On the geometry of graphs associated to infinite-type surfaces
aa r X i v : . [ m a t h . G T ] O c t ON THE GEOMETRY OF GRAPHS ASSOCIATED TOINFINITE-TYPE SURFACES
JAVIER ARAMAYONA & FERR ´AN VALDEZ
Abstract.
Consider a connected orientable surface S of infinite topo-logical type, i.e. with infinitely-generated fundamental group.Our main purpose is to give a description of the geometric structureof an arbitrary subgraph of the arc graph of S , subject to some rathergeneral conditions. As special cases, we recover the main results of J.Bavard [2] and Aramayona-Fossas-Parlier [1].In the second part of the paper, we obtain a number of results on thegeometry of connected, Mod( S )-invariant subgraphs of the curve graphof S , in the case when the space of ends of S is homeomorphic to aCantor set. Introduction
There has been a recent surge of activity around mapping class groups ofinfinite-type surfaces, i.e. with infinitely-generated fundamental group. Themotivation for studying these groups stems from several places, as we nowbriefly describe.First, infinite-type surfaces appear as inverse limits of surfaces of finitetype. In particular, infinite-type mapping class groups are useful in thestudy of asymptotic and/or stable properties of their finite-type counter-parts. This is the approach taken by Funar-Kapoudjian [7], where the au-thors identify the homology of an infinite-type mapping class group with thestable homology of the mapping class groups of its finite-type subsurfaces.In a related direction, a number of well-known groups appear as subgroupsof the mapping class group of infinite-type surfaces. For instance, Funar-Kapoudjian [8] realized Thompson’s group T as a topologically-defined sub-group of the mapping class group of a certain infinitely-punctured sphere.A third piece of motivation for studying mapping class groups of infinite-type surfaces comes from dynamics, as explained by D. Calegari in [3]. Moreconcretely, let S be a closed surface, P ⊂ S a finite subset, and consider thegroup Homeo( S, P ) of those homeomorphisms of S that preserve P setwise.Let G <
Homeo(
S, P ) be a subgroup that acts freely on S − P . Then G Date : May 2016. Revised version: October 2016.The first author was partially funded by grants RYC-2013-13008 and MTM2015-67781-P. The second author was supported by PAPIIT projects IN100115, IN103411 andIB100212. admits a natural homeomorphism to Mod( S − K, P ), where K is either afinite set or a Cantor set. See [3] for more details.1.1. Combinatorial models.
A large number of problems about mappingclass groups of finite-type surfaces may be understood through the varioussimplicial complexes built from curves and/or arcs on surfaces. Notableexamples of these are the curve graph C ( S ) and the arc graph A ( S ); seeSection 4 for definitions. When S has finite type, a useful feature of thesecomplexes is that, with respect to their standard path-metric, they are hy-perbolic spaces of infinite diameter; see [14] and [12], respectively.In sharp contrast, in the case of an infinite-type surface these complexesoften have finite diameter; see Section 4. This obstacle was first overcomeby J. Bavard [2] in the particular case when S is a sphere minus a Cantor setand an isolated point. Indeed, she proved that a certain subgraph of the arcgraph is hyperbolic and has infinite diameter, and used this to construct non-trivial quasi-morphisms from the mapping class group of the plane minusa Cantor set. Subsequently, Aramayona-Fossas-Parlier [1] have producedsimilar graphs for arbitrary surfaces, subject to certain conditions on the setof punctures of S . However, the definition of these subgraphs is surprisinglysubtle, and small variations in the definition may produce graphs that havefinite diameter or are not hyperbolic.1.1.1. Arc graphs.
Our main goal is to give a unified description of the pos-sible geometric structures of an arbitrary subgraph of the arc graph of aninfinite-type surface, subject to some rather natural conditions on the givengraph. First, we will require that it be sufficiently invariant , that is invari-ant under Mod(
S, P ), for some (possibly empty) finite set P of punctures.In addition, we will assume that every such graph satisfies the projectionproperty . This property is needed only for technical reasons, and thus werefer the reader to Section 5 for details. However, we stress that this restric-tion is easy to check, and often automatically satisfied, once one is given anexplicit subgraph G ( S ) of A ( S ). This is the case with the graphs consideredin [1] and [2]; see Remark 5.5 below.Before we state our result, recall from [18] that a witness of a subgraph G ( S ) of A ( S ) is an essential subsurface Y of S such that every vertex of G ( S )intersects Y essentially. Given a witness Y , we denote by G ( Y ) the subgraphof G ( S ) spanned by those vertices of G ( S ) that are entirely contained in Y .We will prove: Theorem 1.1.
Let S be a connected orientable surface of infinite type, and G ( S ) a connected, sufficiently invariant subgraph of A ( S ) with the projectionproperty.(1) If every witness of G ( S ) has infinitely many punctures, then G ( S ) has finite diameter. This definition is due to Schleimer [18], who referred to witnesses as holes . The word“witness” has been suggested to us by S. Schleimer.
RAPHS FOR INFINITE-TYPE SURFACES 3 (2) Otherwise, G ( S ) has infinite diameter. Moreover:(2a) If every two witnesses of G ( S ) intersect, then G ( S ) is hyperbolicif and only if G ( Y ) is uniformly hyperbolic, for every finite-typewitness Y .(2b) If G ( S ) has two disjoint witnesses of finite type, then it is nothyperbolic. We stress that part (2b) of Theorem 1.1 is merely a manifestation ofSchleimer’s
Disjoint Witnesses Principle [18, 14], although we have includeda proof in Section 5 for completeness. In addition, we remark that once oneis given an explicit subgraph G ( S ) of A ( S ), it is trivial to decide what thewitnesses of G ( S ) are and, in particular, where G ( S ) falls in the descriptionoffered by Theorem 1.1; see the various corollaries below. Finally, we willsee in Section 5 that the assumptions that G ( S ) has the projection propertywill not be used in the proof of part (1) of Theorem 1.1, and thus that partholds in slightly more generality; this remark will be useful for the variouscorollaries of Theorem 1.1, see below.As a special case of Theorem 1.1, we recover the main result of Aramayona-Fossas-Parlier [1]; see Section 5 for the necessary definitions: Corollary 1.2 ([1]) . Let S be a connected orientable surface, and P a non-empty finite set of isolated punctures. Then, the relative arc graph A ( S, P ) ⊂A ( S ) is hyperbolic and has infinite diameter. Once again, we stress that this result was first proved by Bavard [2] inthe special case when S is a sphere minus a Cantor set and one isolatedpuncture.We will see in Corollary 5.6 in Section 5 that, on the other hand, if P contains a puncture that is not isolated, Theorem 1.1 implies that A ( S, P )has finite diameter. More drastically, if S has no isolated punctures at all,then there are no geometrically interesting Mod( S )-invariant subgraphs of A ( S ): Corollary 1.3.
Let S be a connected orientable surface with at least onepuncture. If S has no isolated punctures, then any connected Mod( S ) -invariant subgraph of A ( S ) has finite diameter. See Section 5 for some further consequences of Theorem 1.1.1.1.2.
Curve graphs.
In the light of Corollary 1.3, there are no geometri-cally interesting Mod( S )-invariant subgraphs of A ( S ) if S is a puncturedsurface with no isolated punctures. With this motivation we are going tostudy Mod( S )-invariant subgraphs of the curve graph C ( S ) instead. We willrestrict our attention to the case when S has no isolated ends; as we willsee, the situation heavily depends on whether S has finite or infinite genus.Before going any further, we note that the case when S has isolated ends iscovered in the recent preprint [4]; see Remark 1.6 below. JAVIER ARAMAYONA & FERR ´AN VALDEZ
Before we state our results, we denote by NonSep( S ) the non-separatingcurve graph of S , namely the subgraph of C ( S ) spanned by all non-separatingcurves. Further, let NonSep ∗ ( S ) be the augmented nonseparating curvegraph of S , whose vertices are all nonseparating curves on S together withthose curves that cut off a disk containing every puncture of S . Finally,denote by Outer( S ) the subgraph of C ( S ) spanned by all the outer curves on S , namely those curves which cut off a disk containing some, but not all,punctures of S . See Section 4 for further definitions.We start with the case when the genus of S is finite: Theorem 1.4.
Let S a connected orientable surface of infinite type, withfinite genus and no isolated punctures. Then, a Mod( S ) -invariant subgraph G ( S ) ⊂ C ( S ) has infinite diameter if and only if G ( S ) ∩ Outer( S ) = ∅ .Moreover, in this case:(1) If G ( S ) ∩ NonSep( S ) = ∅ then G ( S ) is not hyperbolic.(2) If G ( S ) ∩ NonSep( S ) = ∅ then G ( S ) is quasi-isometric to NonSep( S ) or NonSep ∗ ( S ) . The classification of infinite-type surfaces, stated as Theorem 3.3 in Sec-tion 3, tells us that, under the hypotheses of the theorem, S is homeomorphicto a closed surface with a Cantor set removed.As an immediate consequence of Theorem 1.4, we get that if S has genus 0then any connected, Mod( S )-invariant subgraph of C ( S ) has finite diameter;compare with Corollary 1.3 above.In the light of Theorem 1.4, a natural problem is to decide whetherNonSep( S ) (resp. NonSep ∗ ( S )) is hyperbolic for S a surface of genus g andwith infinitely many punctures. As we will see in Proposition 6.1 below,the answer is positive if and only if NonSep( S ) (resp. NonSep ∗ ( S g,n )) is hy-perbolic uniformly in n ; compare with part (2a) of Theorem 1.1 above. Weremark that NonSep( S g,n ) is known to be hyperbolic by the work of Masur-Schleimer [14] and Hamensd¨adt [10], although the hyperbolicity constantmay well depend on S . Similary, NonSep ∗ ( S g,n ) is conjecturally hyperbolicby the work of Masur-Schleimer [14], since every two of its witnesses inter-sect, see Example 4.3 in Section 4; on the other hand, even if this were thecase, the hyperbolicity constant may well depend on S , again.Next, we deal with the case when the genus of S is infinite: Theorem 1.5.
Let S be a connected orientable surface of infinite genusand no isolated ends. If G ( S ) is a Mod( S ) -invariant subgraph of C ( S ) , then diam( G ( S )) = 2 . Remark 1.6. If S has a finite number ≥ S )-invariant subgraphs of C ( S ) that arehyperbolic and have infinite diameter.The plan of the paper is as follows. Section 2 provides the necessarybackground on δ -hyperbolic spaces and quasi-isometries. In Section 3 we RAPHS FOR INFINITE-TYPE SURFACES 5 recall some facts about the space of ends of a surface. In Section 4 we brieflyintroduce mapping class groups and some of the combinatorial complexesone can associate to a surface. In Section 5 we prove Theorem 1.1 anddiscuss some of its consequences. Finally, Section 6 contains the proofs ofTheorems 1.4 and 1.5, together with some open questions.
Acknowledgements.
This project stemmed out of discussions with Juli-ette Bavard, and the authors are indebted to her for sharing her ideas andenthusiasm. We want to thank LAISLA and CONACYT’s Red tem´aticaMatem´aticas y Desarrollo for its support. This work started with a visit ofthe first named author to the UNAM (Morelia). He would like to thank theCentro de Ciencias Matem´aticas for its warm hospitality. He also thanksBrian Bowditch, Saul Schleimer for conversations. Finally, the authors aregrateful to Federica Fanoni and Nick Vlamis for discussions and for pointingout several errors in an earlier version of this draft.2.
Hyperbolic metric spaces
We briefly recall some notions on large-scale geometry that will be usedin the sequel. For a thorough discussion, see [9].
Definition 2.1 (Hyperbolic space) . Let X be a geodesic metric space. Wesay that X is δ -hyperbolic if there exists δ ≥ T ⊂ X is δ -thin : there exists a point c ∈ X at distance at most δ from everyside of T .We will simply say that a geodesic metric space is hyperbolic if it is δ -hyperbolic for some δ ≥ Definition 2.2 (Quasi-isometry) . Let (
X, d X ) , ( Y, d Y ) be two geodesic met-ric spaces. We say that a map f : ( X, d X ) → ( Y, d Y ) is a quasi-isometricembedding if there exist λ ≥ C ≥ λ d X ( x, x ′ ) − C ≤ d Y ( f ( x ) , f ( x ′ )) ≤ λd X ( x, x ′ ) + C, for all x, x ′ ∈ X . We say that f is a quasi-isometry if, in addition to (1),there exists D ≥ Y is contained in the D -neighbourhood of f ( X ).More concretely, for all y ∈ Y there exists x ∈ X with d Y ( y, f ( x )) ≤ D .We say that two spaces are quasi-isometric if there exists a quasi-isometrybetween them. The following is well-known: Proposition 2.3.
Suppose that two geodesic metric spaces
X, Y are quasi-isometric to each other. Then X is hyperbolic if and only if Y is hyperbolic. The ends of a surface
Let S be a connected orientable surface, possibly of infinite topologicaltype. We will briefly recall the definition of the space of ends of S , and referthe reader to [16] and [17] for a more thorough discussion on the space ofends of topological spaces and surfaces respectively. JAVIER ARAMAYONA & FERR ´AN VALDEZ
Definition 3.1 (Exiting sequence) . An exiting sequence is a collection U ⊇ U ⊇ . . . of connected open subsets of S , such that:(1) U n is not relatively compact, for any n ;(2) The boundary of U n is compact, for all n ;(3) Any relatively compact subset of S is disjoint from U n , for all butfinitely many n .We deem two exiting sequences to be equivalent if every element of thefirst sequence is contained in some element of the second, and vice-versa.An end of S is defined as an equivalence class of exiting sequences, and wewrite Ends( S ) for the set of ends of S . We put a topology on Ends( S ) byspecifying the following basis: given a subset U ⊂ S with compact boundary,let U ∗ be the set of all ends of S that have a representative exiting sequencethat is eventually contained in U .The following theorem is a special case of Theorem 1.5 in [16]: Theorem 3.2.
Let S be a connected orientable surface. Then Ends( S ) istotally disconnected, separable, and compact; in particular, it is a subset ofa Cantor set. We now proceed to describe the classification theorem for connected ori-entable surfaces of infinite type [17]. Before this, we need some notation.Say that an end of S is planar if it has a representative exiting sequencewhose elements are eventually planar; otherwise it is said to be non-planar .We denote by Ends p ( S ) and Ends n ( S ), respectively, the subspaces of planarand non-planar ends of S . Clearly Ends( S ) = Ends p ( S ) ⊔ Ends n ( S ). In [17],Richards proved: Theorem 3.3 ([17]) . Let S and S be two connected orientable surfaces.Then S and S are homeomorphic if and only if they have the same genus,and Ends n ( S ) ⊂ Ends( S ) is homeomorphic to Ends n ( S ) ⊂ Ends( S )) as nested topological spaces. That is, there exist a homeomorphism h :Ends( S ) → Ends( S ) whose restriction to Ends n ( S ) defines a homeomor-phism between Ends n ( S ) and Ends n ( S ) . We remark that this theorem was later extended by Prishlyak and Mis-chenko [15] to surfaces with non-empty boundary.4.
Arcs, curves, and witnesses
In this section we will introduce the necessary definitions about arcs andcurves that appear in our results. Throughout, let S be a connected, ori-entable surface of infinite topological type. Let Π be a (possibly empty)set of marked points on S , which we feel free to regard as marked points,punctures, or (planar) ends of S .4.1. Mapping class group.
The mapping class group Mod( S ) is the groupof self-homeomorphisms of S that preserve Π setwise, up to isotopy preserv-ing Π setwise. Given a (possibly empty) finite subset P of Π, we define RAPHS FOR INFINITE-TYPE SURFACES 7
Mod(
S, P ) to be the subgroup of Mod( S ) whose every element preserves P setwise. Observe that Mod( S, ∅ ) = Mod( S ).4.2. Arcs and curves.
By a curve on S we mean the isotopy class of asimple closed curve on S that does not bound a disk with at most onepuncture. An arc on S is the isotopy class of a simple arc on S with bothendpoints in Π.The arc and curve graph AC ( S ) of S is the simplicial graph whose verticesare all arcs and curves on S , and where two vertices are adjacent in AC ( S ) ifthey have disjoint representatives on S . As is often the case, we turn AC ( S )into a geodesic metric space by declaring the length of each edge to be 1.Observe that Mod( S ) acts on AC ( S ) by isometries. As mentioned in theintroduction, we will concentrate in subgraphs of A ( S ) that are invariantunder big subgroups of Mod( S ). More concretely, we have the followingdefinition: Definition 4.1 (Sufficient invariance) . We say that a subgraph G ( S ) of AC ( S ) is sufficiently invariant if there exists a (possibly empty) subset P ofΠ such that Mod( S, P ) acts on G ( S ).We will be interested in various standard Mod( S )-invariant subgraphs of AC ( S ), whose definition we now recall.The arc graph A ( S ) is the subgraph of AC ( S ) spanned by all verticesof AC ( S ) that correspond to arcs on S ; note that A ( S ) = ∅ if and only ifΠ = ∅ . Observe that if S has infinitely many punctures then A ( S ) has finitediameter.Similarly, the curve graph C ( S ) is the subgraph spanned by those verticesthat correspond to curves on S . Note that C ( S ) has diameter 2 for everysurface of infinite type.The nonseparating curve graph NonSep( S ) is the subgraph of C ( S ) spannedby all nonseparating curves on S . A related graph is the augmented non-separating curve graph NonSep ∗ ( S ), whose vertices are curves that either donot separate S , or else bound a disk containing every puncture of S . Notethat these graphs have diameter 2 if S has infinite genus.Finally, the outer curve graph Outer( S ) is the subgraph of C ( S ) spannedby those curves α that bound a disk with punctures on S , and such thatboth components of S − α contain at least one puncture of S . Observe thatOuter( S ) = ∅ if S is closed or has exactly one puncture, and that Outer( S )has finite diameter if S has infinitely many punctures.As the reader may suspect at this point, these observations constitute themain source of inspiration behind the statements of Theorems 1.4 and 1.5.4.3. Witnesses.
Let S be a connected orientable surface of infinite type,and G ( S ) a connected subgraph of AC ( S ). As mentioned in the introduction,we will use the following notion, originally due to Schleimer [18]: Definition 4.2 (Witness) . A witness of G ( S ) is an essential subsurface Y ⊂ S such that every vertex of G ( S ) intersects Y essentially. JAVIER ARAMAYONA & FERR ´AN VALDEZ
Observe that if Y is a witness of G ( S ) and Z is a subsurface of S suchthat Y ⊂ Z , then Z is also a witness. Example 4.3.
For the sake of concreteness, let S be a connected orientablesurface of finite genus g , possibly with infinitely many punctures.(1) If G ( S ) = A ( S ), then Y ⊂ S is a witness if and only if Y containsevery puncture of S .(2) If G ( S ) = C ( S ), then Y ⊂ S is a witness if and only if Y = S .(3) If G ( S ) = NonSep( S ), then Y ⊂ S is a witness if and only if Y hasgenus g .(4) Let G ( S ) = NonSep ∗ ( S ), and suppose S has at least two puncturesso that NonSep ∗ ( S ) = NonSep( S ). Then Y ⊂ S is a witness if andonly if Y has genus g and at least one puncture.5. Subgraphs of the arc graph
In this section we give a proof of Theorem 1.1. The main tool is thefollowing variant of Masur-Minsky’s subsurface projections [13]:
Subsurface projections.
Let Y be a witness of G ( S ), and suppose Y isnot homeomorphic to an annulus. There is a natural projection π Y : G ( S ) → A ( Y )defined by setting π Y ( v ) to be any connected component of v ∩ Y . In par-ticular, π Y ( v ) = v for every v ⊂ Y ; in other words, the restriction of π Y to G ( Y ) is the identity. Observe that the definition of π Y involves a choice,but any two such choices are disjoint and therefore at distance at most 1 in A ( Y ). The same argument gives: Lemma 5.1.
Let S be a surface and Y an essential subsurface not homeo-morphic to an annulus. If u, v are disjoint arcs which intersect Y essentially,then π Y ( u ) and π Y ( v ) are disjoint. For technical reasons, which will become apparent in the proof of Lemma5.3 below, we will be interested in subgraphs of A ( S ) for which the subsur-face projections defined above satisfy the following property: Definition 5.2 (Projection property) . We say that a subgraph G ( S ) ⊂ A ( S )has the projection property if, for every finite-type witness Y of G ( S ), thegraphs π Y ( G ( S )) and G ( Y ) are quasi-isometric via a quasi-isometry that isthe identity on G ( Y ) and whose constants do not depend on Y .As mentioned in the introduction, we remark that deciding whether agiven explicit subgraph of A ( S ) has the projection property is normallyeasy to check; see Remark 5.5 below.The following lemma, which is a small variation of Corollary 4.2 in [1], isthe main ingredient in the proof of Theorem 1.1. We note that this is thesole instance in which we will make use of the assumption that G ( S ) has theprojection property. RAPHS FOR INFINITE-TYPE SURFACES 9
Lemma 5.3.
Let S be a surface of infinite type, and G ( S ) ⊂ A ( S ) a con-nected subgraph with the projection property. Then, for every finite-typewitness Y of G ( S ) , the subgraph G ( Y ) is uniformly quasi-isometrically em-bedded in G ( S ) .Proof. Let u, v be arbitrary vertices of G ( Y ). First, observe that since G ( Y ) ⊂ G ( S ), we have d G ( S ) ( u, v ) ≤ d G ( Y ) ( u, v ) . To show a reverse coarse inequality, we proceed as follows. Consider ageodesic γ ⊂ G ( S ) between u and v . The projected path π Y ( γ ) is a path in π Y ( G ( S )) between u = π Y ( u ) and v = π Y ( v ), andlength π Y ( G ( S )) ( π Y ( γ )) ≤ length G ( S ) ( γ ) , by Lemma 5.1. In particular, d π Y ( G ( S )) ( u, v ) ≤ d G ( S ) ( u, v ) . Since G ( S ) has the projection property, there exist constants L ≥ C ≥ S ) such that d G ( Y ) ( u, v ) ≤ L · d π Y ( G ( S )) ( u, v ) + C, and thus the result follows by combining the above two inequalities. (cid:3) We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Let S be a connected, orientable surface of infinitetype, and denote by Π the set of marked points of S . Let G ( S ) be a con-nected subgraph of A ( S ) with the projection property, and invariant underMod( S, P ) for some P ⊂ Π finite (possibly empty).We first prove part (1); in fact, we will show that the diameter of G ( S ) isat most 4. Let u, v be two arbitrary distinct vertices of G ( S ). We first claimthat there exists w ∈ G ( S ) that intersects both u and v a finite number oftimes. To see this, observe that if u and v have no endpoints in common, thentheir intersection number is finite and thus we may take w = u . Supposenow that u and v share two distinct endpoints p, p ′ ∈ Π. Then there existsan element h in the subgroup of Mod( S, P ) whose every element fixes p and p ′ , such that w = h ( u ) intersects both u and v a finite number of times, asdesired. The rest of cases are dealt with in a similar fashion. This finishesthe proof of the claim.Continuing with the proof, we now claim that there is a vertex z ∈ G ( S )that is disjoint from v and w . Indeed, consider the surface F ( v, w ) filled by v and w , which has finite type since v and w intersect finitely many times.Since every witness of G ( S ) has infinitely many punctures, by assumption,we deduce that F ( v, w ) is not a witness, and therefore there exists a vertex z ∈ G ( S ) that does not intersect F ( v, w ). Using the same reasoning, thereexists a vertex z ′ ∈ G ( S ) that is disjoint from u and w . Thus, u → z ′ → w → z → v is a path of length at most 4 in G ( S ) between u and v , as desired.We now proceed to prove part (2), arguing along similar lines to [1]. Toshow that G ( S ) has infinite diameter we proceed as follows. By assumption,there exists a witness Y of G ( S ) with finitely many punctures. After re-placing Y by a finite-type surface containing every puncture of Y , we mayassume that Y has finite type and Mod( Y ) contains a pseudo-Anosov. Es-sentially by Luo’s argument proving that the curve graph of a finite-typesurface has infinite diameter (see the comment after Proposition 3.6 of [12]),we deduce that G ( Y ) has infinite diameter. Since G ( Y ) is quasi-isometricallyembedded in G ( S ), by Lemma 5.3, it follows that G ( S ) has infinite diameter,as desired.Next, we establish part (2a) Assume that every two witnesses of G ( S )intersect, and suppose first that there exists δ = δ ( S ) such that G ( Y ) is δ -hyperbolic, for every finite-type witness Y . We will prove that G ( S ) is δ -hyperbolic. To this end, consider a geodesic triangle T ⊂ G ( S ), and let Z bea witness of G ( S ) containing every vertex of T , so that T may be viewed asa triangle in G ( Z ). First, if Z has infinitely many punctures then G ( Z ) hasdiameter ≤
4, by the proof of part (1). Therefore T has a 4-center in G ( Z ),and thus also in G ( S ), as desired. Assume now that Z has finitely manypunctures; in this case, again up to replacing Z by a connected, finite-typesurface containing every puncture of Z , we may in fact assume that Z isconnected and has finite type. Since G ( Z ) is δ -hyperbolic, by assumption, T has a δ -center in G ( Z ), and thus also in G ( S ). Since T is arbitrary anduniformly thin, we obtain that G ( S ) is hyperbolic, as claimed.Using a very similar argument to the one just given, we also deduce thatthe hyperbolicity of G ( S ) implies that of G ( Y ), for every finite-type witness Y of G ( S ). This finishes the proof of part (2a).It remains to show part (2b). Assume that G ( S ) has two disjoint witnesses Y, Z ⊂ S , each of finite type. As remarked above, after enlarging Y and/or Z if necessary we may assume that G ( Y ) and G ( Z ) have infinite diameter.Since Y and Z are witnesses, the projection maps π Y and π Z are well-defined.Therefore there is a projection map π : G ( S ) → A ( Y ) × A ( Z )which is simply the map π Y × π Z . Using this projection and the same argu-ments as in the proof of Lemma 5.3, the fact that G ( S ) has the projectionproperty implies that G ( S ) contains a quasi-isometrically embedded copyof G ( Y ) × G ( Z ). By choosing a bi-infinite quasi-geodesic in G ( Y ) and in G ( Z ), we obtain G ( S ) contains a quasi-isometrically embedded copy of Z ,as claimed. This finishes the proof of part (2b), and hence of Theorem1.1. (cid:3) Remark 5.4.
As mentioned in the introduction, the proof of part (1) ofTheorem 1.1 does not use that G ( S ) has the projection property; this willbe crucial for Corollaries 1.3 and 5.6 below. RAPHS FOR INFINITE-TYPE SURFACES 11
Consequences.
We proceed to discuss some of the consequences ofTheorem 1.1 mentioned in the introduction, starting with Corollary 1.2.Before doing so, we need some definitions from [1]. Let Π be the set ofmarked points of S , where we assume that Π = ∅ . As always, we will feelfree to view the elements of Π as marked points, punctures, or (planar) endsof S .We say that a marked point p ∈ Π is isolated if it is isolated in Π, wherethe latter is equipped with the subspace topology (here we are viewing Π asa set of marked points on S ). Let P ⊂ Π be a non-empty finite subset ofmarked points on S . Define A ( S, P ) as the subgraph of A ( S ) spanned bythose arcs that have at least one endpoint in P . Note that Mod( S, P ) actson A ( S, P ), and hence A ( S, P ) is sufficiently invariant.
Remark 5.5.
The graphs A ( S, P ) have the projection property: if Y isa finite-type witness of S then π Y ( A ( S, P )) is uniformly quasi-isometric to A ( Y, P ), which is G ( Y ) for G ( S ) = A ( S, P ). The proof that both graphsare quasi-isometric boils down to the fact that, for v ∈ A ( S, P ), there is atleast one component of v ∩ Y that has an endpoint in P , which we can useto define a subsurface projection map with nice properties.We are now in a position to prove Corollary 1.2: Proof of Corollary 1.2.
Since P is finite and every puncture is isolated, thereexists a witness containing only finitely many punctures (any finite-typesurface containing P will do). Now, part (2) of Theorem 1.1 applies with G ( S ) = A ( S, P ), and thus A ( S, P ) has infinite diameter. Moreover, if Y isa finite-type witness of A ( S, P ) then G ( Y ) = A ( Y, P ), which is 7-hyperbolicby [11]. (cid:3)
We now prove Corollary 1.3:
Proof of Corollary 1.3.
Let S be as in the statement, and G ( S ) be a con-nected, Mod( S )-invariant subgraph of A ( S ). As S has no isolated punc-tures, using for instance the classification theorem for infinite-type surfaces[17] we deduce that there exists an infinite sequence of distinct and pairwisedisjoint vertices of G ( S ) such that every two distinct arcs in the sequencehave no endpoints in common. In particular, every witness of G ( S ) musthave an infinite number of punctures, and so the result follows from part (1)of Theorem 1.1. (cid:3) In addition, we recover the following observation due to Bavard (statedas Proposition 3.5 of [1]):
Corollary 5.6.
Suppose P ⊂ Π contains a puncture that is not isolated.Then A ( S, P ) has finite diameter. Finally, one could define A ( S, P, Q ) to be, for disjoint finite subsets
P, Q of isolated punctures, the subgraph of A ( S ) spanned by those arcs that have one endpoint in P and the other in Q . In this situation we have the followingresult, also due to Bavard (unpublished): Corollary 5.7.
The graph A ( S, P, Q ) is not hyperbolic.Proof. Observe that Y is a witness of A ( S, P, Q ) if and only if it contains P or Q . In particular, there are two disjoint witnesses of finite type, and part(2b) of Theorem 1.1 applies. (cid:3) Subgraphs of the curve graph
In this section we deal with connected, Mod( S )-invariant subgraphs ofthe curve graph, proving Theorems 1.4 and 1.5. As mentioned in the intro-duction, we restrict our attention to the case when S has no isolated ends,which in turn implies that Ends( S ) is homeomorphic to a Cantor set, by theclassification theorem for infinite-type surfaces [17] described in Section 3.We first prove Theorem 1.4. The arguments we will use are similar inspirit to those used in the previous section, but adapted to this particularsetting. Proof of Theorem 1.4.
Let S be a connected, orientable surface of infinitetype, with finite genus and no isolated ends. Let G ( S ) be a connected,Mod( S )-invariant subgraph of C ( S ).Suppose first that G ( S ) ∩ Outer( S ) = ∅ . We want to conclude thatdiam( G ( S )) = 2. To this end, let α and β be arbitrary vertices of G ( S ).If α and β are disjoint, there is nothing to prove, so assume that i ( α, β ) = 0.Let F ( α, β ) be the subsurface of S filled by α and β , which has finite topo-logical type since α and β are compact. Therefore, there exists a connectedcomponent Y of S − F ( α, β ) that has infinitely many punctures. Now, thefact that Ends( S ) is a Cantor set and the classification theorem for infinite-type surfaces, together imply that Mod( S ) acts transitively on Outer( S ).Thus there exists h ∈ Mod( S ) and γ ∈ Outer( S ) such that h ( γ ) ⊂ Y . Inparticular, h ( γ ) is disjoint from both α and β and hence d G ( S ) ( α, β ) = 2.Hence from now on, we assume that G ( S ) ∩ Outer( S ) = ∅ . Suppose firstthat, in addition, G ( S ) ∩ NonSep( S ) = ∅ , and so every element of G ( S ) is acurve that either separates S into two surfaces of positive genus, or cuts offa disk containing every puncture of S . We claim that G ( S ) has two disjointwitnesses, and thus fails to be hyperbolic. To construct these witnesses,consider a multicurve M consisting of genus(S) + 1 non-separating curveson S such that S − M = W ⊔ W , with W i a surface of genus 0 for i = 1 , S . By construction, W and W arewitnesses for G ( S ). Let P i be the finite subset of punctures of W i comingfrom the elements of M . Using subsurface projections as in the previoussection gives a quasi-isometric embedding A ( W , P ) × A ( W , P ) → G ( S ) , thus obtaining a quasi-isometrically embedded copy of Z inside G ( S ). Inparticular, G ( S ) is not hyperbolic and has infinite diameter. RAPHS FOR INFINITE-TYPE SURFACES 13
Hence, from now on we assume that G ( S ) ∩ NonSep( S ) = ∅ , which inparticular implies that NonSep( S ) ⊂ G ( S ), since Mod( S ) acts on G ( S ).There are two cases to consider: Case I.
No vertex of G ( S ) bounds a disk with punctures. In this case, we claim:
Claim.
The inclusion map NonSep( S ) ֒ → G ( S ) is a quasi-isometry. Proof of Claim.
We begin by showing that the inclusion map is a quasi-isometric embedding. In fact, more is true: we will prove that, given α, β ∈ NonSep( S ) and a geodesic σ in G ( S ) between them, we can modify σ to ageodesic σ ′ in NonSep( S ) of the same length. (We remark that this argumentis contained in the proof that the nonseparating curve complex is connected;see Theorem 4.4 of [5].) Let γ ∈ σ be a curve in G ( S ) − NonSep( S ). Byhypothesis, S − γ = Y ∪ Z , where Y and Z both have positive genus. Let γ L and γ R be the vertices of σ preceding (resp. following) γ . The assumptionthat σ is geodesic implies that either γ L , γ R ⊂ Y or γ L , γ R ⊂ Z ; supposefor the sake of concreteness that we are in the former case. Since Z haspositive genus, it contains a nonseparating curve γ ′ which, by construction,is disjoint from γ L and γ R . Replacing γ by γ ′ on σ produces a geodesic in G ( S ) with a strictly smaller number of separating curves.At this point, we know that the inclusion map NonSep( S ) ֒ → G ( S ) isa (quasi-)isometric embedding. To see that it is a quasi-isometry, observethat every element of G ( S ) is at distance at most 1 from an element ofNonSep( S ). This finishes the proof of the claim. (cid:3) Case II.
There is a vertex of G ( S ) which bounds a disk with punctures. Since G ( S ) ∩ Outer( S ) = ∅ , we get an inclusion NonSep ∗ ( S ) ⊂ G ( S ).Using the same arguments as in the previous claim, we obtain: Fact.
The inclusion map NonSep ∗ ( S ) ֒ → G ( S ) is a quasi-isometry.In the light of the claims above, in order to finish the proof of the theoremit suffices to show: Claim.
The graphs NonSep( S ) and NonSep ∗ ( S ) have infinite diameter. Proof of Claim.
We prove the result for NonSep( S ), as the case of NonSep ∗ ( S )is totally analogous.In a similar fashion to what we did in the previous section, we are goingto prove that, for every finite-type witness Y , the subgraph NonSep( Y ) isquasi-isometrically embedded in NonSep( S ); once this has been done, theclaim will follow since NonSep( Y ) has infinite diameter, which again maybe deduced using Luo’s argument showing that the curve graph has infinitediameter; see Proposition 3.6 of [12].In this direction, let Y be a finite-type witness of NonSep( S ); in otherwords, Y is a finite-type subsurface of S of the same genus as S , see Ex-ample 4.3 above. Let A ( Y, ∂Y ) be the subgraph of A ( Y ) spanned by those vertices that have both endpoints on ∂Y . Similarly, let A NonSep( Y ) be thesubgraph of AC ( Y ) spanned by the vertices of NonSep( Y ) ∪ A ( Y, ∂Y ). Theinclusion map NonSep( Y ) ֒ → A NonSep( Y )is a quasi-isometry, where the constants do not depend on Y ; to see this,one may use the standard argument to show that the embedding of C ( Y )into AC ( Y ) is a uniform quasi-isometry (see for instance Exercise 3.15 of[18]). Now, as in the previous section there is a subsurface projection π Y : NonSep( S ) → A NonSep( Y )that associates, to an element of NonSep( S ), its intersection with Y . Usingan analogous reasoning to that of Lemma 5.3, we obtain that NonSep( Y )is uniformly quasi-isometrically embedded in NonSep( S ), as desired. Thisfinishes the proof of the claim, and thus that of Theorem 1.4. (cid:3)(cid:3) The graphs NonSep( S ) and NonSep ∗ ( S ) have an intriguing geometricstructure. Indeed, using a small variation of the proof of Theorem 1.1, weobtain: Proposition 6.1.
Let S be a connected surface of finite genus g and with in-finitely many punctures. Then NonSep( S ) (resp. NonSep ∗ ( S ) ) is hyperbolicif and only if NonSep( S g,n ) (resp. NonSep ∗ ( S g,n ) ) is hyperbolic uniformlyin n . In the light of Example 4.3, the finite-type witnesses of NonSep( S ) andNonSep ∗ ( S ) are precisely the subsurfaces of the form S g,n ; compare withpart (3) of Theorem 1.1. Proof of Proposition 6.1.
Again, we argue only for NonSep( S ), as the othercase is very similar. Let T be a geodesic triangle in NonSep( S ). Since T has finitely many vertices and curves are compact, there exists a finite-typesubsurface Y of S that contains every element of T . Thus we can view T as ageodesic triangle in NonSep( Y ). If NonSep( S g,n ) is hyperbolic uniformly in n , there is δ = δ ( g ) such that T has a δ -center α ∈ NonSep( Y ) (with respectto the distance function in NonSep( Y )). In particular, α is at distance atmost δ from the sides of T , where distance is measured in NonSep( Y ), andhence is a δ -centre for T in NonSep( S ). Thus, NonSep( S ) is δ -hyperbolic.The other direction is analogous. (cid:3) As mentioned in the introduction, it is known that NonSep( S g,n ) is hy-perbolic [10, 14], but in principle the hyperbolicity constant may well de-pend on n . Similarly, NonSep ∗ ( S g,n ) is conjecturally hyperbolic by Masur-Schleimer’s principle that every two witnesses intersect [14], but even in thiscase the hyperbolicity constant could again depend on n . Thus we ask: RAPHS FOR INFINITE-TYPE SURFACES 15
Question 6.2.
For fixed g , are NonSep( S ) and NonSep ∗ ( S g,n ) hyperbolicuniformly in n ? More generally, are they hyperbolic uniformly in both g and n ? Finally, we prove Theorem 1.5:
Proof of Theorem 1.5.
Let S be a connected orientable surface of infinitegenus with no isolated ends; in other words, Ends( S ) is homeomorphic to aCantor set. Consider a Mod( S )-invariant subgraph G ( S ) of C ( S ). Let α and β be arbitrary vertices of G ( S ), noting again that the subsurface F ( α, β )filled by them has finite type. Using the classification theorem for infinite-type surfaces, there exists h ∈ Mod( S ) such that h ( α ) ⊂ S − F ( α, β ), thusgiving a path of length 2 in G ( S ) between α and β . (cid:3) References [1] J. Aramayona, A. Fossas, H. Parlier. Arc and curve graphs for infinite-type surfaces.
Preprint 2015 .[2] J. Bavard, Hyperbolicit´e du graphe des rayons et quasi-morphismes sur un gros groupemodulaire.
Geometry and Topology
20 (2016).[3] D. Calegari.
Big mapping class groups and dynamics.
Geometry and the imagina-tion, h ttp://lamington.wordpress.com/2009/06/ 22/big-mapping-class-groups-and-dynamics/, 2009[4] M. Durham, F. Fanoni, N. Vlamis, Graphs of curves on infinite-type surfaces withmapping class group actions. Preprint (2016).[5] B. Farb, D. Margalit, A primer on mapping class groups . Princeton MathematicalSeries, 49.[6] A. Fossas, H. Parlier. Curve graphs on surfaces of infinite type.
Ann. Acad. Sci. Fenn.Math.
40 (2015).[7] L. Funar, C. Kapoudjian. An Infinite Genus Mapping Class Group and Stable Coho-mology.
Commun. Math. Physics
287 (2009).[8] L. Funar, C. Kapoudjian, The braided Ptolemy-Thompson group is finitely presented,
Geom. Topol.
12 (2008).[9] E. Ghys, P. de la Harpe (editors),
Sur les groupes hyperboliques d’apr`es MikhaelGromov.
Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988.Progress in Mathematics, 83. Birkhuser Boston, Inc., Boston, MA, 1990.[10] U. Hamenst¨adt. Hyperbolicity of the graph of nonseparating multicurves.
Algebr.Geom. Topol.
14 (2014).[11] S. Hensel, P. Przytycki, R. Webb,
J. Eur. Math. Soc. (JEMS) 17 (2015)[12] H. A. Masur, Y. N. Minsky. Geometry of the complex of curves. I. Hyperbolicity.
Invent. Math.
138 (1999).[13] H. A. Masur, Y. N. Minsky. Geometry of the complex of curves. II. Hierarchicalstructure.
Geom. Funct. Anal.
10 (2000).[14] H. Masur, S. Schleimer. The geometry of the disk complex,
Journal of the AmericanMathematical Society , 26 (2013).[15] A. O. Prishlyak, K. I. Mischenko. Classification of noncompact surfaces with bound-ary.
Methods Funct. Anal. Topology
13 (2007)[16] F. Raymond. The end point compactification of manifolds.
Pacific J. Math.
10 (1960).[17] I. Richards. On the classification of noncompact surfaces,
Trans. Amer. Math. Soc.
106 (1963). [18] S. Schleimer,
Notes on the curve complex . Available from h ttp://homepages.warwick.ac.uk/ masgar/math.html Departamento de Matem´aticas, Universidad Aut´onoma de Madrid &Instituto de Ciencias Matem´aticas, CSIC. [email protected]