Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings
aa r X i v : . [ m a t h . G T ] J a n Random subgroups of acylindrically hyperbolicgroups and hyperbolic embeddings
Joseph Maher and Alessandro SistoNovember 11, 2018
Abstract
Let G be an acylindrically hyperbolic group. We consider a randomsubgroup H in G , generated by a finite collection of independent randomwalks. We show that, with asymptotic probability one, such a randomsubgroup H of G is a free group, and the semidirect product of H actingon E ( G ) is hyperbolically embedded in G , where E ( G ) is the uniquemaximal finite normal subgroup of G . Contents Introduction
Acylindrically hyperbolic groups have been defined by Osin, who showed in[Osi16] that several approaches to groups that exhibit rank one behaviour[BF02, Ham08, DGO11, Sis16b] are all equivalent; see Section 2 for theprecise definition. Acylindrically hyperbolic groups form a very large classof groups that vastly generalises the class of non-elementary hyperbolicgroups and includes non-elementary relatively hyperbolic groups, mappingclass groups [MM99, Bow06, PS16], Out( F n ) [BF14], many groups actingon CAT(0) spaces [BHS14,CM16,Gen16,Hea16,Sis16b], and many others,see for example [GS16, MO15, Osi15].Acylindrical hyperbolicity has strong consequences: For example, ev-ery acylindrically hyperbolic group is SQ-universal (in particular it hasuncountably many pairwise non-isomorphic quotients) and its boundedcohomology is infinite dimensional in degrees 2 [HO13] and 3 [FPS15].These results all rely on the notion of hyperbolically embedded subgroup ,as defined in [DGO11] (see Section 2 for the definition), and in fact, onvirtually free hyperbolically embedded subgroups. Hyperbolically embed-ded subgroups are hence very important for the study of acylindricallyhyperbolic groups, and in fact they enjoy several nice properties such asalmost malnormality [DGO11] and quasiconvexity [Sis16a].In this paper we show that, roughly speaking, a random subgroup H ofan acylindrically hyperbolic group is free and virtually hyperbolically em-bedded. We now give a slightly simplified version of our main theorem, seeSection 2 for a more refined statement. We shall write E ( G ) for the max-imal finite normal subgroup of G , which for G acylindrically hyperbolicexists by [DGO11, Theorem 6.14], and given a subgroup H < G , we shallwrite HE ( G ) for the subset of G consisting of { hg | h ∈ H, g ∈ E ( G ) } ,which in this case is a subgroup, as E ( G ) is normal. We say that a prop-erty P holds with asymptotic probability one if the the probability P holdstends to one as n tends to infinity. Theorem 1.
Let G be an acylindrically hyperbolic group, with maximalfinite normal subgroup E ( G ) , and let µ be a probability measure on G whose support is finite and generates G as a semigroup. For k, n positiveintegers, let H k,n denote the subgroup of G generated by k independentrandom walks generated by µ , each of length n , which we shall denote by w i,n .Then for each fixed k , the probability that each of the following eventsoccurs with asymptotic probability one.1. The subgroup H is freely generated by the { w ,n , . . . w k,n k } andquasi-isometrically embedded. . The subgroup HE ( G ) is a semidirect product H ⋉ E ( G ) , and is hy-perbolically embedded in G . The first part of Theorem 1 was previously shown by Taylor and Tiozzo[TT16], and they apply this result to study random free group and surfacegroup extensions. The second part is definitely the main contribution ofthis paper. For the experts, we note that we can fix the generating setwith respect to which H ⋉ E ( G ) is hyperbolically embedded, see Theorem8. The study of generic properties of groups in geometric group theorygoes back at least to Gromov [Gro87, Gro03], and we make no attempt tosurvey the substantial literature on this topic, see for example [GMO10]for a more thorough discussion, though we now briefly mention someclosely related results. This model of random subgroups is used in Guiv-arc’h’s [Gui90] proof of the Tits alternative for linear groups, and is alsodeveloped by Rivin [Riv10] and Aoun [Aou11], who proves that a randomsubgroup of a non-virtually solvable linear group is free and undistorted.Gilman, Miasnikov and Osin [GMO10] consider subgroups of hyperbolicgroups generated by k elements arising from nearest neighbour randomwalks on the corresponding Cayley graph, and they show that the prob-ability that the resulting group is a quasi-isometrically embedded freegroup, freely generated by the k unreduced words of length n , tends toone exponentially quickly in n . The fact that the k elements freely gener-ate a free group as n becomes large was shown earlier for free groups byJitsukawa [Jit02] and Martino, Turner and Ventura, [MTV], and for braidgroups by Myasnikov and Ushakov [MU08]. Our argument makes useof particular group elements which we call strongly asymmetric, namelyloxodromic elements g contained in maximal cyclic subgroups which areequal to h g i × E ( G ). We say a loxodromic element g is weakly asymmet-ric if it is contained in a maximal cyclic subgroup which is a semidirectproduct h g i ⋉ E ( G ), see Section 2 for full details. Masai [Mas14] haspreviously shown that random elements of the mapping class group arestrongly asymmetric, and the argument we present uses similar methodsin the context of acylindrically hyperbolic groups. Mapping class groupshave trivial maximal finite normal subgroups, except for a finite list ofsurfaces in which E ( G ) is central, see for example [FM12, Section 3.4], soin the case of the mapping class groups there is no distinction betweenweakly and strongly asymmetric elements.Theorem 1 is used in [HS16] to study the bounded cohomology ofacylindrically hyperbolic groups. .1 Acknowledgments The first author acknowledges the support of PSC-CUNY and the SimonsFoundation.This material is based upon work supported by the National ScienceFoundation under Grant No. DMS-1440140 while the authors were inresidence at the Mathematical Sciences Research Institute in Berkeley,California, during the Fall 2016 semester.
We say a geodesic metric space (
X, d X ), which need not be proper, is Gromov hyperbolic , δ -hyperbolic or just hyperbolic , if there is a number δ > X satisfies the δ -slim triangle condition, i.e. for any geodesic triangle, any side is contained in the δ -neighbourhood of the other two sides.Let G be a countable group which acts on a hyperbolic space X byisometries. We say the action of G on X is non-elementary if G containstwo hyperbolic elements with disjoint pairs of fixed points at infinity.We say a group G acts acylindrically on a Gromov hyperbolic space X ,if there are real valued functions R and N such that for every number K >
0, and for any pair of points x and y in X with d X ( x, y ) > R ( K ),there are at most N ( K ) group elements g in G such that d X ( x, gx ) K and d X ( y, gy ) K . We shall refer to R and N as the acylindricalityfunctions for the action. This definition is due to Sela [Sel97] for trees,and Bowditch [Bow06] for general metric spaces.We say a group G acts acylindrically hyperbolically on a space X , if X is hyperbolic, and the action is non-elementary and acylindrical. A groupis acylindrically hyperbolic if it admits an acylindrically hyperbolic actionon some space X .A finitely generated subgroup H in G is quasi-isometrically embedded in X , if for any choice of word metric d H , and any basepoint x ∈ X ,there are constants K and c such that for any two elements h and h in H , 1 K d X ( h x , h , x ) − c d H ( h , h ) Kd X ( h x , h , x ) + c. We say that a subgroup H of G is geometrically separated in X , if for each x ∈ X and R ≥ B ( R ) so that for each g ∈ G \ H , we havethat the diameter of N R ( gHx ) ∩ N R ( Hx ) is bounded by B , where N R denotes the metric R -neighborhood in X .For the remainder of this paper fix an ayclindrically hyperbolic group G . We shall write E ( G ) for the maximal finite normal subgroup of G , hich exists and is unique by [DGO11, Theorem 6.14]. Given an element g ∈ G , let E ( g ) be the maximal virtually cyclic subgroup containing g ,which is well-defined by work of Bestvina and Fujiwara [BF02]. For ahyperbolic element g , let Λ( g ) = { λ + ( g ) , λ ( g ) } be the set consisting ofthe pair of attracting and repelling fixed points for g in ∂X . We shallwrite stab(Λ( g )) for the stabilizer of this set in G . Dahmani, Guirardeland Osin [DGO11, Corollary 6.6] show that in fact E ( g ) = stab(Λ( g )) . For any hyperbolic element g , the group E ( g ) is always quasi-isometricallyembedded and geometrically separated.The subgroup E ( G ) acts trivially on the Gromov boundary ∂X , soin many applications it may be natural to consider G/E ( G ), which willhave a trivial maximal finite subgroup, and a reader interested in this caseshould feel free to assume E ( G ) is trivial, which simplifies the argumentsand statements in many places. We shall write h g , . . . , g k i for the sub-group of H generated by { g , . . . g k } , and in particular, h g i denotes thecyclic group generated by g . Recall that given a subgroup H < G , we willwrite HE ( G ) for the subset of G consisting of { hg | h ∈ H, g ∈ E ( G ) } ,which is a subgroup, as E ( G ) is normal. If H ∩ E ( G ) = { } , then thegroup HE ( G ) is a finite extension of H by E ( G ), but in general need notbe either a product H × E ( G ), or a semidirect product, which we shallwrite as H ⋉ E ( G ). The following observation is elementary, but we recordit as a proposition for future reference. Proposition 2.
Let G be a countable group acting acylindrically hyper-bolically on X , and let H be a subgroup of G with trivial intersection with E ( G ) , i.e. H ∩ E ( G ) = { } . Then the subgroup HE ( G ) is a semidirectproduct H ⋉ E ( G ) .Proof. The quotient ( HE ( G )) /E ( G ) corresponds to the set of cosets hE ( G ).If h is a non-trivial element of H then hE ( G ) = E ( G ), as H ∩ E ( G ) = { } .Therefore, ( HE ( G )) /E ( G ) is isomorphic to H , and the inclusion of H into HE ( G ) gives a section H → HE ( G ), i.e. a homomorphism whosecomposition with the quotient map is the identity on H . This impliesthat HE ( G ) is a split extension of ( HE ( G )) /E ( G ) by E ( G ), and hence asemidirect product H ⋉ E ( G ).If g is a hyperbolic element, then h g i is an infinite cyclic group, andthe subgroup E ( g ) always contains h g i E ( G ) = h g i ⋉ E ( G ), but maybe larger. For example, E ( g ) = E ( g ) but E ( g ) cannot be equal to h g i E ( G ), as E ( g ) contains g . Furthermore, the subgroup h g i E ( G )is quasi-isometrically embedded, but not geometrically separated, as g oarsely stabilizes this subgroup. We say that a group element g ∈ G is weakly asymmetric if E ( g ) is equal to h g i ⋉ E ( G ), and strongly asymmet-ric when E ( g ) is actually the product h g i × E ( G ). Strongly asymmetricelements are sometimes called special, though we do not use this termi-nology in this paper. We note that strongly asymmetric elements alwaysexist by [DGO11, Lemma 6.18]. Example 3.
Let S be a closed genus 2 surface, and let e S → S be a degree2 cover, so e S is a closed genus 3 surface. Let G be the mapping class groupof e S , which has trivial maximal finite normal subgroup, and let T ∼ = Z / Z be the subgroup of G consisting of covering transformations. Any pseudo-Anosov map g : S → S has a power which lifts to a map e g : e S → e S , whichcommutes with T , so h e g i × T < E ( e g ), and so e g is not weakly asymmetric.We now describe the particular model of random subgroups which weshall consider. A random subgroup of G with k generators is a subgroupwhose generators are chosen to be independent random walks of length n i on G . We will require the following restrictions on the probabilitydistributions µ i generating the random walks. We say a probability dis-tribution µ on G is non-elementary if the group generated by its supportis non-elementary. Definition 4.
Let G act acylindrically hyperbolically on X . We say thatthe probability distribution µ on G is ( G y X ) -admissible if the supportof µ generates a non-elementary subgroup of G containing a weakly asym-metric element, and furthermore, the support of µ has bounded image in X . The set of admissible measures depends on the action of G on X ,though we shall suppress this from our notation and just write admissible for ( G y X )-admissible. We shall write ˇ µ for the reflected probabilitydistribution ˇ µ ( g ) = µ ( g − ), and ˇ µ is admissible if and only if µ is admis-sible.We may now give a precise definition of our model for random sub-groups. Let µ , . . . , µ k be a finite collection of admissible probabilitydistributions on G . We shall write H ( µ , . . . , µ k , n , . . . , n k ) to denotethe subgroup generated by h w ,n , . . . w k,n k i , where each w i,n i is a groupelement arising from a random walk on G of length n i generated by µ i .To simplify notation we shall often just write H ( µ i , n i ), or just H , for H ( µ , . . . , µ k , n , . . . , n k ). We may now state our main result. Theorem 5.
Let G be a countable group acting acylindrically hyperboli-cally on the separable space X , and let E ( G ) be the maximal finite normalsubgroup of G . Let H ( µ , . . . , µ k , n , . . . , n k ) be a random subgroup of G ,where the µ i are admissible probability distributions on G . Then the prob-ability that each of the following three events occurs tends to one as min n i ends to infinity.1. All of the w i,n i are hyperbolic and weakly asymmetric.2. The subgroup H is freely generated by the { w i,n i } and quasi-isometricallyembedded in X , and so in particular HE ( G ) is a semidirect product H ⋉ E ( G ) .3. The subgroup H ⋉ E ( G ) is geometrically separated in X . In general the semidirect product H ⋉ E ( G ) need not be the product H × E ( G ) for random subgroups H , as shown below. Example 6.
Let G be a group acting acylindrically hyperbolically on X with trivial maximal finite normal subgroup E ( G ), which admits a splitextension 1 → F → G + → G → , which is not a product, for some finite group F . Such a split extensionis determined by a homomorphism φ : G → Aut( F ), where Aut( F ) is theautomorphism group of F . The maximal finite normal subgroup E ( G ) isequal to F . A random walk on G + pushes forward to a random walk on G , and then to a random walk on φ ( G ) < Aut( F ). As φ ( G ) is finite, therandom walk is asymptotically uniformly distributed. A hyperbolic groupelement g has E ( g ) = h g i × F if and only if the image of g in φ ( G ) istrivial, which happens with asymptotic probability 1 / | φ ( G ) | .In the next section, Section 2.1, we recall the definition of a hyperboli-cally embedded subgroup, and show how Theorem 1 follows from Theorem5. Osin [Osi16] showed that if a group is acylindrically hyperbolic, then thereis a (not necessarily finite) generating set Y , such that the Cayley graphof G with respect to Y , which we shall denote Cay( G, Y ), is hyperbolic,and the action of G on Cay( G, Y ) is acylindrical and non-elementary. Ingeneral, there are many choices of Y giving non-quasi-isometric acylindri-cally hyperbolic actions, for which different collections of subgroups willbe hyperbolically embedded, but for the remainder of this section we shallassume we have chosen some fixed Y .Let H be a subgroup of G ; we will write Cay( G, Y ⊔ H ) for the Cayleygraph of G with respect to the disjoint union of Y and H (so it mighthave double edges). The Cayley graph Cay( H, H ) is a complete subgraphof Cay(
G, Y ⊔ H ). We say a path p in Cay( G, Y ⊔ H ) is admissible ifit does not contain edges of Cay( H, H ), though it may contain edges ofnon-trivial cosets of H , and may pass through vertices of H . We define a estriction metric ˆ d H on H be setting ˆ d H ( h , h ) to be the minimal lengthof any admissible path in Cay( G, Y ⊔ H ) connecting h and h . If no suchpath exists we set ˆ d H ( h , h ) = ∞ .We say a finitely generated subgroup H of a finitely generated acylin-drically hyperbolic group G is hyperbolically embedded in G with respectto a generating set Y ⊂ G if the Cayley graph Cay( G, Y ⊔ H ) is hyperbolic,and ˆ d H is proper. We shall denote this by H ֒ → h ( G, Y ).We shall use the following sufficient conditions for a subgroup to behyperbolically embedded, due to Hull [Hul13, Theorem 3.16] and Antolin,Minasyan and Sisto [AMS16, Theorem 3.9, Corollary 3.10] (both refine-ments of Dahmani, Guirardel and Osin [DGO11, Theorem 4.42]), whichwe now describe.
Theorem 7. [Hul13, AMS16]
Suppose that G acts acylindrically hyper-bolically on X . Let H be a finitely generated subgroup of G , which isquasi-isometrically embedded and geometrically separated in X . Then H is hyperbolically embedded in G . Moreover, if X = Cay ( G, Y ) for some Y ⊆ G , then H ֒ → h ( G, Y ) . Theorem 1 now follows immediately from Theorem 5 and Theorem 7,choosing X to be Cay( G, Y ). In fact, we have the following refinement,which we record for future reference:
Theorem 8.
Let the finitely generated group G act acylindrically hyper-bolically on its Cayley graph Cay ( G, Y ) , and let E ( G ) be the maximalfinite normal subgroup of G . Let H ( µ , . . . , µ k , n , . . . , n k ) be a randomsubgroup of G , where the µ i are admissible probability distributions on G .Then the probability that each of the following events occurs tends to oneas min n i tends to infinity.1. The subgroup H is freely generated by the { w i,n i } , and in particular HE ( G ) is a semidirect product H ⋉ E ( G ) .2. H ⋉ E ( G ) ֒ → h ( G, Y ) . We conclude this section with a brief outline of the rest of the paper, usingthe notation of Theorem 5. In the final part of this section, Section 2.3,we recall some basic concepts and define some notation. In Section 3 wereview some estimates for the behaviour of random walks. In Section 3.1we review some exponential decay estimates that we will use, includingthe fact that a random walk makes linear progress in X , with the prob-ability of a linearly large deviation tending to zero exponentially quickly,an estimate for the Gromov product of the initial and final point of a andom walk, based at an intermediate point, and the property that thehitting measure of a shadow set in X decays exponentially in its distancefrom the basepoint.In Section 3.2 we review some matching estimates, which we now de-scribe. We say two geodesics α and β in X have an ( A, B )-match, if thereis a subgeodesic of one of length A which has a translate by some elementof G which B -fellow travels the other one; we say that a single geodesic γ has an ( A, B )-match if there is a subgeodesic of γ of length A which B -fellow travels a disjoint subgeodesic. If the constant B can be chosen toonly depend on δ , the constant of hyperbolicity, then we may refer to an( A, B )-match as a match of length A . Let γ n be a geodesic in X from x to w n x . For any geodesic η , the probability that η has a match of length | η | with γ n decays exponentially in | η | , and this can be used to show thatthe probability that γ n has a match of linear length (with itself) tendsto zero as n tends to infinity. We will also use the facts that if γ ω is thebi-infinite geodesic determined by a bi-infinite random walk, and α n is anaxis for w n , assuming w n is hyperbolic, then the probability that γ n , γ ω and α n have matches of a size which is linear in n , tends to 1 as n tends toinfinity. Finally, we also use the fact that for any group element g in thesupport of µ with axis γ g , ergodicity implies that the bi-infinite geodesic γ ω has infinitely many matches with γ g of arbitrarily large length.In Section 4 we recall some standard results about free subgroups ofa group G acting by isometries on a Gromov hyperbolic space X . Inparticular, as shown by, e.g., Taylor and Tiozzo [TT16], if a subgroup H has a symmetric generating set A = { a , a − , . . . a k , a − k } , for which thedistances d X ( x , ax ) are large, for all a in A , and the Gromov products( ax · bx ) x are small, for all distinct a and b in A , then { a , . . . , a k } freelygenerates a free group H , which is quasi-isometrically embedded in X .We show that furthermore, if Γ H is a rescaled copy of the Cayley graph,in which an edge corresponding to a ∈ A has length d X ( x , ax ), thenΓ H is quasi-isometrically embedded in H , with quasi-isometry constantsdepending only on δ and the size of the largest Gromov product ( ax · bx ) x , and not on the lengths of the geodesics [ x , ax ], for a ∈ A .In Section 5 we prove a version of Theorem 5 in the case that the grouphas a single generator, i.e. k = 1, which is equivalent to showing that theprobability that w n is hyperbolic and weakly asymmetric tends to onewith asymptotic probability one. In Section 5.1 we define coarse analoguesof the following properties of group elements: being primitive and beingasymmetric, and we show that these conditions are sufficient to show thata group element is weakly asymmetric, as long as it is not conjugate toits inverse. Then in Section 5.2, we use the matching estimates to show hat the coarse analogues hold with asymptotic probability one, as doesthe property that w n is not conjugate to its inverse.A key step is to use the fact that the support of µ contains a weaklyasymmetric element, g say, with axis α g . A result of Bestvina and Fuji-wara [BF02] says that if a group element h coarsely stabilizes a sufficientlylong segment of α g , then in fact h lies in E ( g ). Ergodicity implies thatthe bi-infinite geodesic γ ω fellow travels infinitely often with long seg-ments of translates of α g , and the matching estimates then imply thatthe axis α n for w n also fellow travels with long segments of translatesof α g , with asymptotic probability one. Therefore an element h ∈ E ( g )which coarsely fixes α n pointwise must also stabilize disjoint translatesof long segments of α g , h α g and h α g say. This implies that h lies in( h h g i h − ⋉ ( E ( g )) ∩ ( h h g i h − ⋉ E ( g )) = E ( g ), and so w n is weaklyasymmetric.Finally, in Section 6, we extend this result to finitely generated randomsubgroups. Let w i,n i be the generators of H , and let γ i,n i be a geodesicfrom x to w i,n i x . The random walk corresponding to each generatormakes linear progress, and pairs of independent random walks satisfy anexponential decay estimate for the size of their Gromov products basedat the basepoint x , so this shows that H is asymptotically freely gener-ated by the locations of the sample paths w i,n i , and is quasi-isometricallyembedded in X . If H is not geometrically separated, then there is anarbitrarily large intersection of N R ( gH ) and N R ( H ), which implies thatthere is a pair of long geodesics γ and γ ′ with endpoints in H such that gγ ′ fellow travels with γ , for g ∈ G \ ( H ⋉ E ( G )). The probability that γ i,n i matches any combination of shorter generators tends to zero, so forsome i , the group element g takes some translate h γ i,n i say, to anothertranslate h γ i,n i . This implies that h − gh coarsely stabilizes γ i,n i , andso g lies in h ( h w i,n i i ⋉ E ( G )) h ⊂ H ⋉ E ( G ), by the fact that each indi-vidual random walks gives weakly asymmetric elements with asymptoticprobability one. This contradicts our initial assumption that g did not liein H ⋉ E ( G ). Throughout the paper we fix a group G acting acylindrically hyperboli-cally on a separable hyperbolic space X . We will always assume that thehyperbolic space X is geodesic, but it need not be locally compact. Wedenote the distance in X by d X , and δ will refer to the constant of hyper-bolicity for the Gromov hyperbolic space X . We will write O ( δ ) to referto a constant which only depends on δ , through not necessarily linearly.We shall write | γ | for the length of a path γ . If γ is a geodesic, then | γ | is qual to the distance between its endpoints. Geodesics will always haveunit speed parameterizations, and γ ( t ) will denote a point on γ distance t from its initial point, and we will write [ γ ( t ) , γ ( t ′ )] for the subgeodesicof γ from γ ( t ) to γ ( t ′ ).In all statements about a single random walk on G , we will assume thatthe random walk is generated by an admissible probability measure µ anddenote the position of the walk at time n by w n , while the correspondingnotations for multiple random walks will be µ i for the admissible measuresand w i,n i for the locations of the random walks.If we say that a constant A depends on an admissible probabilitymeasure µ , then as the set of admissible measures depends on the actionof G on X , we allow that A may also depend on the action, and also onthe constant of hyperbolicity δ , and the acylindricality functions R ( K )and N ( K ). If we say that a constant A depends on the collection ofprobability distributions µ , . . . , µ k corresponding to a random subgroup H , this includes the possibility that the constant may depend on thenumber k of probability distributions. We will occasionally recall some ofthese assumptions and notations. Let µ be an admissible probability distribution on G . We will make use ofthe following exponential decay estimates, as shown by Maher and Tiozzo[MT14] and Mathieu and Sisto [MS14]. We denote the Gromov productby ( x · y ) w , which by definition is( x · y ) w = ( d X ( w, x ) + d X ( w, y ) − d X ( x, y )) . Furthermore, as the distance the sample path has moved in X is subad-ditive, the limit L = lim n →∞ n d X ( x , w n x )exists almost surely, and L is the same for almost all sample paths, byergodicity. If µ is admissible then L is positive, and we say that therandom walk has positive drift, or makes linear progress.Given x , x ∈ X and R >
0, the shadow S x ( x, R ) is defined to be S x ( x, R ) = { y ∈ X : ( x · y ) x ≥ d X ( x , x ) − R } . Proposition 9.
Let G be a countable group which acts acylindricallyhyperbolically on a separable space X with basepoint x , and let µ be anadmissible probability distribution on G . Then the following exponentialdecay estimates hold: .1 Positive drift in X with exponential decay.There is a positive drift constant L > such that for any ε > thereare constants K > and c < , depending on µ and x , such that P ((1 − ε ) Ln d X ( x , w n x ) (1 + ε ) Ln ) > − Kc n , (1) for all n .9.2 Exponential decay for Gromov products in X .There are constants K > and c < , depending on µ and x , suchthat for all i, n and r , P (( x · w n x ) w i x > r ) Kc r (2) X .There is a constant R > , which only depends on the action of G on X , and constants K > and c < , which depend on µ and x ,such that for all g and R > R , P ( w n ∈ S x ( gx , R )) Kc d X ( x ,gx ) − R . (3) A match for a pair of geodesics in X , is a subsegment of one geodesic,which may be translated by an element of G to fellow travel with a sub-segment of the other one. We now give a precise definition.We say that two geodesics γ and γ ′ in X have an ( A, B ) -match if thereare disjoint subgeodesics α ⊂ γ and α ′ ⊂ γ ′ of length at least A , and agroup element g ∈ G , such that the Hausdorff distance between gα and α ′ is at most B . We may choose γ and γ ′ to be the same geodesic, oroverlapping geodesics. If γ and γ ′ are the same geodesic, then we will justsay that γ has an ( A, B )-match.Given w n , a random walk of length n on G , we shall write γ n for ageodesic in X from x to w n x . As sample paths converge to the Gromovboundary ∂X almost surely [MT14], a bi-infinite sample path { w n x } n ∈ Z determines a bi-infinite geodesic in X almost surely, which we shall denote γ ω . In an arbitrary non-locally compact Gromov hyperbolic space, pairsof points in the boundary need not be connected by bi-infinite geodesics,however, they are always connected by (1 , O ( δ ))-quasigeodesics. With aslight abuse of language, we will call any bi-infinite (1 , O ( δ ))-quasigeodesicconnecting the limit points of g an axis for the hyperbolic element g . roposition 10. Let G be a countable group which acts acylindricallyhyperbolically on the separable space X , and let µ be an admissible proba-bility distribution on G . Then the there is a constant K , depending onlyon δ , such that for any K > K , the following matching estimates hold.10.1 There are constants B and c , depending on µ and K , such thatfor any geodesic segment η and any constant t , the probability that atranslate of η is contained in a K -neighbourhood of [ γ n ( t ) , γ n ( t + | η | )] is at most Bc | η | .10.2 For any ε > the probability that γ n has an ( ε | γ n | , K ) -match tendsto zero as n tends to infinity.10.3 For any ε > , the probability that the γ n contains a subsegment oflength at least (1 − ε ) | γ n | which is contained in a K -neighbourhood of γ ω tends to one as n tends to infinity. In particular, the probabilitythat γ n and γ ω have an ((1 − ε ) | γ n | , K ) -match tends to one as n tends to infinity.10.4 Let g be a hyperbolic isometry with axis α g which lies in the supportof µ . Then for any constants < ε < and L > , the probabil-ity that γ − n and α g have an ( L, K ) -match tends to one as n tendsto infinity, where γ − n is the subgeodesic of γ n obtained by removing ε | γ n | -neighbourhoods of its endpoints.10.5 For any ε > , the probability that w n is hyperbolic with axis α n , and γ n and α n have a ((1 − ε ) | γ n | , K ) -match tends to one as n tends toinfinity. Propositions 10.1, 10.2 and 10.3 are shown by Calegari and Maher[CM15]. Proposition 10.1 is not stated explicitly, but follows directlyfrom the proof of [CM15, Lemma 5.26].Proposition 10.5 is shown for µ with finite support by Dahmani andHorbez [DH15, Proposition 1.5]. However, they only need finite supportto ensure linear progress with exponential decay, and exponential decayfor shadows, and so their argument also works for µ with bounded supportin X .Finally, a version of Proposition 10.4 is shown for the mapping classgroup acting on Teichm¨uller space, for µ with finite support, by Gadre andMaher [GM16], and independently by Baik, Gekhtman and Hamenst¨adt[BGH16]. A significantly simpler version of these arguments works inthe setting of acylindrically hyperbolic groups, but we present the detailsbelow for the convenience of the reader. As a side remark, we note that itcan be shown that, in fact, the largest match of γ n and α g has logarithmicsize in n [ST16]. roof (of Proposition 10.4). Let g be a hyperbolic element which lies inthe support of µ , and let α g be an axis for g . Let γ ω be the bi-infinitegeodesic determined by a bi-infinite random walk generated by µ . Weshall write ν for the harmonic measure on ∂X , and ˇ ν for the reflectedharmonic measure, i.e the harmonic measure arising from the randomwalk generated by the probability distribution ˇ µ ( g ) = µ ( g − ).By assumption, the group element g lies in the support of µ , and sothe group element g − lies in the support of ˇ µ . Given a constant L > m sufficiently large such that any geodesic from S x ( g m x , R )to S x ( g − m x , R ) has a subsegment of length L which K = O ( δ )-fellowtravels with α g . The following result of Maher and Tiozzo [MT14] guar-antees that the harmonic measures of these shadow sets are positive. Proposition 11. [MT14, Proposition 5.4]
Let G be a countable groupacting acylindrically hyperbolically on a separable space X , and let µ bea non-elementary probability distribution on G . Then there is a number R such that for any group element g in the semigroup generated by thesupport of µ , the closure of the shadow S x ( gx , R ) has positive hittingmeasure for the random walk determined by µ . Therefore ν ( S x ( g m x , R )) > ν ( S x ( g − m x , R )) >
0, and sothere is a positive probability p say that γ ω has a subsegment of length atleast L which lies in a K -neighbourhood of γ g . Ergodicity now implies thatthe proportion of times in {⌊ n ⌋ , . . . , ⌊ n ⌋} for which γ ω has a subsegmentof length at least L which lies in a K -neighbourood of w m γ g tends to p as n tends to infinity, for almost all sample paths ω . Proposition 10.3 thenimplies that the probability that γ n has an ( L, K )-match with γ g tendsto one. In this section we collect together some standard results about free sub-groups of a group G acting by isometries on a hyperbolic space X , seefor example Bridson and Haefliger [BH99] for a thorough discussion. Forcompleteness, we present a mild generalization of an argument due to Tay-lor and Tiozzo [TT16], and show that one may rescale the Cayley graphΓ of a Schottky group so that the quasi-isometric embedding constants ofΓ into X depend only on δ , the constant of hyperbolicity for X , and thesize of the Gromov products between the generators.A relation g = g g . . . g n between elements of G may be thought ofas a recipe for assembling a path from x to gx as a concatenation oftranslates of paths from x to g i x . The following proposition gives an stimate for the distance of the endpoints of the total path in terms ofthe lengths of the shorter segments, and the Gromov products betweenadjacent segments.Let η be a path which is a concatenation of k geodesic segments { η i } ki =1 , and label the endpoints of η i as x i − and x i , such that the com-mon endpoint of η i and η i +1 is labelled x i . For 2 i k , let p i be thenearest point projection of x i − to η i , and for 1 i k −
1, let q i be thenearest point projection of x i +1 to η i . We define p = x and q k = x k +1 .We will call the subsegment [ p i , q i ] ⊂ η i the persistent subgeodesic of η i .This is illustrated below in Figure 1. x i − η i x i p i +1 x i +1 η i +1 q i − x i − η i − p i q i Figure 1: A concatenation of geodesic segments.
The length of the persistent subgeodesic may be estimated in terms ofGromov products.
Proposition 12.
There is a constant C , which only depends on δ suchthat if η is a concatenation of geodesic segments η i , with persistent sub-geodesics [ p i , q i ] , then d X ( p i , q i ) d X ( x i − , x i ) − ( x i − · x i ) x i − − ( x i − · x i +1 ) x i + C, and d X ( p i , q i ) > d X ( x i − , x i ) − ( x i − · x i ) x i − − ( x i − · x i +1 ) x i − C. (4)We omit the proof of Proposition 12, which is a straight forward ap-plication of thin triangles and the definition of the Gromov product.We now show that if each persistent subsegment is sufficiently long,then the distance between x and x k is equal to the sum of the lengthsof the persistent subsegments, up to an additive error proportional to thenumber of geodesic segments. roposition 13. There exists a constant
C > , which depends only on δ , such that if η is a concatenation of geodesic segments η i for i k ,with persistent subgeodesics [ p i , q i ] with d X ( p i , q i ) > C, (5) for all i k , then k X i =1 d X ( p i , q i ) − Ck d X ( x , x k ) k X i =1 d X ( p i , q i ) + 2 Ck. (6)
Furthermore, any geodesic from x to x k passes within distance C of both p i and q i .Proof. For any three points x, y and z , determining a triangle in in X ,there is a point m , known as the center of triangle, such that m is distanceat most δ from each of the three sides of the triangle. Furthermore, thereis a C , which only depends on δ , such that if p is a closest point on [ y, z ]to x , then d X ( p, m ) C . This implies that d X ( q i , p i +1 ) C , usingthe triangle with vertices x i − , x i and x i +1 . The upper bound d X ( x , x k ) d X ( q i , p i +1 ) + 2 C k then follows from the triangle inequality.There are constants C and C , which only depend on δ , such that forany point y in X , whose nearest point projection to η i − is distance at least C away from η i , the nearest point projection of y to η i is distance at most C from p i . As this also holds for η i − , the nearest point projection of x i − to η i − is within distance C of p i − , and so the nearest point projectionof x i − to η i is within distance C of p i . By induction, the nearest pointprojection of x to η i is within distance C of p i , and similarly, the nearestpoint projection of x k to η i is within distance C of q i .There are constants C and C , depending only on δ , such that if twopoints x and y in X have nearest point projections p and q onto a geodesic α , and d X ( p, q ) > C , then any geodesic from x to y passes within distance C of both p and q .In particular, there exists C depending only on δ so that if γ is ageodesic from x to x k , then for each i there is a subsegment of γ of lengthat least d X ( p i , q i ) − C which is contained in a C +4 δ -neighbourhood of thepersistent subsegment [ p i , q i ], and is disjoint from C +4 δ -neighbourhoodsof the other persistent subsegments [ p j , q j ] for i = j . Therefore d X ( x , x k ) > k X i =1 ( d X ( p i , q i ) − C ) , giving the required lower bound. e now use Proposition 13 to give a lower bound on the translationlength of group elements. Proposition 14.
There exists a constant
C > , which depends only on δ , such that if g is an isometry of a hyperbolic space X with basepoint x ,which is a product of isometries g = g g . . . g n , where the g i satisfy thefollowing collection of inequalities d X ( x , g i x ) > ( g − i − x · g i x ) x + ( g − i x · g i +1 x ) x + C, (7) where g n +1 = g and g = g n , then the translation length of g is at least τ ( g ) > n X i =1 (cid:0) d X ( x , g i x ) − ( g − i − x · g i x ) x − ( g − i x · g i +1 x ) x − C (cid:1) , (8) and furthermore, any geodesic from x to gx , and any axis γ for g passeswithin distance ( g − i − x · g i x ) x +( g − i x · g i +1 x ) x + C of each g . . . g i x .Proof. We first define a sequence of points { x i } ni =0 , and a sequence ofgeodesic segments { η i } ni =1 , following the index conventions of Proposition12. Let x be the basepoint of X , and for 1 i n let x i = g . . . g i x .For 1 i n let η i be a geodesic from x i − to x i , and let η be the pathformed from the concatenation of the geodesic segments η i .We may now consider the bi-infinite sequences obtained from all g -translates of the points x i and the geodesics η i , labelled such that x jn + i = g j x i and η jn + i = g j η i , for j ∈ Z and 1 i n . The terminal point x n of η n is equal to g . . . g n x = gx , which is the same as the initial point of η n +1 = gη = gx , so the concatenation of the geodesics η i is a bi-infinite g -equivariant path in X , which we shall denote η .If we choose C > C , where C is the constant from Proposition13 then the assumption (7), together with the estimate for persistentlength in terms of Gromov products (4), implies that any subpath { η i } bi = a of η satisfies the hypothesis of Proposition 13, and so the conclusion ofProposition 13 implies that d X ( x a − , x b ) > C ( b − a ). In particular, thisimplies that the translation length τ ( g ), which by definiton is equal to τ ( g ) = lim m →∞ m d X ( x , g m x ) , is given by τ ( g ) = lim m →∞ m d X ( x , x mn ) > C n > . Therefore the translation length τ ( g ) is positive, and so g is hyperbolic,and η is a quasi-axis for g . The estimate for translation length (8) thenfollows by combining (4) and (6), and the statements about the distancefrom x i to any geodesic γ from x a to x b , for a i b , and the distancefrom x i to any axis for g follow from thin triangles and the definition ofthe Gromov product, for C = 3 C + O ( δ ). e now give conditions on the generators of a subgroup which ensurethat the generators freely generate a subgroup which is quasi-isometricallyembedded in X .Given a symmetric generating set A = { a , a − , . . . , a k , a − k } gener-ating a subgroup H of G , let F A be the free group generated by A = { a , . . . , a k } , and let Γ H be a rescaled copy of the Cayley graph for F A ,with respect to the generating set A , where an edge in Γ H correspondingto a generator a i has length equal to d X ( x , a i x ). We shall refer to Γ H asthe rescaled Cayley graph for F A , which is quasi-isometric to the standardunscaled Cayley graph in which every edge has length one. The map fromΓ H to X which sends a vertex h to hx , and an edge from h to h ′ toa geodesic from hx to h ′ x is continuous, can be made H -equivariant,and is an isometric embedding on each edge. The conditions we give be-low will in fact show that Γ H is quasi-isometrically embedded in X , withquasi-isometry constants independent of the lengths of the edges. Proposition 15.
There is a constant K , which only depends on δ , suchthat for any K > K , if H is a subgroup generated by the symmetric gen-erating set A = { a , a − , . . . a k , a − k } , satisfying the following conditions, d X ( x , ax ) > K for all a ∈ A ( ax · bx ) x K for all a = b in A ) (9) then H is isomorphic to the free group F k , freely generated by the gen-erating set A = { a , . . . , a k } , and furthermore, the subgroup H is quasi-isometrically embedded in X , and the rescaled Cayley graph Γ H is (6 , O ( δ, K )) -quasi-isometrically embedded in X .Proof. We shall choose K > C , where C is the constant from Proposi-tion 14. Let g = g . . . g n be a reduced word in the generating set A . The g i satisfy the Gromov product inequalities from (7). Proposition 14 thenimplies that τ ( g ) > Cn , so in particular all reduced words are non trivial,so A freely generates a free group.We now show that Γ H is quasi-isometrically embedded in X , for quasi-isometry constants that are independent of the lengths of the g i . Thetranslation length τ ( g ) is a lower bound for d X ( x , gx ), and as g . . . g n is a reduced word d Γ H ( x , gx ) = n X i =1 d X ( x , g i x ) . Therefore conclusion (8) of Proposition 14 implies the left hand boundbelow d Γ H ( x , gx ) − Kk d X ( x , gx ) d Γ H ( x , gx ) , here K > K , for the choice of K given above. The right hand boundfollows immediately from the triangle inequality. As each geodesic seg-ment of Γ H has length at least 6 K , this implies d Γ H ( x , gx ) d X ( x , gx ) d Γ H ( x , gx ) . Finally, using thin triangles, we may extend this estimate to all points x and y in Γ H to obtain d Γ H ( x, y ) − K + O ( δ ) d X ( x , gx ) d Γ H ( x , gx ) + 2 K + O ( δ ) . as required. The probability that w n is hyperbolic tends to one, so in particular, if X =Cay( G, Y ), then the probability that E ( w n ) is hyperbolically embeddedin ( G, Y ) tends to one. In this section we show that the probabilitythat w n is weakly asymmetric tends to one, i.e. the probability that E ( w n ) = h w n i ⋉ E ( G ) tends to one, and this is precisely the special caseof Theorem 5 when k = 1. Proposition 16.
Let G be a countable group acting acylindrically hyper-bolically on the separable space X , and let µ be an admissible probabilitydistribution on G . Then the probability that w n is hyperbolic and weaklyasymmetric tends to one as n tends to infinity. We start in Section 5.1 by giving some geometric conditions which aresufficient to show that a group element is weakly asymmetric. In Section5.2 we show that the probability that these conditions are satisfied by arandom element w n tends to one as n tends to infinity. Let g be a hyperbolic isometry. Recall that a group element g ∈ G is primitive if there is no element h ∈ G such that h n = g for n >
1. Wenow define a notion of coarse primitivity for group elements.
Definition 17.
Let γ be an axis for g , let p i be the projection of g i x to γ , and set P = S i ∈ Z p i . We say that g is K -primitive if any element h ∈ E ( g ) K -stabilizes P , i.e. the Hausdorff distance d Haus ( P, hP ) K .If g is K -primitive, then g is primitive, for K = O ( δ ) sufficiently large,and if the translation distance τ ( g ) satisfies τ ( g ) > K + O ( δ ).Recall that for a hyperbolic element g , Λ( g ) = { λ + ( g ) , λ − ( g ) } is the setconsisting of the pair of attracting and repelling fixed points for g in ∂X , nd E ( g ) = stab(Λ( g )). We shall write E + ( g ) for the subgroup of E ( g )which preserves Λ( g ) pointwise, i.e. E + ( g ) = stab( λ + ( g )) ∩ stab( λ − ( g )).This subgroup has index at most 2 in E ( g ). Definition 18.
We say a hyperbolic isometry g is reversible if there is anelement in E ( g ) which switches the fixed points of g , i.e. E + ( g ) ( E ( g ).Otherwise g is irreversible and E + ( g ) = E ( g ).We say that the K -stabilizer of a geodesic γ = [ p, q ], consists of allgroup elements g such that if d X ( p, gp ) K and d X ( q, hq ) K . Definition 19.
Let G be a countable group acting acylindrically hy-perbolically on a separable space X . We say that a group element g is K -asymmetric if g is hyperbolic with axis α g , and if p is a closest pointon α g to the basepoint x , then the K -stabilizer for the geodesic [ p, gp ] isequal to E ( G ).We first show that every non-elementary subgroup H of G containinga weakly asymmetric element, also contains a K -asymmetric element. Proposition 20.
Let G be a countable group acting acylindrically hyper-bolically on a separable space X , and let H be a non-elementary subgroup H of G , which contains a weakly asymmetric element. Then for any con-stant K > , the subgroup H contains a K -asymmetric element g . We will use the following lemma, which follows from work of Bestvinaand Fujiwara.
Lemma 21. [BF02, Proposition 6]
Let g be a hyperbolic isometry withaxis α g . Then for any number K > there is a D , depending on g, δ and K , such that if h K -coarsely stabilizes a segment of α g of length at least D , then h lies in E ( g ) .Proof (of Proposition 20). Let h be a weakly asymmetric element in H ,with axis γ h . As H is non-elementary, and E ( h ) is virtually cyclic, thereis a hyperbolic element f in H which does not lie in E ( h ). In particular, h and f are independent, i.e. their fixed point sets in ∂X are disjoint.Consider the group element g = h a f b h a . For all a and b sufficiently large,the translation lengths of h a and f b are much larger than twice any ofthe Gromov products between distinct elements of { h ± a , f ± b } , so we mayapply Proposition 14, which in particular implies that g is hyperbolic.Furthermore, for any constant D >
0, there is an a sufficiently large suchthat the axis γ g of g has a subsegment γ of length at least D whichis contained in an O ( δ )-neighbourhood of γ h , and a disjoint subsegment γ of length at least D which is contained in an O ( δ )-neighbourhood of h a f b γ h . We shall choose an a sufficiently large such that this holds for D > D h + O ( K, δ ), where D h is the constant from Lemma 21 applied tothe hyperbolic element h with constant K + O ( δ ). Finally, we may choose to be much larger than b , so that D is at least three times as large asthe distance between γ and γ . This is illustrated in Figure 2 below. x h a x h a f b x h a f b h a x = gx γ h γ h a f b γ h γ p γ g gp Figure 2: The axis γ g fellow travels two translates of γ h . Let p be a closest point on γ g to the basepoint x . If an element g ′ in G K -coarsely stabilizes [ p, gp ], then g ′ ( K + O ( δ ))-stabilizes γ and γ .The segments γ and γ fellow travel axes of two distinct translates of γ h ,say u γ h and u γ h , and so g ′ ( K + O ( δ ))-stabilizes segments of these axesof length at least D h . Therefore by Lemma 21, g ′ lies in E ( u hu − ) ∩ E ( u hu − ) , which is equal to (cid:0) u h h i u − ⋉ E ( G ) (cid:1) ∩ (cid:0) u h h i u − ⋉ E ( G ) (cid:1) , as h is weakly asymmetric. Hyperbolic elements in each of these subgroupshave distinct fixed points in ∂X , and so cannot be equal. The set of non-hyperbolic elements is equal to E ( G ), therefore the intersection of the twosubgroups is exactly E ( G ), and so g ′ ∈ E ( G ), as required.Finally, we show that these geometric conditions are sufficient to showthat a group element g is weakly asymmetric. Proposition 22.
Let G be a countable group acting acylindrically hyper-bolically on the separable space X . Then there is a constant K , dependingonly on δ , such that if g is an element which is hyperbolic, K -primitive, K -asymmetric and irreversible, then g is weakly asymmetric. roof. Let g be a group element in G which is hyperbolic, irreversible, K -primitive and K -asymmetric, and let h be an element of E ( g ). Let α g be an axis for g , and let p be a closest point on α g to the basepoint x . As g is K -primitive, we may multiply by a power of g , so that g n hK -coarsely fixes [ p, gp ]. As g is K -asymmetric, this implies that g n h liesin E ( G ), and so h lies in h g i E ( G ). Finally, as g is hyperbolic, h g i E ( G ) isa semidirect product h g i ⋉ E ( G ), by Proposition 2. In this section we show that the geometric properties defined in the pre-vious section hold for random elements w n with asymptotic probabilityone.We start by showing that the translation length τ ( w n ) also growslinearly, using Proposition 14. Lemma 23.
Let G be a countable group acting acylindrically hyperboli-cally on the separable space X , and let µ be an admissible probability dis-tribution on G . For any < ε < the probability that τ ( w n ) > (1 − ε ) | γ n | goes to . Notice that, in the notation of the lemma, | γ n | > τ ( w n ) always holds. Proof.
We shall apply Proposition 14 with g = w n , considered as aproduct of g = w m and g = w − m w n , where m = ⌊ n/ ⌋ . Recall that w n = s . . . s n , where the s i are the steps of the random walk, and areindependent µ -distributed random variables.By linear progress, Proposition 9.1, there exists L > P ( d X ( x , w m x ) > Ln ) and P ( d X ( x , w − m w n x ) > Ln ) tend to one as n tends to infinity (the L here is smaller than the L in Proposition 9.1).By Proposition 9.2, the probability that the Gromov product ( w − m x · w − m w n x ) x = ( x · w n x ) w m x is bounded above by εLn/ n tends to infinity. For the other Gromov product (( w − m w n ) − x · w m x ) x , the two random variables ( w − m w n ) − = s − n . . . s − m +1 and w m = s , . . . s m are independent, and so the distribution of(( w − m w n ) − x · w m x ) x = ( s − n . . . s − m +1 x · s . . . s m x ) x is the same as the distribution of( s − n − m . . . s − x · s n − m +1 . . . s n x ) x = ( x · w n x ) w n − m x , and so again by Proposition 9.2, the probability that this Gromov productis bounded above by εLn/ n tends to infinity.Therefore, the probability that the two inequalities (7) are satisfiedtends to one as n tends to infinity. Hence, by Proposition 14, for n ufficiently large we have τ ( w n ) > d X ( x , w m x ) + d X ( w m x , w n x ) − εLn ≥ (1 − ε ) | γ n | with probability that tends to 1 as n tends to infinity,as required.We now show that the probability that w n is irreversible tends to oneas n tends to infinity. Proposition 24.
Let G be a countable group acting acylindrically hyper-bolically on the separable space X , and let µ be an admissible probabilitydistribution on G . Then for any K , the probability that w n is irreversibletends to one as n tends to infinity.Proof. We can assume that w n is hyperbolic, with axis α n . Now suppose h ∈ E ( w n ) is an element which reverses the endpoints of w n . Since α n and hα n are O ( δ )-fellow travelers, this gives a ( τ ( w n ) − O ( δ ) , O ( δ ))-matchfor any subsegment of α n of length τ ( w n ).Propositions 10.2 and 10.5 (in view of Lemma 23) then show that theprobability that this occurs tends to zero as n tends to infinity.In fact, informally, if α n had a match of size approximately τ ( w n ) / γ n , but this is ruledout by Proposition 10.2 since Lemma 23 says that τ ( w n ) is approximatelyequal to | γ n | .We now show that random walks give K -primitive elements with asymp-totic probability one. Proposition 25.
Let G be a countable group acting acylindrically hyper-bolically on the separable space X , and let µ be an admissible probabilitydistribution on G . Then for any K , the probability that w n is K -primitivetends to one as n tends to infinity.Proof. Let α g be an axis for a hyperbolic element with τ ( g ) > K + O ( δ ),and suppose there is an element h in E ( g ) which does not K -stabilize P . Up to replacing h with some g k h , we can assume d X ( p , hp ) ≤ d X ( p , gp ) + O ( δ ). As h moves p distance at least K , h is hyper-bolic by applying Proposition 14, in the case where n = 1, g = g = h and the basepoint x = p . Therefore, there is a power of h such that d X ( p , gp ) − O ( δ ) d X ( p , h a p ) d X ( p , gp ) + O ( δ ) . As α g and h a α g are O ( δ )-fellow travelers, this gives a ( τ ( g ) − O ( δ ) , O ( δ ))-match for any subsegment of α g of length τ ( g ). Proposition 10.5 thenimplies that the probability that γ n has a ( τ ( g ) − O ( δ ) , O ( δ ))-matchtends to one as n tends to infinity, and the probability that this occurstends to zero as n tends to infinity, by Proposition 10.2. e now show that the probability that w n is K -asymmetric tends toone as n tends to infinity. Proposition 26.
Let G be a countable group acting acylindrically hyper-bolically on the separable space X , and let µ be an admissible probabilitydistribution on G . Then for any constant K > the probability that w n is K -asymmetric tends to one as n tends to infinity.Proof. By Proposition 25 the probability that w n is hyperbolic and K -primitive tends to one as n tends to infinity. By Proposition 20 there is anelement h in the support of µ which is ( K + O ( δ ))-asymmetric. Let α h bean axis for h , and let p be a closest point on α h to the basepoint x . ThenProposition 10.4 implies that the probability that w n is hyperbolic withaxis α n , and α n has a subsegment of length at least 2 τ ( h ) which O ( δ )-fellow travels with a translate of α h tends to one as n tends to infinity. Ifthis happens, then if an element g ∈ G K -stabilizes [ x , w n x ], then it also( K + O ( δ ))-stabilizes a translate of [ p, hp ]. As h is ( K + O ( δ ))-asymmetric,this implies that g ∈ E ( G ), so w n is K -asymmetric, as required.This completes the proof of Proposition 16: we have shown that all ofthe geometric hypotheses of Proposition 22 hold with asymptotic prob-ability one, so Proposition 22 implies that w n is hyperbolic and weaklyasymmetric with asymptotic probability one.Although we have completed the proof of the special case of Theorem5 in the case k = 1, we now conclude this section by showing a slightlystronger result, which we will need for the general case. Proposition 27.
Let G be a countable group acting acylindrically hyper-bolically on the separable space X , and let µ be an admissible probabilitydistribution on G with positive drift L > . Let < ε < . Then theprobability that w n is ( εLn ) -asymmetric tends to as n tends to infinity.Proof. Let h be a hyperbolic element in the support of µ which is K = O ( δ )-asymmetric, with axis α h , and let p be a closest point on α h to thebasepoint x .The probability that w n is hyperbolic tends to one, so we may assumethat w n is hyperbolic with axis α n . Let q be a closest point on α n to x , let γ be a geodesic from q to w n q , and let g be a group elementwhich ( εLn )-coarsely stabilizes γ . We have already shown the result forgroup elements g which K -stabilize γ for fixed K , so we may assume that d X ( q, gq ) and d X ( w n q, gw n q ) are both at least K = O ( δ ).We now show that there is a subgeodesic γ − of γ for which all points aremoved a similar distance by g . Define γ − to be γ \ ( B X ( q, εLn ) ∪ B X ( w n q, εLn )). laim 28. For all s and t in γ − , | d X ( s, gs ) − d X ( t, gt ) | O ( δ ) . q w n qγgq gw n qgγsu gs tv gt Figure 3: Points on γ − are moved a similar distance. Proof. As g is an isometry d X ( s, t ) = d X ( gs, gt ). Let u be a closest pointon gγ to s , and let v be a closest point on gγ to t , then d X ( u, v ) = d X ( s, t ) + O ( δ ). This implies that d X ( u, gs ) = d X ( v, gt ) + O ( δ ), and as d X ( u, s ) δ and d X ( v, t ) δ , thus implies that d X ( s, gs ) = d X ( t, gt ) + O ( δ ), as required.By Propositions 9.1 and 10.3 the length of γ is at least (1 − ε ) Ln ,and so the length of γ − is at least (1 − ε ) Ln . Therefore by Proposition10.4 the probability that γ − has a subsegment of length at least 2 τ ( h )which O ( δ )-fellow travels with γ h tends to 1 as n tends to infinity. If d X ( s, gs ) K = O ( δ ) for s ∈ γ − , then g ( K + O ( δ ))-stabilizes a translateof [ p, hp ], and so g ∈ E ( G ), which implies that w n is K -asymmetric,as required. Therefore the final step is to eliminate the case in which d X ( s, gs ) > K = O ( δ ) for s ∈ γ − , which we now consider.Let s be a point on γ − , let t be a nearest point to gs on γ , and let u be a nearest point on γ to gt . This is illustrated below in Figure 4. q w n qγgp gw n qgγs tgs ugtg s Figure 4: The image of s under g and g . The distance from gs to t is at most 2 δ , and the distance from g s to u is at most 4 δ . As d X ( s, gx ) > K = O ( δ ), this gives an upper bound n the Gromov product ( s · g s ) gs = ( g − s · gs ) s of at most O ( δ ), andso we may apply Proposition 15. Therefore, g is hyperbolic, and the axis α g for g passes within distance O ( δ ) of t . Furthermore, this holds for all t ∈ γ − , so the axis α g for g O ( δ )-fellow travels with γ − . The axis α g is τ ( g ) periodic, and τ ( g ) εLn , this means that γ − , and hence γ n has an( εLn + O ( δ ) , O ( δ ))-match, which contradicts Proposition 10.2. We briefly recall the notation we use for a random subgroup H = H ( µ i , n i ).The µ , . . . , µ k are admissible probability distributions on G , and the n , . . . , n k are positive integers. We write w i,n i for a random walk oflength n i generated by the probability distribution µ i , and γ i for a geodesicin X from x to w i,n i x . We shall write H for the subgroup generatedby { w ,n , . . . , w k,n k } , and set n = min n i . Recall that the random walkgenerated by an admissible probability distribution µ i has positive drift,i.e. there is a constant L i such that n d X ( x , w i,n i x ) → L i as n i → ∞ ,almost surely. We shall set L = min L i , so in particular L >
0, and weshall reorder the µ i so that Ln L n · · · L k n k , as we shall needto keep track of the expected lengths of the generators in the subsequentargument. Finally, it will be convenient to have notation for paths whichtravel along a geodesic γ i in the reverse direction, so we will extend ourindex set from I = { , . . . , k } to ± I = {± , . . . , ± k } , and write γ − i fora geodesic in X from x to w − i,n i x , which is a translate by w − i,n i of thereverse path along γ i .In order to show that H is hyperbolically embedded in G we shall showthat H is freely generated by { w ,n , . . . w k,n k } , H is quasi-isometricallyembedded in X , and H ⋉ E ( G ) is geometrically separated, with asymptoticprobability one.We start by showing some generalizations of the properties that holdfor individual random walks to the case of multiple random walks. Eachindividual random walk makes linear progress with exponential decay. Wenow show that the collection of k random walks also makes linear progresswith exponential decay. Definition 29.
Given 0 < ε <
1, and a random subgroup H , we say that H satisfies ε -length bounds if(1 − ε ) L i n i d X ( x , w i,n i x ) (1 + ε ) L i n i . (10)for all 1 i k . Proposition 30.
Let H be a random subgroup, and let ε > . Thenthere are constants K and c , depending only on ε , and the probability dis- ributions µ i , such that the probability that a random subgroup H satisfies ε -length bounds is at least − Kc n .Proof. By Proposition 9.1, for any ε >
0, for each µ i there are constants L i , K i and c i such that P ((1 − ε ) L i n i d X ( x , w i,n i x ) (1 + ε ) L i n i ) > − K i c n i i . If K ′ = max K i , c = max c i and n = min n i , then the probability thatthese inequalities are satisfied simultaneously for all i is at least 1 − kKc n .Therefore the required estimate holds, with K = kK ′ , and the previouschoice of c .We now show that the collection of k random walks satisfies the fol-lowing estimates on their mutual Gromov products. Definition 31.
We say a random subgroup H satisfies K -Gromov productbounds if ( ax · bx ) x K. for all distinct a and b in the symmetric generating set A = { w ± ,n , . . . w ± k,n k } for H . Proposition 32.
Let H be a random subgroup. Given < ε < there areconstants K and c , depending only on ε , and the probability distributions µ i , such that the probability that H satisfies ( εLn ) -Gromov product boundsis at least − Kc n .Proof. If ( ax · bx ) x εLn , then, by definition of shadows, ax ∈ S x ( bx , d X ( x , bx ) − εLn ). By Proposition 9.3, the random walk de-termined by each µ i satisfies exponential decay for shadows, i.e. there areconstants R , K i and c i < R > R , and all g ∈ G , P ( w i,n i ∈ S x ( gx , R )) K i c d X ( x ,gx ) − Ri . (11)We shall use (11) with g = b . If d X ( x , bx ) − εLn > R , (12)then (11) implies that the probability that ( ax · bx ) x εLn is at most K i c εLni .In order to apply the estimate (11), we need to check that (12) holdswith asymptotic probability one. Using linear progress, Proposition 9.1, P ( d X ( x , bx ) (1 − ε ) Ln ) K ′ i c ′ in , for some constants K ′ i and c ′ i depending on ε and µ i . Therefore P ( d X ( x , bx ) − εLn (1 − ε ) Ln ) K ′ i c ′ in . s we have chosen ε < , this implies that P ( d X ( x , bx ) − εLn R ) K ′ i c ′ in . for all n > R / ( L (1 − ε )).Therefore, the probability that ( ax · bx ) x εLn is as at most K ′ i c ′ in + K i c εLni . As there are at most 2 k choices for each of a and b in A , the probability that any of these events occurs is at most 4 k K ′′ c n ,where K ′′ = max { K i , K ′ i } and c i = max { c i , c ′ i } . The result then holdswith K = 4 k K ′′ , and the previous choice of c , as required.If H satisfies ε -length bounds and ( εLn )-Gromov product bounds,then the conditions (9) are satisfied in Proposition 15, so the rescaledCayley graph Γ H is (6 , O ( εLn ))-quasi-isometrically embedded in X . Inparticular, this implies that H is freely generated by { w ,n , . . . w k,n k } ,and HE ( G ) is a semidirect product H ⋉ E ( G ). As well as these properties,it will be convenient to know certain matching properties for the geodesicsdefined by H , which we now describe. Definition 33.
We say that a random subgroup H has an ε -large match if a translate of [ γ j ( εLn ) , γ j ( | γ j |− εLn )] is contained in a 2 δ -neighbouroodof γ i , for some i < j . Proposition 34.
Let H be a random subgroup, and let < ε < .Then there are constants K and c , depending on ε and the probabilitydistributions µ i , such that the probability that H has an ε -large match isat most Kc n .Proof. We may assume that H satisfies ε -length bounds, which by Propo-sition 30, happens with probability at least 1 − K ′ c ′ n , for some K ′ and c ′ <
1, depending on the µ i and ε . By ε -length bounds, the length of γ j is at least (1 − ε ) L j n j , and the length of γ i is at most (1 + ε ) L i n i .Let γ − j be the subgeodesic of γ j given by [ γ j ( εLn ) , γ j ( | γ j | − εLn )]. Itwill be convenient to consider a discrete set of points γ j ( ℓ ) along γ j , where ℓ ∈ N . If γ − j is contained in a 2 δ -neighbourhood of [ γ i ( t ) , γ i ( t + (cid:12)(cid:12) γ − j (cid:12)(cid:12) )],then γ − j is contained in a (2 δ + 1)-neighbourhood of [ γ i ( ℓ ) , γ i ( ℓ + (cid:12)(cid:12) γ − j (cid:12)(cid:12) )]for some ℓ ∈ N .By Proposition 10.1, there are constants K i and c i < γ − j is contained in a (2 δ +1)-neighbourhoodof γ i starting at γ i ( ℓ ) is at most K i c (1 − ε ) L j n j − εLni Kc (1 − ε ) L j n j , where the inequality above holds with K = max K i , c = max c i , and Ln L j n j . Given the length estimates for γ i and γ j , the number of ossible values of ℓ is at most(1 + ε ) L i n i − (1 − ε ) L j n j + 2 εLn εL j n j , where the inequality holds as L i n i L j n j , and negative terms on the lefthand side are discarded.Therefore, the probability that a translate of γ − j is contained in a2 δ -neighbourhood of γ i is at most3 εL j n j Kc (1 − ε ) L j n j K ′′ c ′′ n , for some constants K ′′ and c ′′ , where the inequality above holds as thefunction f ( x ) = xc x is decreasing for all x sufficient large, and boundedabove by a constant multiple of an exponential function. As there are atmost 2 k choices of indices for each of i and j , the result follows.Finally, we give an estimate for the probability that a geodesic γ j hasan initial segment which matches a terminal segment of γ i , concatenatedwith an initial segment of γ i ′ , for some i j and i ′ j .Given a collection of geodesics { γ i } i ∈± I , and a number K , define acollection of geodesic segments { η ( i, i ′ , K, ℓ ) | i, i ′ ∈ ± I, i = − i ′ , ℓ ∈ N , ℓ | γ i |} as follows. Let i and i ′ be indices in ± I with the property that i = − i ′ , and let 0 ℓ | γ i | be an integer. Let p be a point on γ i distance ℓ from its endpoint, and let q be a point on w i γ i ′ distance K from theinitial point of w i γ i ′ . Define η ( i, i ′ , K, ℓ ) to be a geodesic from p to q . γ i w i x w i γ i ′ p ℓ K qη ( i, i ′ , K, ℓ )Figure 5: A geodesic η ( i, i ′ , K, ℓ ). Definition 35.
We say that a random subgroup H is K -unmatched if forall i j , i ′ j and for 0 t K , no geodesic η ( i, i ′ , K, ℓ ), is containedin a 2 δ -neighbourhood of a subgeodesic of γ j starting at γ j ( t ). Proposition 36.
Let H be a random subgroup, and let < ε < . Thenthere are constants K and c , depending on ε and the µ i , such that theprobability that H is (3 εLn ) -unmatched is at least − Kc n .Proof. We shall assume that the random subgroup H satisfies the ε -lengthbounds and ( εLn )-Gromov product bounds, which happens with proba-bility at least 1 − K ′ c ′ n , for some constants K ′ and c ′ , depending on ε and the µ i . irst consider a fixed collection of indices i, i ′ , j in ± I , with i j , i ′ j and i = − i ′ . The Gromov product bound for i and i ′ implies thatthe length of η = η ( i, i ′ , εLn, ℓ ) is at least ℓ + 3 εLn − εLn = ℓ + εLn .If a translate of η ( i, i ′ , εLn, ℓ ) is contained in a 2 δ neighbourhood of[ γ j ( t ) , γ j ( t + | η | )], then it is contained in a (2 δ + 1)-neighbourhood of[ γ j ( m ) , γ j ( m + | η | )], for some m ∈ N .By Proposition 10.1, the probability that a translate of η ( i, i ′ , εLn, ℓ )is contained in a (2 δ + 1)-neighbourhood of [ γ j ( m ) , γ j ( m + | η | )] is at most Kc ℓ + εLn . As there are at most 3 εLn choices for m , the probability thatthis occurs for some 0 m εLn is at most 3 εLnKc ℓ + εLn . The sum ofthese probabilities over all values of ℓ is at most (3 εLnKc εLn ) / (1 − c ) K ′ c ′ n for different constants K ′ and c ′ .There are at most 2 k possible admissible choices for each of the indices i, i ′ and j , and so assuming ε -length bounds and Gromov product bounds,the probability that the geodesics are not (3 εLn )-unmatched is at most(2 k ) K ′ c ′ n . Therefore, the probability that ε -length bounds, Gromovbounds and (3 εLn )-unmatching all hold simultaneously is at least 1 − Kc ′ n , for K = (2 k ) K ′ .In order to show Theorem 5 it therefore suffices to show: Proposition 37.
Let H be a random subgroup of G , and let < ε < .If H satisfies ε -length bounds, ( εLn ) -Gromov product bounds, has no ε -large match and is (3 εLn ) -unmatched, then Γ H is (6 , O ( δ, εLn )) -quasi-isometrically embedded in X and H ⋉ E ( G ) is geometrically separated in X . We now prove Theorem 5, assuming Proposition 37.
Proof (of Theorem 5).
The first property in Theorem 5, the fact that eachgenerator w i,n i is hyperbolic and asymmetric, follows from Proposition 16applied to each of the random walks w i,n i .The second property, that Γ H is a quasi-isometrically embedded fol-lows (as we have already observed) if H satisfies ε -length bounds and( εLn )-Gromov product bounds, which hold with probabilities at least1 − Kc n , by Propositions 30 and 32, for constants K and c < ε and the µ i . This then implies that H is freely generated by itsgenerators w i,n i and HE ( G ) = H ⋉ E ( G ).The final property, that H ⋉ E ( G ) is geometrically separated, holds if H satisfies the four conditions, ε -length bounds, ( εLn )-Gromov productbounds, no ε -large match and being (3 εLn )-unmatched, and these holdwith probability at least 1 − K ′ c ′ n , by Propositions 30, 32, 34 and 36,for some constants K ′ and c ′ <
1, depending only on ε and the µ i , asrequired. he final step is to prove Proposition 37. We shall use the follow-ing properties of geodesics and quasigeodesics in a hyperbolic space X ,see for example Bridson and Haefliger [BH99, III.H.1]. If two geodesicsin X are A -fellow travellers, then they are in fact O ( δ )-fellow travellers,outside balls of radius A about their endpoints. Similarly, if two ( A, B )-quasigeodesics are C -fellow travellers, then they are O ( δ, A, B )-fellow trav-ellers outside C -neighbourhoods of their endpoints. Proof (of Proposition 37).
Recall that the (image in X of the) rescaledCayley graph Γ H is the union of translates of geodesic segments γ i from x to w i,n i x by elements of H . Let γ be a geodesic in X connectingtwo points h x and h x of Hx . These two points are also connectedby a path b γ in Γ H , which is a concatenation of geodesic segments γ i ,corresponding to the reduced word determined by h − h in H . The path b γ is an (6 , O ( δ, εLn ))-quasigeodesic in X , which by the Morse property iscontained in an O ( δ, εLn )-neighbourhood of γ .We will show that geometric separation holds for a constant B ( R ) =4 R + O ( δ, εLn ). Let γ and γ ′ be geodesics in X of length at least B , withendpoints in H , and an element g ∈ G , such that gγ is an (2 R + O ( δ ))-fellow traveller with γ ′ . In order to show geometric separation, it sufficesto show that g in fact lies in H ⋉ E ( G ).Let b γ and b γ ′ be the corresponding paths in Γ H connecting the end-points of γ and γ ′ . The quasigeodesics b γ and b γ ′ are (2 R + O ( δ, εLn ))-fellowtravellers in X , and we shall denote their endpoints by b γ (0) and b γ ( T ) for b γ , and b γ ′ (0) and b γ ′ ( T ′ ) for b γ ′ . Therefore, if we set b γ − and b γ ′− to bethe largest union of segments which are translates of the γ i contained in b γ \ ( B X ( b γ (0) ∪ b γ ( T ) , R + O ( δ, εLn )) and b γ ′ \ ( B X ( b γ ′ (0) ∪ b γ ′ ( T ′ ) , R + O ( δ, εLn )), then b γ − and b γ ′− are O ( δ, εLn )-fellow travellers. By a suffi-ciently large choice of B we may assume that the lengths of b γ and b γ ′ areat least 4 R + (1 + ε ) Ln + O ( δ ), and so both b γ − and b γ ′− are non-empty, aswe have assumed that the γ i satisfy ε -length bounds and Gromov productbounds.Each path b γ − or b γ ′− is a concatenation of geodesic segments which aretranslates of the γ i . Let j be the largest index of any path segment whosetranslate appears in either of b γ − or b γ ′− . If the largest index j does notappear in both paths, then up to relabelling, we may assume that j occursin b γ − , and let hγ j be a corresponding geodesic segment in the path b γ − ,for some h ∈ H .We now consider two cases. Either the nearest point projection of hγ j to b γ − is contained in the translate of a single γ i for i j , or hγ j ⊂ b γ − contains a point within distance εLn of some point of the orbit Hx . If thefirst case occurs with i < j , then H has an ε -large match, which we have ssumed does not happen, so ghγ j in fact ( εLn )-fellow travels a translateof itself in b γ − . This means that the translate ghγ j ( εLn )-fellow travels h ′ γ j for some h ′ ∈ H , and so h ′− gh ( εLn + O ( δ ))-stabilizes [ p, w j,n j p ],where p is a nearest point projection of the basepoint x to the axis α j for w j,n j . By Proposition 24, w j,n j is irreversible with asymptotic probabilityone, so h ′− gh does swap the endpoints of the geodesic [ p, w j,n j p ], and byProposition 27, we may assume that w j,n j is ( εLn + O ( δ ))-asymmetric,and so this implies that h ′− gh ∈ h w j,n j i ⋉ E ( G ) ⊂ H ⋉ E ( G ). As both h and h ′ lie in H , this implies that g lies in H ⋉ E ( G ), with asymptoticprobability one, as required.It remains to show that if the second case occurs then H is (3 εLn )-unmatched, as we now explain. Let p be a point in Γ H closest to theinitial point of g ′ γ j , and let q be the point in Γ H closest to the terminalpoint of g ′ γ j . Let hx be the first point of Hx occurring between p and q . Let hw − i γ i be the geodesic segment of Γ H containing p , and let hγ i ′ be the next geodesic segment of Γ H along the geodesic in Γ H from p to q .Finally, let q ′ be a point on hγ i ′ distance εLn from hx . This is illustratedbelow in Figure 6. g ′ γ j hw − i,n i γ i hx hγ i ′ B X ( hx , εLn ) p q ′ Figure 6: A subsegment of the geodesic g b γ − fellow travels Γ H . We now observe that the geodesic in X from p to q ′ is the geodesic η ( i, i ′ , εLnℓ ) used in Definition 35, and so if the second case occurs,then H is not ( εLn )-unmatched, contradicting our initial assumptions on H . References [AMS16] Yago Antolin, Ashot Minasyan, and Alessandro Sisto,
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Alessandro SistoETH Z¨urich [email protected]@math.ethz.ch