾intersects the geo- desic multicurve𝐶in precisely two points. Let𝛾1be the component of𝛾 − 𝐶 of smaller length. Then𝛾1∪ 𝜎(𝛾1)is a simple closed curve freely homotopic to 𝛾, and its length is at most the length of𝛾. But𝛾 is the unique simple closed curve of minimal length in its free homotopy class and hence𝛾 = 𝛾1∪ 𝜎(𝛾1).

Thus𝛾intersects𝐶orthogonally, and𝛾 ∩ 𝑋is a component of the arc system

̂𝜁. The claim now follows from the well known fact that the geodesic represen- tative of a simple closed multicurve on a hyperbolic surface is a simple closed multicurve.

As a consequence of the above discussion, the order of the endpoints of the components of𝛽 −int(𝐵)on ̂𝑐1∪ ̂𝑐2coincides with the order of the endpoints of the collection of geodesic arcs ̂𝜁which meet ̂𝑐1∪ ̂𝑐2orthogonally at their end- points and are freely homotopic to the components of𝛽 − int(𝐵). This implies that a diskbounding simple closed curve𝛽on𝜕𝐻can be homotoped to a curve

̂𝛽of the following form.

(i) The restriction of ̂𝛽to𝜕𝐻 − int(𝐵)consists of a finite collection of pair- wise disjoint geodesic arcs which meet ̂𝑐𝑖 orthogonally at their end- points. Some of these arcs may occur more than once.

(ii) The restriction of ̂𝛽to the once punctured annulus𝐵consists of a finite non-empty collection of arcs connecting ̂𝑐1 to ̂𝑐2 and perhaps a finite number of arcs which go around the puncture and return to the same boundary component of𝐵. Distinct such arcs have disjoint interiors.

The curve ̂𝛽is uniquely determined by𝛽and the choice of the hyperbolic metric on𝜕𝐻up to a homotopy of the components of ̂𝛽 ∩𝐵with fixed endpoints (note that the above construction does not determine uniquely the intersection of ̂𝛽 with𝐵). This completes the construction of a normal form for a diskbounding simple closed curve𝛽on𝜕𝐻.

The goal is to use this normal form to construct a Lipschitz retraction of 𝒟𝒢as stated in the proposition by associating to a diskbounding simple closed curve𝛽in𝒟𝒢a pairΨ⁻¹(Θ(𝛽), 𝑘)whereΨis as in Lemma2.4, whereΘis as in (1) and where𝑘is a twist parameter, read off from the intersection of the normal form with the once punctured annulus𝐵. We first check compatibility of this twist parameter construction with the twist parameter stemming from the decompositionΩ = ∪𝑘Λ^𝑘𝜄(ℛ𝒟(𝑐)).

By construction of the map𝜄, if𝛽 = 𝜄(𝛽^′) ∈ 𝜄(ℛ𝒟(𝑐))then ̂𝛽 ∩ 𝜕𝐻 − int(𝐵) is just the lift of the geodesic representative of𝛽^′to𝜕𝐻 − int(𝐵)for the follow- ing hyperbolic metric on𝜕𝐻0− 𝑐. Recall that the metric on𝜕𝐻was chosen in such a way that there exists an orientation reversing involution𝜌which maps

̂𝑐1to ̂𝑐2. Cuttingint(𝐵)off𝜕𝐻and gluing𝑐1to𝑐2with the isometric involution 𝜌constructs from𝜕𝐻 − int(𝐵)a hyperbolic surface which can be viewed as a hyperbolic metric on𝜕𝐻0. Using this metric for the construction of the embed- ding𝜄 ∶ ℛ𝒟(𝑐) → ℛ𝒟(𝑐1, 𝑐2), we conclude that the intersections with𝐵of the representatives ̂𝛽of the elements𝛽 ∈ 𝜄(ℛ𝒟(𝑐))are pairwise disjoint.

Define a map

Ξ ∶ 𝒟𝒢 → ℤ

as follows. Let ̂𝛽be a closed piecewise geodesic curve with properties (i),(ii) above which is constructed from the simple closed diskbounding curve𝛽. Let 𝑏be one of the components of ̂𝛽 ∩ 𝐵with endpoints on ̂𝑐₁and ̂𝑐₂, oriented in such a way that it connects ̂𝑐1to ̂𝑐2. Such a component exists since otherwise the image of 𝛽 under the spot removal map is homotopic to a curve disjoint from the diskbusting curve𝑐on𝜕𝐻0. Let𝑥1, 𝑥2be the endpoints of𝑏on ̂𝑐1, ̂𝑐2. Let𝑎 = 𝛼(𝑥1, 𝑥2); then𝑏, 𝑎are simple arcs in𝐵with the same endpoints which intersect some core curve of the annulus𝐵 in precisely one point. As- sume that ̂𝑐1, ̂𝑐2are oriented and define the boundary orientation of𝐵. Then 𝑏 is homotopic with fixed endpoints to the arc ̂𝑐^𝑘₁ ⋅ 𝑎 ⋅ ̂𝑐₂^𝓁for unique𝑘, 𝓁 ∈ ℤ (read from left to right). In other words, if we denote by𝜏𝑖 the positive Dehn twist about ̂𝑐𝑖, viewed as a diffeomorphism of the punctured disk𝐵with fixed boundary, then𝑏is homotopic with fixed endpoints to the arc𝜏^𝑘₁𝜏₂^−𝓁𝑎. Define Ξ(𝛽) = 𝑘.

Although this definition depends on the choice of the arcs𝛼(𝑥1, 𝑥2)and on the choice of the component𝑏 of𝐵 ∩ ̂𝛽, the mapΞis coarsely well defined.

Namely, let𝑏^′be a second component of ̂𝛽 ∩ 𝐵, with endpoints𝑥^′₁, 𝑥^′₂on ̂𝑐1, ̂𝑐2

and distinct from𝑏. Then the interior of𝑏^′is disjoint from the interior of𝑏. In particular, if𝑎^′is an arc in𝐵with the same endpoints as𝑏^′whose interior is disjoint from𝑎, then𝑏^′ is homotopic with fixed endpoints to𝜏₁^𝑞𝜏^−𝑟₂ 𝑎^′ for|𝑞 −

𝑘| ≤ 1, |𝑟 − 𝓁| ≤ 1. On the other hand, the arcs𝑎 = 𝛼(𝑥1, 𝑥2), 𝛼(𝑥^′₁, 𝑥₂^′)do not have an essential intersection with a fixed arc connectinĝ𝑐₁to ̂𝑐₂and hence𝑎^′= 𝜏^𝑠₁𝜏^−𝑢₂ 𝛼(𝑥^′₁, 𝑥₂^′)for some|𝑠| ≤ 1, |𝑢| ≤ 1. This shows that the multiplicity𝑘^′of the curvê𝑐₁in the description of𝑏^′relative to𝛼(𝑥^′₁, 𝑥^′₂)satisfies|𝑘−𝑘^′| ≤ 2. The same reasoning yields that the mapΞis coarsely two-Lipschitz. Furthermore, we haveΞ(𝜄(ℛ𝒟(𝑐))) ⊂ [−2, 2]. Namely, recall that we chose the geodesic arc 𝛼in the beginning of this proof to be contained in one of the curves𝜄(ℛ𝒟(𝑐)) (which is nothing else but a normalization assumption).

To summarize, the map

(Θ, Ξ) ∶ 𝒟𝒢 → ℛ𝒟(𝑐) × ℤ

is coarsely Lipschitz, and its composition with the inverse of the mapΨfrom Lemma2.4is a coarse Lipschitz retraction of𝒟𝒢ontoΩprovided that the map Ξmaps a point inΛ^𝑘𝜄(ℛ𝒟(𝑐))into a uniformly bounded neighborhood of𝑘.

However, if𝛽0 ∈ 𝜄(ℛ𝒟(𝑐))and if𝛽 = Λ^𝑘(𝛽0) ∈ Λ^𝑘𝜄(ℛ𝒟(𝑐)), then the inter- sections with𝐻 − int(𝐵)of the representatives ̂𝛽, ̂𝛽0of𝛽, 𝛽0constructed above coincide. This implies that up to homotopy with fixed endpoints, ̂𝛽 ∩ 𝐵 = Λ^𝑘( ̂𝛽₀∩ 𝐵).

On the other hand, point-pushing along a simple closed curve𝛾based at𝑝 descends to conjugation by𝛾in𝜋1(𝜕𝐻0, 𝑝). Therefore the image under the map Λof a simple arc𝑏in𝐵with endpoints on the two distinct components of𝜕𝐵 is homotopic with fixed endpoints to𝑐1𝑏𝑐2(recall that we oriented𝑐1, 𝑐2so that they define the boundary orientation of𝐵). AsΞ(𝜄(ℛ𝒟(𝑐))) ⊂ [−2, 2], it follows that|Ξ(𝛽) − 𝑘| ≤ 2. This shows the proposition.

To summarize, we obtain

Corollary 2.6. The disk graph of a handlebody𝐻of genus𝑔 ≥ 2with one spot contains quasi-isometrically embedded copies ofℝ².

Proof. A subgraphΓof a metric graph𝐺is uniformly quasi-isometrically em- bedded if there exists a coarsely Lipschitz retraction𝐺 → Γ. Proposition 2.5 shows that for any𝐼-bundle generator𝑐in𝜕𝐻0, there is a coarse Lipschitz re- traction of𝒟𝒢onto its subgraphΩ = ∪𝑘Λ^𝑘𝜄(ℛ𝒟(𝑐)), and by Lemma2.4,Ωis quasi-isometric to the direct productℛ𝒟(𝑐)×ℤ. Thus as by Lemma2.1,ℛ𝒟(𝑐) is quasi-isometric to the arc graph of the base𝐹of the𝐼-bundle determined by𝑐 and hence has infinite diameter, the product of any biinfinite geodesic inℛ𝒟(𝑐) andℤdefines a quasi-isometrically embeddedℤ²in𝒟𝒢.

Remark 2.7. In [4] we showed that in contrast to handlebodies without spots, the disk graph of a handlebody𝐻 with a single spot on the boundary isnota quasi-convex subgraph of the curve graph of𝜕𝐻. We do not know whether𝒟𝒢 contains quasi-isometrically embedded euclidean spaces of dimension bigger than two.

3. Once spotted doubled handlebodies

In this section we consider the connected sum 𝑀 = ♯_𝑔𝑆² × 𝑆¹ of an even number𝑔 = 2𝑛 ≥ 2of copies of 𝑆² × 𝑆¹ with one spot (marked point). We explain how the construction that led to the proof of Theorem2can be used to show Theorem3: The sphere graph of𝑀contains quasi-isometrically embed- ded copies ofℝ².

Consider the double𝑀0 = ♯𝑔𝑆² × 𝑆¹ of a handlebody 𝐻0 of genus𝑔 ≥ 2 without spots. Let𝑀be the manifold𝑀0equipped with a marked point𝑝. As before, we call𝑝a spot in𝑀. There is a natural spot removing mapΦ ∶ 𝑀 → 𝑀0.

The vertices of thesphere graph 𝒮𝒢of 𝑀 are isotopy classes of embedded spheres in𝑀 which are disjoint from the spot and not isotopic into the spot.

Isotopies are required to be disjoint from the spot as well. Two such spheres are connected by an edge of length one if they can be realized disjointly. Similarly, let𝒮𝒢₀be the sphere graph of𝑀0.

Choose an embedded oriented surface𝐹0 ⊂ 𝑀0of genus𝑛with connected boundary such that the inclusion𝐹0→ 𝑀0induces an isomorphism𝜋1(𝐹0) → 𝜋1(𝑀0). We may assume that the oriented𝐼-bundle𝐻0over𝐹0is an embedded handlebody𝐻0 ⊂ 𝑀0whose double equals𝑀0. Thus every embedded essential arc𝛼in𝐹0with boundary in𝜕𝐹0determines a sphereΥ0(𝛼)in𝑀0as follows.

The interval bundle over𝛼is an embedded essential disk in𝐻0, with boundary in𝜕𝐻0, and we letΥ0(𝛼)be the double of this disk. By construction, the sphere Υ0(𝛼)intersects the surface𝐹0precisely in the arc𝛼. By Lemma 4.17 of [6], distinct arcs give rise to non-isotopic spheres, furthermore the mapΥ0preserves disjointness and henceΥ0is a simplicial embedding of the arc graph𝒜(𝐹0)of 𝐹0into the sphere graph𝒮𝒢0of𝑀0.

Now mark a point𝑝on the boundary𝜕𝐹0of𝐹0and view the resulting spotted surface𝐹 as a surface in the spotted manifold𝑀. Thearc graph𝒜(𝐹)of𝐹 is the graph whose vertices are isotopy classes of essential simple arcs in𝐹with endpoints on the complement of𝑝in the boundary of𝐹. Here we exclude arcs which are homotopic with fixed endpoints to a subarc of𝜕𝐹containing the base point𝑝, and we require that an isotopy preserves the marked point𝑝and hence endpoints of arcs can only slide along𝜕𝐹 − {𝑝}. Two such arcs are connected by an edge if they can be realized disjointly. Note that𝒜(𝐹)isnotthe arc graph of the bordered surface𝐹punctured at an interior point of𝐹. Associate to an arc𝛼in𝐹the doubleΥ(𝛼)of the𝐼-bundle over𝛼.

The spot removal mapΦ ∶ 𝑀 → 𝑀0induces a simplicial surjection𝒮𝒢 → 𝒮𝒢0, again denoted byΦfor simplicity. Similarly, if we let𝜑 ∶ 𝐹 → 𝐹0 be the map which forgets the marked point𝑝 ∈ 𝜕𝐹, then𝜑induces a simplicial surjection𝒜(𝐹) → 𝒜(𝐹0), denoted as well by𝜑. We then obtain a commutative

diagram

𝒜(𝐹) 𝒜(𝐹0)

𝒮𝒢 𝒮𝒢₀

𝜑

Υ Υ0

Φ

(2)

Similar to the case of the handlebody𝑀0without spots and the mapΥ0, we obtain

Lemma 3.1. The mapΥis a simplicial embedding of the arc graph𝒜(𝐹)into the sphere graph.

Proof. We have to show that the mapΥis injective. AsΥ0is injective and as the diagram (2) commutes, it suffices to show the following. Let𝛼 ≠ 𝛽 ∈ 𝒜(𝐹) be such that𝜑(𝛼) = 𝜑(𝛽); thenΥ(𝛼) ≠ Υ(𝛽).

Now𝜑(𝛼) = 𝜑(𝛽)means that up to exchanging𝛼and𝛽, there exists a num- ber𝑘 > 0such that𝛽can be obtained from𝛼by𝑘half Dehn twists about the boundary𝜕𝐹of𝐹. Here the half Dehn twist𝑇(𝛼)of𝛼is defined as follows.

The orientation of𝐹 induces a boundary orientation for𝜕𝐹 which in turn induces an orientation on𝜕𝐹 − {𝑝}. With respect to the order defined by this orientation, let𝑥be the bigger of the two endpoints𝑥, 𝑦of𝛼. Slide𝑥across𝑝 to obtain a new arc𝑇(𝛼), with endpoints𝑥^′, 𝑦. This arc is not homotopic to𝛼.

To see this it suffices to show that the double𝐷𝑇(𝛼)of𝑇(𝛼)in the double𝐷𝐹of 𝐹(which is a surface with one puncture) is not freely homotopic to the double 𝐷(𝛼)of𝛼. This follows since𝐷(𝛼)and𝐷𝑇(𝛼)can be homotoped in such a way that they bound a once punctured annulus in𝐷𝐹.

The same reasoning also shows that the sphereΥ(𝑇(𝛼))is not homotopic to the sphereΥ(𝛼). Namely, let𝜒 ⊂ 𝜕𝐹 ∪ {𝑝}be the oriented embedded arc con- necting the intersection point𝑥of𝛼with𝜕𝐹to the point𝑥^′. This arc contains 𝑝 in its interior. Then the sphereΥ(𝑇(𝛼)) is a connected sum of the sphere Υ(𝛼)with the boundary of a punctured ball which is a thickening of𝜒. Thus Υ(𝛼)andΥ(𝑇(𝛼))can be isotoped in such a way that they bound a subset of𝑀 homeomorphic to the complement of an interior point of𝑆²× [0, 1].

The above construction, applied to the sphereΥ(𝑇(𝛼))instead of the sphere Υ(𝛼)and where the point𝑦takes on the role of the point𝑥in the above discus- sion, shows thatΥ(𝑇²(𝛼))is obtained fromΥ(𝛼)by point-pushing along the ori- ented loop𝜕𝐹with basepoint𝑝. This is a diffeomorphism of𝑀which leaves the complement of a small tubular neighborhood of𝜕𝐹pointwise fixed and pushes the basepoint𝑝along𝜕𝐹. As in the proof of Lemma2.4, this argument can be iterated. It shows that the sphereΥ(𝑇^𝑘(𝛼))intersects the sphereΥ(𝛼)in𝑘 − 1 intersection circles. These circles are essential since they cut bothΥ(𝑇^𝑘(𝛼))and Υ(𝛼)into two disks and𝑘 − 2annuli, where a disk component of𝑇^𝑘(𝛼) − 𝑇(𝛼) bounds together with a disk component of𝑇(𝛼) − 𝑇^𝑘(𝛼)an embedded sphere enclosing the spot. Invoking the proof of Lemma2.4, we conclude that indeed, for𝑘 ≠ 𝓁,Υ(𝑇^𝑘(𝛼))is not homotopic toΥ(𝑇^𝓁(𝛼)).

We showed so far that the mapΥis injective. To complete the proof of the lemma, it suffices to observe that disjoint arcs are mapped to disjoint spheres.

But this is immediate from the construction.

Proposition 4.18 of [6] shows that there is a one-Lipschitz retraction Ψ0∶ 𝒮𝒢0→ Υ0(𝒜(𝐹0))

which is of the formΨ0 = Υ0◦Θ0(read from right to left) whereΘ0 ∶ 𝒮𝒢₀ → 𝒜(𝐹₀)is a one-Lipschitz map. In particular,Υ₀(𝒜(𝐹₀))is a quasi-isometrically embedded subgraph of𝒮𝒢₀ which is quasi-isometric to𝒜(𝐹0). Our goal is to show that there also is a coarse Lipschitz retraction of𝒮𝒢ontoΥ(𝒜(𝐹))of the formΨ = Θ◦Υ whereΘ ∶ 𝒮𝒢 → 𝒜(𝐹)is a coarse Lipschitz map. This then yields Theorem3from the introduction.

To construct the mapΘwe use the method from [6]. We next explain how this method can be adapted to our needs.

Let as before𝐹 ⊂ 𝑀be an embedded oriented surface with connected bound- ary𝜕𝐹so that𝑀is the double of the trivial𝐼-bundle over𝐹. We assume that the marked point𝑝is contained in the boundary𝜕𝐹of𝐹. Furthermore, we assume that the boundary𝜕𝐹of𝐹is a smoothly embedded circle in𝑀 ∪ {𝑝}(that is, an embedded compact one-dimensional submanifold). We use the marked point 𝑝as the basepoint for the fundamental group of𝑀. Then𝜕𝐹equipped with its boundary orientation defines a homotopy class𝛽 ∈ 𝜋1(𝑀, 𝑝) = 𝜋1(𝐹, 𝑝) = ℱ2𝑔

(the free group in2𝑔generators). Since𝛽is the oriented boundary curve of𝐹, it is an iterated commutator in a standard set of generators ofℱ2𝑔and hence𝛽 is not contained in any free factor (Whitehead graphs are a convenient tool to verify this fact). Thus𝜕𝐹intersects every sphere in𝑀. Namely, for any given sphere𝑆in𝑀, the subgroup of𝜋1(𝑀, 𝑝)of all homotopy classes of loops which do not intersect𝑆is a proper free factor of𝜋1(𝑀, 𝑝).

As in [6] and similar to the construction in Lemma 2.2, the strategy is to associate to a sphere𝑆in𝑀a component of the intersection𝐹 ∩ 𝑆. However, unlike in the case of curves on surfaces, there is no suitable normal form for intersections of spheres with the surface𝐹, and the main work in [6] consists of overcoming this difficulty by introducing a relative normal form which allows to associate to a sphere in𝑀0an intersection arc with𝐹0so that the resulting map𝒮𝒢0 → 𝒜(𝐹0)is one-Lipschitz.

For the remainder of this section we outline the main steps in this construc- tion, adapted to the sphere graph𝒮𝒢of𝑀 and the arc graph𝒜(𝐹)of𝐹. This requires modifying spheres with isotopies not crossing through𝑝, and modify- ing the surface𝐹with homotopies leaving the boundary𝜕𝐹pointwise fixed.

For convenience, we record some definitions from [6] (the following com- bines Definition 4.7 and Definition 4.9 of [6]).

Definition 3.2. LetΣbe a sphere or a sphere system.

(1) 𝜕𝐹intersectsΣminimallyif𝜕𝐹intersectsΣtransversely and if no com- ponent of𝜕𝐹−Σnot containing the basepoint𝑝is homotopic with fixed endpoints intoΣ.

(2) 𝐹 is in minimal position with respect toΣif𝜕𝐹 intersectsΣminimally and if moreover each component ofΣ ∩ 𝐹is a properly embedded arc which either is essential or homotopic with fixed endpoints to a subarc of𝜕𝐹containing the marked point.

A version of the easy Lemma 4.6 of [6] states that any closed curve containing the basepoint can be put into minimal position relative to a sphere systemΣas defined in the first part of Definition3.2. The following is a version of Lemma 4.12 of [6]. For its formulation, call a sphere systemΣsimpleif it decomposes 𝑀into a simply connected components.

Lemma 3.3. LetΣbe a simple sphere system in𝑀. Suppose that𝐹is in minimal position with respect toΣ. Let𝜎^′be an embedded sphere disjoint fromΣand letΣ^′ be a simple sphere system obtained fromΣby either adding𝜎^′, or removing one sphere𝜎 ∈ Σ. Then𝐹can be homotoped leaving𝑝fixed to a surface𝐹^′which is in minimal position with respect toΣ^′.

Proof. As in the proof of Lemma 4.12 of [6], removing a sphere preserves min- imal position, so only the case of adding a sphere has to be considered.

Thus letΣbe a simple sphere system and let 𝜎^′ be a sphere disjoint from Σ. Assume that 𝐹 is in minimal position with respect to Σ. Let 𝑊Σ be the complement ofΣin𝑀, that is,𝑊Σ is a compact (possibly disconnected) man- ifold whose boundary consists of2𝑘boundary spheres𝜎⁺₁, 𝜎⁻₁, ⋯ , 𝜎⁺_𝑘, 𝜎⁻_𝑘. The boundary spheres𝜎⁺_𝑖 and𝜎⁻_𝑖 correspond to the two sides of a sphere𝜎𝑖 ∈ Σ.

The surface𝐹intersects𝑊Σin a collection of embedded surfaces with bound- aries. Each such surface is a polygonal disk𝑃𝑖(𝑖 = 1, … , 𝑚). The sides of each such polygon alternate between subarcs of𝜕𝐹and arcs contained inΣ. There is at most one bigon, that is, a polygon with two sides, and this polygon then contains the point𝑝in one of its sides. Each rectangle, if any, is homotopic into

𝜕𝐹.

The proof of Lemma 4.12 of [6] now proceeds by studying the intersection of each polygonal component of 𝐹 − Σ with the sphere𝜎^′. This is done by contracting each such polygonal component𝑃 to a ribbon tree𝑇(𝑃) in such a way that the boundary components inΣ are contracted to single points in 𝑇(𝑃). If𝑃is not a rectangle or bigon, then𝑇(𝑃)has a single vertex which is not univalent. As such ribbon trees are one-dimensional objects, they can be homotoped with fixed endpoints on𝜕𝑊Σto trees which are in minimal position with respect to𝜎^′. This construction applies without change to rectangles and perhaps the bigon which can be represented by an interval with one endpoint at𝑝and the second endpoint on a component of Σ. We refer to the proof of Lemma 4.12 of [6] for details. No adjustment of the argument is necessary.

The above construction is only valid for simple sphere systemsΣand not for individual spheres. Furthermore, it is known that the arc system on𝐹 ∩ Σ obtained by putting𝐹 into minimal position with respect toΣis not uniquely determined byΣ. To overcome this difficulty, the work of [6] uses as an auxiliary

datum a maximal system𝐴0of pairwise disjoint essential arcs on the surface 𝐹0. Here maximal means that any arc which is disjoint from𝐴0is contained in 𝐴0. The system𝐴0thenbinds𝐹0, that is,𝐹 − 𝐴0is a union of topological disks.

Furthermore,𝜕𝐹0and each arc𝛼 ∈ 𝐴0is equipped with an orientation.

Choose an arc system𝐴for𝐹which binds𝐹. If𝐹 ⊂ 𝑀is in minimal position with respect toΣ, then a homotopy assures that no arc from the arc system𝐴 intersects a component of𝐹 − Σwhich is a rectangle or a bigon. Then Lemma 4.12 of [6] and its proof applies without modification and shows that with a homotopy, 𝐹 can be put into normal form with respect to the arc system𝐴, called𝐴-tight minimal positionwith respect toΣ. This then yields the statement of Lemma 4.16 of [6]: if 𝐹 is in𝐴-tight minimal position with respect to the simple sphere systemΣ, then the binding arc systemΣ ∩ 𝐹is determined byΣ.

In particular, two distinct spheres fromΣintersect𝐹in disjoint essential arcs.

There may in addition be inessential arcs, that is, arcs which are homotopic with fixed endpoints to a subsegment of 𝜕𝐹 containing the basepoint 𝑝, but these will be unimportant for our purpose.

Now let𝜎 be an essential sphere in𝑀. LetΣbe a simple sphere system in 𝑀containing𝜎as a component. We put𝐹into𝐴-tight minimal position with respect toΣ. Then 𝜎 ∩ 𝐹consists of a non-empty collection of essential arcs and perhaps some additional non-essential arcs. Choose one of the essential intersection arcs𝛼and defineΘ(𝜎) = 𝛼. As in [6] and Proposition2.5we now obtain

Proposition 3.4. The mapΘis a coarsely Lipschitz map. For each arc𝛼 ∈ 𝒜(𝐹), we haveΘ(Υ(𝛼)) = 𝛼. As a consequence, if𝑔 = 2𝑛is even then the sphere graph 𝒮𝒢of𝑀contains quasi-isometrically embedded copies ofℝ².

Proof. Given the above discussion, the proof thatΘis a coarsely Lipschitz map is identical to the proof that the mapΘ0is a coarsely Lipschitz map in Proposi- tion 4.18 of [6] and will be omitted. Moreover, as for𝛼 ∈ 𝒜(𝐹), the sphereΥ(𝛼) intersects𝐹in the unique arc𝛼, we haveΘ(Υ(𝛼)) = 𝛼.

As a consequence,Θ|Υ(𝒜(𝐹))is a Lipschitz bijection, with inverseΥ. Then the subgraphΥ(𝒜(𝐹))of𝒮𝒢is bilipschitz equivalent to𝒜(𝐹). Furthermore, the mapΥ◦Θis a Lipschitz retraction of𝒮𝒢ontoΥ(𝒜(𝐹)). ThenΥ(𝒜(𝐹))is a quasi- isometrically embedded subgraph of𝒮𝒢which is moreover quasi-isometric to 𝒜(𝐹).

Let as before𝐹0be the surface obtained from𝐹by removing the spot. We are left with showing that𝒜(𝐹)is quasi-isometric to𝒜(𝐹0) × ℤ. However, this was shown in Lemma2.4. Namely, in the terminology used before, the boundary

𝜕𝐹is an𝐼-bundle generator in the trivial interval bundle𝐻over𝐹, and asso- ciating to an arc𝛼the𝐼-bundle over𝛼defines an isomorphism of𝒜(𝐹)with the subgraphΩof the disk graph of𝐻used in Lemma2.4. The statement now

follows from Lemma2.4.

Remark 3.5. Most likely Proposition3.4 holds true as well in the case that 𝑔 = 2𝑛+1is odd, and furthermore this can be deduced with the above argument