Central limit theorems for counting measures in coarse negative curvature
CCENTRAL LIMIT THEOREMS FOR COUNTING MEASURES INCOARSE NEGATIVE CURVATURE
ILYA GEKHTMAN, SAMUEL J. TAYLOR, AND GIULIO TIOZZO
Abstract.
We establish central limit theorems for an action of a group G on ahyperbolic space X with respect to the counting measure on a Cayley graph of G . Our techniques allow us to remove the usual assumptions of properness andsmoothness of the space, or cocompactness of the action. We provide several ap-plications which require our general framework, including to lengths of geodesicsin geometrically finite manifolds and to intersection numbers with submanifolds. Introduction
The goal of this paper is to provide a novel approach to the central limit theoremon groups acting on hyperbolic spaces, for sampling with respect to the word lengthin the group. We shall replace the traditional approach based on thermodynamicformalism with techniques coming from the theory of random walks on groups. Thisallows us to establish new applications, including central limit theorems for lengthsof geodesics in geometrically finite hyperbolic manifolds, for intersection numberswith submanifolds, and for homomorphisms between hyperbolic groups.
Motivation.
The distribution of lengths of closed orbits for smooth flows on man-ifolds has long been a topic of considerable interest. For instance, Sinai [43] andthen Ratner [38] proved a central limit theorem (CLT) for the geodesic flow on ahyperbolic manifold (see also Lalley [27]). One prominent technique, pioneered bySinai [44], Bowen [6], Ruelle [40], Parry–Pollicott [34], and others, uses Markov par-titions to reduce the study of smooth flows to symbolic dynamics to which one canapply tools from thermodynamic formalism. This approach has been successful ina variety of settings, especially applied to Anosov flows and their generalizations.More recently, there has been a renewed interest in statistical properties of ge-odesic length and other geometric quantities with respect to a different sampling,namely according to the counting measure , i.e. uniform measure on spheres in aCayley graph of a finitely generated group G . For instance, Pollicott–Sharp [35]considered the ratio between the word length and the geometric length, while CLTshave been established for quasimorphisms on free groups by Horsham-Sharp [24]and on general hyperbolic groups by Calegari–Fujiwara [7] and Bj¨orklund–Hartnick[5].In [18] the authors, building on [16], settled a conjecture of Chas–Li–Maskit [12]about the distribution of hyperbolic lengths of closed geodesics on compact surfaces Date : April 29, 2020. a r X i v : . [ m a t h . D S ] A p r I. GEKHTMAN, S.J. TAYLOR, AND G. TIOZZO when sampling with respect to word length. Further, a CLT and statistical laws havebeen established for cocompact, proper actions of hyperbolic groups on CAT( − X is CAT( − G (cid:121) X is proper cocompact. While this is the case in the clas-sical setting, they are not satisfied for most actions on Gromov hyperbolic spaces.The goal of this paper is to provide a new approach to the central limit theoremon groups G (cid:121) X acting on hyperbolic spaces, which will allow us to consider inparticular:(1) groups G which are not necessarily word hyperbolic;(2) actions on spaces ( X, d ) which are δ -hyperbolic, but not necessarily CAT( − G (cid:121) X which need not be convex cocompact or even proper;(4) observables φ : G → R which are not necessarily H¨older continuous, and arenot quasimorphisms.For the sake of concreteness, we will now present a version of our main theorem(Theorem 1.1) from which we will then derive several applications. Our discussionhere will be a special case of the most general theorems (Theorems 7.3, 7.4) whichwe will state and prove in Section 7. Main results.
Let G be a finitely generated group acting by isometries on a δ -hyperbolic metric space ( X, d ), and fix a finite generating set S . We require thatthe action is nonelementary in the sense that there are two independent loxodromicelements.Let S n := { g ∈ G : (cid:107) g (cid:107) = n } be the sphere of radius n for the word metric withrespect to S . We denote as N σ the Gaussian measure d N σ ( t ) = √ πσ e − t / σ dt if σ >
0, and the Dirac mass at 0 if σ = 0. We require that G admits a thickbicombing for S and we refer the reader to Section 2.1 for definitions. We note herethat these general conditions are satisfied in a variety of settings; for example, seethe applications below and Lemma 8.1. Theorem 1.1.
Let G be a group which admits a thick bicombing for the generatingset S . Let G (cid:121) X be a nonelementary action by isometries on a δ -hyperbolic space ( X, d ) , and let o ∈ X be a base point. (1) (CLT for displacement) Then there exists (cid:96) > , σ ≥ such that for any a < b we have lim n →∞ S n (cid:26) g ∈ S n : d ( o, go ) − n(cid:96) √ n ∈ [ a, b ] (cid:27) = (cid:90) ba d N σ ( t ) . ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 3 (2) (CLT for translation length)
Moreover, if τ ( g ) denotes the translation lengthof g on X , we also have for any a < b lim n →∞ S n (cid:26) g ∈ S n : τ ( g ) − n(cid:96) √ n ∈ [ a, b ] (cid:27) = (cid:90) ba d N σ ( t ) . (3) Further, σ = 0 if and only if there exists a constant C such that | d ( o, go ) − (cid:96) (cid:107) g (cid:107)| ≤ C for any g ∈ G . We remark that as a consequence of (3), if σ = 0 then the translation length ofany g ∈ G with respect to its action on X is a constant multiple of its translationlength in the word metric. Moreover, if the action G (cid:121) X is not proper, then σ > d ( o, go ) − n(cid:96) √ n −→ N σ , hence from now on we will use the above notation as a shorthand. Applications.
There are a number of applications to the above theorems and wesummarize a few of them here. For the proofs, see Section 8.
Geometrically finite hyperbolic manifolds.
First, let us state an extension of ourprevious work on surfaces [18] to general hyperbolic manifolds, possibly with cusps.If M = H n / Γ is a hyperbolic manifold and γ ∈ Γ = π ( M ), then we set (cid:96) ( γ ) to be thelength of the geodesic freely homotopic to γ unless γ is peripheral (i.e. homotopicinto a cusp), in which case we set (cid:96) ( γ ) = 0. Theorem 1.2.
Suppose that M is a geometrically finite hyperbolic manifold and let S (cid:48) be any generating set for π ( M ) . Then there is a finite generating set S ⊃ S (cid:48) and (cid:96), σ > such that (cid:96) ( γ ) − n(cid:96) √ n −→ N σ , where γ is chosen uniformly at random in the sphere of radius n with respect to S .If moreover π ( M ) is word hyperbolic, then we can take S = S (cid:48) . The statement includes the cases where M is either finite volume or convex co-compact, and is new even when M is a finite area surface. We remark that when M is either convex cocompact or a surface, the above theorem works for any gen-erating set S . In the convex cocompact case, the needed action π ( M ) (cid:121) H n issufficiently tame so that the techniques of thermodynamics may be applicable ([35],[10]). However, this is not the case when the manifold M has cusps.We note that Theorem 1.2 further extends to manifolds of variable negative cur-vature, as long as the peripheral subgroups are virtually abelian, and the same proofapplies. I. GEKHTMAN, S.J. TAYLOR, AND G. TIOZZO
Geometrically infinite 3-manifolds.
In the case of 3-manifolds, the previous resultcan be strengthened further as follows.
Theorem 1.3.
Let M be a hyperbolic –manifold such that π ( M ) is finitely gen-erated and not virtually abelian. Suppose further that M does not have any rank cusps. Then for any finite generating set S of π ( M ) , there are (cid:96), σ > such that (cid:96) ( γ ) − n(cid:96) √ n −→ N σ , where γ is chosen uniformly at random in the sphere of radius n with respect to S .Moreover, if M has rank cusps, the same statement holds after enlarging thegenerating set as in Theorem 1.2. To the authors’ knowledge, this is the first CLT for lengths of closed geodesicsfor possibly geometrically infinite 3–manifolds.
Intersection numbers with a submanifold.
For our next application, the requiredactions are on locally infinite trees, which are nonproper hyperbolic spaces.Let M be a smooth orientable manifold and Σ a smooth orientable codimension − π -injective on each component. We say Σ is fiber-like if foreach boundary component of the cut manifold M | Σ its induced subgroup in thefundamental group of the corresponding component of M | Σ has index at most 2.For γ ∈ π ( M ), let i ( γ, Σ) denote the minimal intersection number of Σ withloops in M freely homotopic to γ . Theorem 1.4.
Suppose that M is a closed orientable hyperbolic manifold and let S be any generating set for π ( M ) . Let Σ be a smooth orientable codimension − submanifold that is π -injective but not fiber-like. Then there are (cid:96), σ > such that i ( γ, Σ) − (cid:96)n √ n −→ N σ , where γ is chosen uniformly at random in the sphere of radius n with respect to S . The theorem is new even for surfaces; in that context, Chas–Lalley [11] proveda CLT for self-intersection numbers of curves with respect to word length. Follow-ing Theorem 1.2, a similar result could be formulated for more general hyperbolicmanifolds.
Homomorphisms between hyperbolic groups.
Our next application is to homomor-phisms between hyperbolic groups. Interestingly, the condition for nonzero variancecan be recast in terms of the induced Patterson–Sullivan measures.
Theorem 1.5.
Suppose that φ : G → G (cid:48) is a homomorphism between hyperbolicgroups such that the image of φ is not virtually cyclic. For any fixed generating sets S and S (cid:48) of G and G (cid:48) , respectively, there are (cid:96) > and σ ≥ such that (cid:107) φ ( g ) (cid:107) S (cid:48) − (cid:96) (cid:107) g (cid:107) S (cid:112) (cid:107) g (cid:107) S −→ N σ , for g ∈ G chosen uniformly at random in the sphere of radius n with respect to S . ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 5
Moreover, σ = 0 (i.e. the Gaussian is degenerate) if and only if φ has finite ker-nel and the induced homeomorphism ∂φ : ∂G → ∂G (cid:48) pushes the Patterson–Sullivanmeasure class for ( G, S ) to the Patterson–Sullivan measure class for ( φ ( G ) , S (cid:48) ) . The above result generalizes [10, Theorem 1.6], who proved a CLT where φ is theabelianization homomorphism, which in turn generalizes work of Rivin [39] for freegroups. This is also a generalization of Calegari-Fujiwara [7, Corollary 4.27]. Hyperplanes crossed in right-angled Artin and Coxeter groups.
Our final applicationis to a collection of groups that is not necessarily relatively hyperbolic.Suppose that G is a right-angled Artin group or right-angled Coxeter group thatis not a direct product. Let V be its set of vertex generators. For each v ∈ V ,define a function v : G → Z that counts the number of occurrences of v ± in ashortest spelling of g ∈ G with respect to V . Equivalently, v ( g ) is the number ofhyperplanes labeled by v separating o and go in the cube complex associated to G . Theorem 1.6.
For G as above, there are (cid:96), σ > such that for any vertex v , v ( g ) − (cid:96)n √ n −→ N σ , where g is chosen uniformly at random in the sphere of radius n with respect to thevertex generators. We conclude by noting that our methods are sufficiently general to apply beyondthe case of ‘nonpositively curved’ groups. Moreover, we do not need to assumethat our counting measures are associated to geodesic combings. See Theorems7.3, 7.4 for the most general result. For example, by using the standard graphstructure associated to the language of geodesics for a free group, we obtain a CLTfor nonbacktracking random walks on any group with a nonelementary action on ahyperbolic space X . From thermodynamics to random walks.
Most central limit theorems forcounting measures established so far use a coding for geodesics with finite paths,and then apply classical results in thermodynamic formalism, like the existence anduniqueness of Gibbs measures for shifts of finite type. There, the observable is as-sumed to be H¨older continuous with respect to the standard metric on the shiftspace.In this paper, instead, we do not assume any good geometric property on theaction. Let us recall that displacement is not a quasimorphism, is in general not weakly combable (in the language of [7]) if the action is not convex cocompact,and it is not a
H¨older weight function in the sense of [35, Proposition 1] if X isnot CAT( − ζ -functions (as in e.g. [34], [10]).Rather, our general strategy is as follows. I. GEKHTMAN, S.J. TAYLOR, AND G. TIOZZO (1) We start with a graph structure , i.e. a graph whose paths parameterize thegroup elements we want to count. We first consider a vertex v of this graph,and consider a random walk on the semigroup Γ v of loops based at thisvertex. Here, we apply the CLT for cocycles for groups acting on hyperbolicspaces, as devised by Benoist-Quint [4] and generalized by Horbez [23] toactions on nonproper spaces.(2) Then, we consider the set of paths in a maximal component for the graph asa suspension on the space of loops at v , and we apply results of Melbourne-T¨or¨ok [28] to “lift” the CLT to the suspended transformation. To be precise,we need to consider a skew product over the shift space.(3) Now, we note that a thick graph structure is almost semisimple , hence thereexists a power p for which the transition matrix M p is semisimple. We usethis to prove that the counting measure starting at an initial vertex convergesto a convex combination of stationary measures for the Markov chains onthe maximal components.(4) Using biautomaticity, we show that all the CLTs for all Markov chains havethe same mean and variance. This implies a CLT for the counting measureon the set of paths starting at any vertex in a semisimple structure.(5) Finally, for a general thick structure of period p we condition on the firstprefix of length r ; since all these distributions for the conditional measuresconverge to the same law (by (4) above), the CLT for the entire sequenceholds. Acknowledgments.
Gekhtman is partially supported by NSERC. Taylor is par-tially supported by NSF grant DMS-1744551 and the Sloan Foundation. Tiozzo ispartially supported by NSERC and the Sloan Foundation.2.
Background
Graph structures for countable groups.
Given a countable group G , wedefine a graph structure on G as a triple (Γ , v , ev), where Γ is finite, directed graph, v is a vertex of Γ which we call its initial vertex , and ev : E (Γ) → G is a map thatlabels the edges of Γ with group elements. Given this data, we extend the map ev bydefining for each finite path g = g . . . g n the group element ev( g ) = ev( g ) . . . ev( g n ).To simplify notation, we will use g = ev( g ) to denote the group element associatedto the path g . We denote as (cid:107) g (cid:107) the length of the path g . Throughout the paper, weassume that the graph structure is proper in the sense that for each group elementthere are at most finitely many paths in the graph that evaluate to it.For a graph structure Γ, we define Ω to be the set of all infinite paths start-ing at any vertex of Γ and σ : Ω → Ω to be the shift map. Given a path ω =( g , . . . , g n , . . . ), we denote as w n := g . . . g n its prefix of length n .We define two vertices v i , v j to be equivalent if there is a path from v i to v j anda path from v j to v i , and the components of Γ as the equivalence classes for thisrelation.We will denote by M the transition matrix for Γ. By Perron-Frobenius, M hasa real eigenvalue of largest modulus, which we will denote by λ . Moreover, such a ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 7 matrix is almost semisimple if for any eigenvalue of maximal modulus, its geometricand algebraic multiplicity agree. Furthermore, such a matrix is semisimple if its onlyeigenvalue of maximal modulus is real positive. We call a graph structure (almost)semisimple if its associated transition matrix is.Let Γ be almost semisimple, and let λ be the leading eigenvalue of M . Then wedefine a vertex v to be of large growth iflim n →∞ n log { paths of length n starting at v } = λ and of small growth otherwise (in which case, the limit above is < λ ). Furthermore,a component C is maximal iflim n →∞ n log { paths of length n inside C } = λ. As discussed in [17], the global structure of Γ is as follows: there is no pathbetween maximal components and vertices of large growth are precisely the oneswhich have a path to a maximal component.Given a vertex v , we denote as Γ v the loop semigroup of v , i.e. the set of all finitepaths from v to itself. This is a semigroup under concatenation, and all its elementslie entirely in the component of v . We denote as Γ v the image of Γ v in G under theevaluation map. Definition 2.1 (Thick graph structure) . A graph structure Γ is thick if for anyvertex v in a maximal component, there exists a finite set B ⊆ G such that G = B · Γ v · B where the equality is in the group G .In what follows, we often make the evaluation map implicit in our notation. Inparticular, if G acts on a metric space ( X, d ), o ∈ X is a base point, and g is a finitepath in Γ, we will often write go to mean the point go ∈ X .The next lemma summarizes some properties of thick graph structures that wewill need in the sequel. Lemma 2.2.
A thick graph structure Γ is almost semisimple. Moreover, if G (cid:121) X is a nonelementary action on a hyperbolic space, then the actions of both semigroups Γ v and Γ − v on X are also nonelementary, for each vertex v contained in a maximalcomponent.Proof. If the transition matrix M of Γ is not almost semisimple, then M has a Jordanblock for an eigenvalue of modulus λ of size k ≥ n k − λ n . However, thisis impossible if Γ is thick because in this case the growth of closed paths is no morethan the growth of closed paths in Γ v for v in a maximal component. This, in turn,is bounded by a constant times λ n since it is no more than the growth of closedpaths in its maximal component.The statement that the action of Γ v (and hence Γ − v ) on X is nonelementary isproven in [17, Proposition 6.3]. (cid:3) I. GEKHTMAN, S.J. TAYLOR, AND G. TIOZZO
Bicombings.
For particular applications, it is also useful to define the notion of ageodesic graph structure. A graph structure Γ is geodesic if the length (cid:107) g (cid:107) of anypath g is equal the word length of g in the subgroup generated by the edge labels,using edge labels as the (finite) generating set. A geodesic graph structure is calleda geodesic combing if, in addition, the evaluation map is a bijection from the set offinite paths starting at v to the set of elements of G . We say that Γ is a geodesiccombing associated to a finite generating set S if, up to adding inverses, S is the setof edge labels for the graph structure. In this case, (cid:107) g (cid:107) is equal to the word lengthof g with respect to S .We will make use of the following notion of biautomatic. See, for example, [32]and [14, Lemma 2.5.5]. First, fix a word metric d G on G . For a finite path g in Γ,we denote by g ( i ) the length i prefix of g when i ≤ (cid:107) g (cid:107) and set g ( i ) = g otherwise. Definition 2.3 (Biautomatic graph structure) . A graph structure Γ for G is biau-tomatic if the following holds. For any finite set B ⊆ G there exists C ≥ g and h are finite length paths in Γ, and g = b hb in G , with b , b ∈ B , then d G ( g ( i ) , b h ( i )) ≤ C, for all i ≥ Definition 2.4. A bicombing for the generating set S on a group G is a geodesiccombing whose graph structure Γ is biautomatic, and is thick if the graph structureis thick.We emphasize that the geodesic condition is used in the applications of our maintheorem (as in Theorem 1.1–1.6), but is not required in the proof of the most generalresults, Theorems 7.3, 7.4.2.2. Cocycles and horofunctions.
Let (
X, d ) be a metric space, and let o ∈ X be a base point. Given z ∈ X , we define the Busemann function ρ z : X → R as ρ z ( x ) := d ( x, z ) − d ( o, z ) . Thus, setting Φ( z ) := ρ z defines a map Φ : X → Lip o ( X )where Lip o ( X ) is the space of 1-Lipschitz functions on X which vanishes at o .We define the horofunction compactification X h as the closure of Φ( X ) in Lip o ( X ),with respect to the topology of pointwise convergence. Elements of X h will be called horofunctions . We denote as X h ∞ the space of infinite horofunctions , i.e. the set of h ∈ X h such that inf x ∈ X h ( x ) = −∞ .For any ξ ∈ X h , the Busemann cocycle is defined as β ξ ( x, y ) := lim z n → ξ [ d ( y, z n ) − d ( x, z n )]= h ξ ( y ) − h ξ ( x ) , ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 9 where h ξ is the horofunction associated to ξ . This has the usual cocycle property β ξ ( x, z ) = β ξ ( x, y ) + β ξ ( y, z ). Remark 2.5.
Benoist-Quint [4] and Horbez [23] define B : G × X h → R by B ( g, ξ ) = h ξ ( g − o ) . To compare their definition with ours: B ( g, ξ ) = h ξ ( g − o ) = lim z n → ξ [ d ( g − o, z n ) − d ( o, z n )] = β ξ ( o, g − o ) . CLT for random walks on the loop semigroups
Let Γ be a graph structure for G , and let v be a vertex in a maximal component.Recall that a loop is prime if it is not itself a product of nontrivial loops, and thatprime loops generate Γ v as a semigroup.Given a probability measure µ on the set of edges of Γ, one defines the first returnmeasure µ v on Γ v as follows: if l = g . . . g n is a prime loop in Γ v , then we set µ v ( l ) := µ ( g ) · · · µ ( g n ) . We set µ v ( l ) = 0 for all other loops. Note that inversion defines a map Γ v → Γ − v andwe define the measure ˇ µ on Γ − v by ˇ µ ( l ) = µ ( l − ). These measures push forward tomeasures on the group G under the evaluation map. We say that µ v is nondegenerate if it gives positive measure to any prime loop of Γ v .Let M be a metric space on which G acts by homeomorphisms. A measure ν on M is µ -stationary if ν = (cid:82) G g (cid:63) ν dµ ( g ), and µ -ergodic if it cannot be written as anontrivial convex combination of µ -stationary measures.3.1. Central limit theorems for cocycles.
Recall that a cocycle is a function σ : G × M → R such that σ ( gh, x ) = σ ( g, hx ) + σ ( h, x ) , ∀ g, h ∈ G, ∀ x ∈ M . A cocycle σ : G × M → R has constant drift λ if there exists λ ∈ R such that (cid:90) G σ ( g, x ) dµ ( g ) = λ for any x ∈ M . A cocycle σ : G × M → R is centerable if it can be written as σ ( g, x ) = σ ( g, x ) + ψ ( x ) − ψ ( g · x )where σ is a cocycle with constant drift and where ψ : M → R is a bounded,measurable function. In this case, we say that σ is the centering of σ ; note that λ = (cid:82) G ×M σ ( g, x ) dµ ( g ) dν ( x ) for any µ -stationary ν . We say that the cocycle σ has finite second moment with respect to a measure µ on G if (cid:90) G sup x ∈M | σ ( g, x ) | dµ ( g ) < + ∞ . We now use the following CLT for centerable cocycles: as remarked in [23, Remark1.7], the proof is exactly the same as the proof of [4, Theorem 4.7].
Theorem 3.1 (Central limit theorem for cocycles) . Let G be a discrete group, M be a compact metrizable G -space and µ a probability measure on G . Let ν be a µ -ergodic, µ -stationary probability measure on M , and let M be a G -invariant subsetof M of full ν -measure. Let σ : G × M → R be a centerable cocycle with drift λ and finite second moment. Then there exist σ ≥ such that for any continuous F : R → R with compact support, we have for ν -a.e. x ∈ M , lim n →∞ (cid:90) G F (cid:18) σ ( g, x ) − nλ √ n (cid:19) dµ ∗ n ( g ) = (cid:90) R F ( t ) d N σ ( t ) . We now apply this result to the loop semigroup. Let F v be the group freelygenerated by the prime loops in Γ v .Let N : Γ v → Z be the semigroup homomorphism N ( g ) := −(cid:107) g (cid:107) , where (cid:107) g (cid:107) is thelength in Γ of the loop g . There is a natural inclusion Γ v → F v as a subsemigroupand we can extend the semigroup homomorphism above to a group homomorphism N : F v → Z . Moreover, we also extend the natural semigroup homomorphism Γ v → G , induced by evaluation, to a group homomorphism e : F v → G . Now, using thehomomorphism e : F v → G , the free group F v has a nonelementary action on X ,and moreover µ ∗ nv is supported on Γ v ⊆ F v for all n ≥ (cid:96) ∈ R to be specified below, we define η : F v × X h → R as η ( g, ξ ) := β ξ ( o, g − o ) − (cid:96)N ( g ) . Lemma 3.2.
Suppose that the action of Γ v on X is nonelementary and µ v is non-degenerate. Then for any (cid:96) ∈ R , the restriction of η : F v × X h → R to F v × X h ∞ isa centerable cocycle.Proof. We have η ( gh, ξ ) = β ξ ( o, h − g − o ) − (cid:96)N ( gh )= β ξ ( o, h − o ) + β ξ ( h − o, h − g − o ) − (cid:96)N ( g ) − (cid:96)N ( h )= β ξ ( o, h − o ) + β hξ ( o, g − o ) − (cid:96)N ( g ) − (cid:96)N ( h )= η ( h, ξ ) + η ( g, hξ )hence η is a cocycle. Moreover, by [23, Proposition 1.5], using [23, Corollary 2.7] and[23, Proposition 2.8], the cocycle B ( g, ξ ) = β ξ ( o, g − o ) is centerable on F v × X h ∞ .Then, since η ( g, ξ ) − B ( g, ξ ) = (cid:96)N ( g ) is a homomorphism and depends only on g ,we have that η ( g, ξ ) is also centerable on F v × X h ∞ . (cid:3) Thus, as a consequence of Theorem 3.1, we obtain the following.
Corollary 3.3.
Let Γ be a thick structure, let v be a vertex in a maximal componentof Γ . Suppose that the first return measure µ v is nondegenerate, and let ν v be a ˇ µ v -ergodic, ˇ µ v -stationary measure on X h . Then there exist (cid:96), σ ≥ such that for anycontinuous F : R → R with compact support, we have for ν v -a.e. ξ , lim n →∞ (cid:90) G F (cid:18) β ξ ( o, go ) − (cid:96) (cid:107) g (cid:107)√ n (cid:19) dµ ∗ nv ( g ) = (cid:90) R F ( t ) d N σ ( t ) . ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 11
Proof.
We apply Theorem 3.1 to the measure ˇ µ v , supported on Γ − v , where (cid:96) ischosen so that λ = (cid:82) F v × X h η ( g, ξ ) d ˇ µ v ( g ) dν v ( ξ ) = 0. Note that by [28, Proposition4.4] and the fact that Γ − v is nonelementary, we have ν v ( X h ∞ ) = 1. Moreover, forany g ∈ Γ v we have η ( g − , ξ ) = β ξ ( o, go ) − (cid:96) (cid:107) g (cid:107) . (cid:3) Skew products and invariance on the loop semigroup.
Let M be acompact metric space with a continuous G -action. We define the skew product T : Ω × M → Ω × M as T ( ω, ξ ) := ( σ ( ω ) , g − ξ )where ω = ( g , g , . . . ).A graph structure Γ is primitive if its associated transition matrix M is primitive,i.e. has a positive power. Now let Γ be a primitive graph structure, let v be a vertexof Γ, let Γ v be the loop semigroup, and let µ v be the first return measure.Finally, let Ω v = (Γ v ) N with shift map σ v . To highlight the difference, we denotethe elements of Γ N v as ( l , l , . . . ), since each element of the sequence is a loop, whilethe elements of Ω will be denoted as ω = ( g , g , . . . ), since its elements are edges.Let us define the map T v : Ω v × M → Ω v × M as T v ( ω, ξ ) = ( σ v ( ω ) , l − ξ ) . Lemma 3.4.
A measure ν on M is ˇ µ v -stationary if and only if µ N v ⊗ ν is T v -invariant.Proof. Fix C ⊂ Ω v measurable and let C l ⊂ Ω v be the subset consisting of sequencesbeginning with l ∈ Γ v such that σ v ( C l ) = C . Then for any A ⊂ M measurable, T − v ( C × A ) = (cid:91) l C l × lA. Since µ N v ⊗ ν (cid:32)(cid:91) l C l × lA (cid:33) = µ ( C ) (cid:88) l µ ( l ) ν ( lA )= µ ( C ) (cid:88) l ˇ µ ( l ) l ∗ ν ( A ) , the lemma follows. (cid:3) Lemma 3.5.
There exists an ergodic ˇ µ v -stationary measure ν v on M such that theproduct measure µ N v ⊗ ν v is T v -invariant and ergodic.Proof. Since M is a compact metric space, there exists a ˇ µ v -stationary measure ν on M . Then by Lemma 3.4 the measure λ := µ N v ⊗ ν is T v -invariant. If λ isnot ergodic, let us consider its ergodic decomposition, and take one of its ergodiccomponents λ v . By definition, λ v (cid:28) λ and λ v is T v -invariant and ergodic. Thenby [31, Corollary 3.1], λ v is of the form λ v = µ N v ⊗ ν v for some measure ν v on M .Finally, again by Lemma 3.4, the measure ν v is ˇ µ v -stationary. (cid:3) Lemma 3.6.
Consider the function f : Ω × X h → R defined as f ( ω, ξ ) := β ξ ( o, g o ) . Then for any n we have (1) n − (cid:88) j =0 f ( T j ( ω, ξ )) = β ξ ( o, w n o ) . Proof.
The cocycle property implies β ξ ( o, w n o ) = n − (cid:88) j =0 β ξ ( w j o, w j +1 o ) = n − (cid:88) j =0 β w − j ξ ( o, g j +1 o )for any ξ ∈ X h . Moreover, by definition and G -equivariance we have f ( T j ( ω, ξ )) = β w − j ξ ( o, g j +1 o )and the claim follows. (cid:3) An analogous statement holds by replacing T, Ω by T v , Ω v .4. CLT for Markov chains of primitive graph structures
We begin by recalling the following: If Γ is a directed graph whose transitionmatrix M is primitive with leading eigenvalue λ , thenlim n →∞ M n λ n = ρu T , where M ρ = λρ , u T M = λu T , and u T ρ = 1. Then the stationary measure for thecorresponding Markov chain is given by setting the starting probability at vertex v i as π i = ρ i u i and assigning to an edge from v i to v j the transition probability p ij = ρ j λρ i . This gives the measure of maximal entropy P for the path space Ω ofΓ, also called the Parry measure [33]. For a vertex v of Γ, we use P v to denotethe measure on the space of paths Ω v ⊂ Ω starting at v obtained by beginning theMarkov chain at v and using the above transition probabilities. From now on weuse these edge probabilities to define the first return measure µ v as in Section 3.In this section we prove the following result. Theorem 4.1.
Suppose that Γ is a primitive graph structure and let µ n be the n -thstep distribution of the Markov chain on Γ . There are constants (cid:96) and σ such thatfor any continuous function F : R → R with compact support, we have lim n →∞ (cid:90) G F (cid:18) d ( o, go ) − n(cid:96) √ n (cid:19) dµ n ( g ) = (cid:90) R F ( t ) d N σ ( t ) . The main technique to obtain the CLT for the Markov chain as above from the onefrom the random walk on the loop semigroup is using a suspension flow , adaptingthe approach of Melbourne-T¨or¨ok [30] for dynamical systems.
ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 13
Suspension flows.
Let S : ( X , λ ) → ( X , λ ) be a measure-preserving dynam-ical system, and let r : X → N be a measurable, integrable function, which we callthe roof function . Then the discrete suspension flow of S with roof function r is thedynamical system given by the map (cid:98) S : (cid:98) X → (cid:98) X where (cid:98) X := { ( x, n ) ∈ X × N : 0 ≤ n ≤ r ( x ) − } with measure (cid:98) λ := r ( λ ⊗ δ ), where δ is the counting measure on N and r := (cid:82) X r dλ .Then, the map (cid:98) S is defined as (cid:98) S ( x, n ) = (cid:26) ( x, n + 1) if n ≤ r ( x ) − S ( x ) ,
0) if n = r ( x ) − . Since in this case the system has discrete time, the above construction is alsocalled a
Kakutani skyscraper .The main theorem of Melbourne-T¨orok [30, Theorem 1.1] is the following.
Theorem 4.2.
Let S : ( X , λ ) → ( X , λ ) be an ergodic, measure-preserving trans-formation, and let (cid:98) S : ( (cid:98) X , (cid:98) λ ) → ( (cid:98) X , (cid:98) λ ) be the suspension flow with roof function r .Let φ : (cid:98) X → R be such that (cid:82) φ d (cid:98) λ = 0 , and define Φ( x ) := (cid:80) r ( x ) − k =0 φ ( x, k ) . Let φ ∈ L b ( (cid:98) X ) and let r ∈ L a ( X ) be the roof function, with (1 − /a )(1 − /b ) ≥ / .Suppose that Φ and r satisfy a CLT. Then φ satisfies a CLT.Moreover, if the CLT for Φ has variance σ , then the CLT for φ has variance σ = σ r . Invariant measure on the suspended space.
For any ω ∈ Ω v , let r ( ω ) bethe length in Γ of l ( ω ). This is the first return time for the loop determined by ω .Let us define the suspension of the skew productΩ ( s ) := { ( ω, k, ξ ) ∈ Ω v × N × M : 0 ≤ k ≤ r ( ω ) − } and (cid:98) T ( ω, k, ξ ) = (cid:26) ( ω, k + 1 , ξ ) if k ≤ r ( ω ) − σ v ( ω ) , , l − ξ ) if k = r ( ω ) − . Let us now denote R := (cid:82) Γ v (cid:107) g (cid:107) dµ v ( g ) = (cid:82) r ( ω ) d P v ( ω ) and define the probabilitymeasure ν ( s ) := R (cid:0) µ N v ⊗ δ ⊗ ν v (cid:1) on Ω ( s ) . Lemma 4.3.
Let ν v be ˇ µ v -stationary measure constructed in Lemma 3.5. Then ν ( s ) on Ω ( s ) is (cid:98) T -invariant and ergodic.Proof. It suffices to check invariance of the measure using cylinder sets C l ,...,l n consisting of loops beginning with l . . . l n . We have (cid:98) T − ( C l ,...,l n × { k } × A ) = (cid:26) C l ,...,l n × { k − } × A if k > (cid:70) l ∈ P v C l,l ,...,l n × {(cid:107) l (cid:107) − } × lA if k = 0where P v ⊆ Γ v is the set of prime loops. Hence in the first case, the equality ν ( s ) ( (cid:98) T − ( C l ,...,l n × { k } × A )) = ν ( s ) ( C l ,...,l n × { k } × A ) is obvious. In the second case, ν ( s ) ( (cid:98) T − ( C l ,...,l n × { k } × A )) = 1 R (cid:88) l ∈ P v µ v ( l ) µ v ( l ) . . . µ v ( l n ) ν v ( lA )= 1 R µ v ( l ) . . . µ v ( l n ) (cid:88) l ∈ P v µ v ( l ) ν v ( lA )= 1 R µ v ( l ) . . . µ v ( l n ) ν v ( A )= ν ( s ) ( C l ,...,l n × { k } × A )hence ν ( s ) is (cid:98) T -invariant. Moreover, the suspension of an ergodic measure is ergodic,see e.g. [41, Proposition 1.11]. (cid:3) Pushforward of the (cid:98) T -invariant measure to Ω × M . Recall that Ω is thespace of all infinite sample paths in Γ starting at any vertex. Let us define theprojection π : Ω ( s ) → Ω × M as π ( ω, k, ξ ) = ( σ k ( ω ) , ( g . . . g k ) − ξ )and recall the skew product T : Ω × M → Ω × M is T ( ω, ξ ) := ( σ ( ω ) , g − ξ ) . Lemma 4.4.
The following diagram commutes: (Ω ( s ) , ν ( s ) ) (cid:98) T (cid:11) (cid:11) π (cid:47) (cid:47) (cid:15) (cid:15) Ω × M T (cid:7) (cid:7) (Ω v × M , µ N v ⊗ ν v ) As a consequence, in the hypotheses of the previous lemmas, the measure ν := π (cid:63) ν ( s ) is T -invariant and ergodic.Proof. We show that the horizontal arrow is equivariant for the shifts. This followsfrom the fact that if we write l ( ω ) for the first return loop of ω then l ( ω ) = g ( ω ) . . . g r ( ω ) ( ω ). Hence, π ◦ (cid:98) T ( ω, r ( ω ) − , ξ ) = π (( σ v ( ω ) , , l − ξ ))= ( σ r ( ω ) ( ω ) , ( g . . . g r ( ω ) ) − ξ ) , which is equal to T ◦ π (( ω, r ( ω ) − , ξ )). The other cases being trivial, this provesthe first statement.Finally, since ν is the pushforward of an ergodic measure, it is ergodic. (cid:3) ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 15
Return times and invariant measures for the Markov chain.
Recallthat in the previous section we produced a measure ν v on M which is ˇ µ v -stationaryand such that the product measure µ N v ⊗ ν v is T v -invariant and ergodic. Then, bylifting it to the suspension and pushing it forward to Ω × M , we have an ergodic, T -invariant measure ν on Ω × M .Now, for any vertex w other than v we define the measure ν w on M as(2) ν w ( A ) = (cid:88) γ ∈ Γ v,w µ ( γ ) ν v ( γA )where the sum is over the set Γ v,w of all paths γ from v to w which do not passthrough v in their middle, and µ ( γ ) is the product of the measures of the edges of γ . Recall also we denote as P w the Markov chain measure on the space of infinitesample paths starting at w . Lemma 4.5.
We have ν := 1 R (cid:88) w P w ⊗ ν w . Proof.
Let w be a vertex, and let g , g , . . . , g n be a finite path starting from w . Wehave for any measurable A ⊆ M π − ( C g ,...,g n × A ) = (cid:26) C g ,...,g n × { } × A if w = v (cid:70) γ ∈ Γ v,w C γ,g ,...,g n × {| γ |} × γA if w (cid:54) = v where the union is over the set Γ v,w of all paths γ from v to w = s ( g ) which do notpass through v in their middle. Thus we have ν ( C g ,...,g n × A ) = 1 R (cid:88) γ ∈ Γ v,w µ ( γ ) µ ( g ) . . . µ ( g n ) ν v ( γA )= 1 R µ ( g ) . . . µ ( g n ) ν w ( A )= 1 R P w ( C g ,...,g n ) ν w ( A )which proves the claim, since both measures agree on all rectangles. (cid:3) Recall that R = (cid:82) r ( ω ) d P v ( ω ), and set n w = ν w ( M ). Here we show Lemma 4.6.
We have the identities: (1) R = π v (2) π w = n w R for any vertex w of Γ . Note that if we replace Γ with the graph Γ obtained by reversing the directionof each edge, then the transition matrix for Γ is M T and so we have that ρ and u switch roles. In particular, new transition probabilities on edges from v i to v j are (interms of the quantities defined in Section 4) p ij = u j λu i but the stationary measureon vertices is unchanged. Proof of Lemma 4.6. (1) is well-known. To prove (2), recall that Γ v,w is the set ofall paths γ from v to w which do not pass through v in their middle. Hence, if wereverse all the paths in this set, we obtain Γ v,w the set of all paths γ from w to v which do not pass through v in their middle. Note that since almost every pathstaring at w passes through v (cid:88) γ ∈ Γ v,w µ ( γ )= u v u w (cid:88) γ ∈ Γ v,w λ −| γ | , where µ ( γ ) is the product of the measures of the edges of γ with respect to p andwe have used our previous observation about p .Using this and the fact that (cid:88) γ ∈ Γ v,w λ −| γ | = (cid:88) γ ∈ Γ v,w λ −| γ | , we compute, n w = ν w ( M )= (cid:88) γ ∈ Γ v,w µ ( γ ) = ρ w ρ v (cid:88) γ ∈ Γ v,w λ −| γ | = ρ w ρ v · u w u v = π w π v . Hence, the lemma follows from (1). (cid:3)
The Central Limit Theorem for the Markov chain.
We are now in aposition to prove Theorem 4.1. By Melbourne-T¨or¨ok ([30], Theorem 1.1), we have:
Proposition 4.7.
Let φ : Ω × X h → R belong to L b (Ω × X h , ν ) for some b > , andlet m := (cid:82) φ dν . Define Φ : Ω v × X h → R as Φ( ω, ξ ) := (cid:80) r ( ω ) − k =0 φ ( T k ( ω, ξ )) − mr ( ω ) ,and suppose that (cid:80) n − j =0 Φ ◦ T jv √ n converges to a normal distribution in probability on (Ω v × X h , µ N v ⊗ ν v ) . Then thesequence (cid:80) n − j =0 φ ◦ T j − nm √ n converges to a normal distribution in probability on (Ω × X h , ν ) .Proof. Note that since r has exponential tail, it belongs to L a (Ω v ) for any a ≥ − /a )(1 − /b ) ≥ / b >
2. Moreover,( r ◦ T nv ( ω )) n is a sequence of independent random variables and so it satisfies a CLT.Hence, we can apply Theorem 4.2 to obtain a central limit theorem for the observable ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 17 φ ◦ π − m and the system (cid:98) T , with measure ν ( s ) . Moreover, since φ ◦ π ◦ (cid:98) T n = φ ◦ T n ◦ π by Lemma 4.4, this is equivalent to a central limit theorem for the observable φ onthe system T with the measure π ∗ ( ν ( s ) ) = ν . (cid:3) Proposition 4.8.
There exist (cid:96), σ such that for any continuous, compactly supported F : R → R one has (cid:90) Ω × X h F (cid:18) β ξ ( o, g . . . g n o ) − n(cid:96) √ n (cid:19) dν ( ω, ξ ) → (cid:90) R F ( t ) d N σ ( t ) , as n → ∞ .Proof. Let us apply the previous Proposition with φ = f where f : Ω × X h → R isdefined as f ( ω, ξ ) := β ξ ( o, g o ). Then by definition of Φ and f , Lemma 3.6 givesthat for every ω ∈ Ω v Φ( ω, ξ ) = r ( ω ) − (cid:88) k =0 f ( T k ( ω, ξ )) − (cid:96)r ( ω )= β ξ ( o, w r ( ω ) o ) − (cid:96)r ( ω ) =: f v ( ω, ξ ) , where (cid:96) = m = (cid:82) β ξ ( o,go ) dµ v ( g ) dν v ( ξ ) (cid:82) (cid:107) g (cid:107) dµ v ( g ) . Now, by Corollary 3.3, integrating in dν v wehave for some σ ≥ (cid:90) G × X h F (cid:18) β ξ ( o, go ) − (cid:96) (cid:107) g (cid:107)√ n (cid:19) dµ ∗ nv ( g ) dν v ( ξ ) → (cid:90) R F ( t ) d N σ ( t ) . Note moreover that β ξ ( o, l . . . l n o ) − (cid:96) (cid:107) l . . . l n (cid:107) = (cid:80) n − j =0 f v ( T jv ( ω, ξ )), hence we canrewrite the above equation as (cid:90) Ω v × X h F (cid:32) (cid:80) n − j =0 f v ( T jv ( ω, ξ )) √ n (cid:33) d ( µ N v ⊗ ν v )( ω, ξ ) → (cid:90) R F ( t ) d N σ ( t ) . Thus, by Proposition 4.7 and the above calculation, we also have (for some different σ ) (cid:90) Ω v × X h F (cid:32) (cid:80) n − j =0 f ◦ T j ( ω, ξ ) − n(cid:96) √ n (cid:33) dν ( ω, ξ ) → (cid:90) R F ( t ) d N σ ( t ) . The claim follows by again using that by Lemma 3.6, we have (cid:80) n − j =0 f ◦ T j ( ω, ξ ) = β ξ ( o, g . . . g n o ). (cid:3) Now, we will need to go from the CLT for the Busemann cocycle to the one fordisplacement. To do so, we use the following variation of [4, Proposition 3.3].By [28, Proposition 4.4] and the fact that Γ − v is nonelementary, we have ν v ( X h ∞ ) =1 for any vertex v . Lemma 4.9.
For any (cid:15) > there exists T such that for all vertices w in Γ , all ξ ∈ X h ∞ and all n ≥ we have P w ( ω : | d ( o, g . . . g n o ) − β ξ ( o, g . . . g n o ) | ≤ T ) ≥ − (cid:15). Proof.
Recall that by [28, Section 3.3] there exists a G -equivariant map π : X h ∞ → ∂X , where ∂X is the Gromov boundary. Then, by definition of Gromov productand δ -hyperbolicity, we have(3) d ( o, go ) − β ξ ( o, go ) = 2( go, π ( ξ )) o + O ( δ )for any ξ ∈ X h ∞ . Now, since the pushforward of the stationary measure P w forthe Markov chain starting at w to the Gromov boundary of X is not atomic ([17,Lemma 4.2]), we have that for every (cid:15) > T such that P w ( ω ∈ Ω w : sup n ≥ ( w n o, π ( ξ )) o ≤ T ) ≥ − (cid:15) for all ξ ∈ X h ∞ and for all w . This, combined with eq. (3), yields the desiredestimate. (cid:3) Proof of Theorem 4.1.
Let F : R → R be continuous with compact support. Since F is uniformly continuous and by Lemma 4.9, for any η > n suchthat for any n ≥ n , any w and any ξ ∈ X h ∞ one has (cid:12)(cid:12)(cid:12)(cid:12) F (cid:18) d ( o, w n o ) − n(cid:96) √ n (cid:19) − F (cid:18) β ξ ( o, w n o ) − n(cid:96) √ n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < η with probability P w at least 1 − (cid:15) . Thus, since ν = R (cid:80) w P w ⊗ ν w , for any η > n such that for any n ≥ n we have(4) (cid:12)(cid:12)(cid:12)(cid:12) F (cid:18) d ( o, g . . . g n o ) − n(cid:96) √ n (cid:19) − F (cid:18) β ξ ( o, g . . . g n o ) − n(cid:96) √ n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < η on a subset of Ω × X h of ν -measure ≥ − (cid:15) . On the other hand, by Proposition 4.8,we have (cid:90) Ω × X h F (cid:18) β ξ ( o, g . . . g n o ) − n(cid:96) √ n (cid:19) dν ( ω, ξ ) → (cid:90) R F ( t ) d N σ ( t ) . Now, by (4), for any η > n such that for any n ≥ n we have (cid:12)(cid:12)(cid:12)(cid:12) F (cid:18) d ( o, g . . . g n o ) − n(cid:96) √ n (cid:19) − F (cid:18) β ξ ( o, g . . . g n o ) − n(cid:96) √ n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < η on a set of ν -measure ≥ − (cid:15) , hence (cid:90) Ω × X h F (cid:18) d ( o, g . . . g n o ) − n(cid:96) √ n (cid:19) dν ( ω, ξ ) → (cid:90) R F ( t ) d N σ ( t ) . Since the integrand does not depend on ξ , then we also have (cid:90) Ω F (cid:18) d ( o, g . . . g n o ) − n(cid:96) √ n (cid:19) dP ( ω ) → (cid:90) R F ( t ) d N σ ( t ) , where P is the pushforward of ν to Ω. Finally, since ν = R (cid:80) w P w ⊗ ν w , thepushforward of ν to Ω equals P = (cid:80) w n w R P w , where n w = ν w ( M ). Hence Lemma ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 19 P = (cid:80) w π w P w = P , thus we also have (cid:90) Ω F (cid:18) d ( o, g . . . g n o ) − n(cid:96) √ n (cid:19) d P ( ω ) → (cid:90) R F ( t ) d N σ ( t )as required. (cid:3) Uniqueness of drift and variance
Now suppose that Γ is a semisimple graph structure. In particular, each maximalcomponent C i of Γ gives a primitive graph structure (without an initial vertex) on G to which the results of the previous section (in particular, Theorem 4.1) applies.Hence for each maximal component C i of Γ, Theorem 4.1 gives constants (cid:96) i and σ i for the associated CLT.In this section, we show that the CLTs for the recurrent components of Γ arecompatible in the sense that they have the same drift and variance. This is theprimary place where we will use thickness and biautomaticity of Γ. Remark 5.1.
Our standing assumption until Section 7 is that Γ is a semisimplegraph structure on G . This implies that the transition matrix for each componentof maximal growth is aperiodic.5.1. Uniformly bicontinuous functions.
Let us begin by introducing a class offunctions that are well behaved under bounded perturbations in the group.Let Ω ∗ be the set of finite length paths in Γ starting at any vertex. Throughoutthis section, we fix a word metric d G on G . Definition 5.2.
A function f : Ω ∗ → R is uniformly bicontinuous if for any finiteset B ⊆ G and any η >
0, there exists N ≥ (cid:107) g (cid:107) ≥ N and b gb = h in G for some b , b ∈ B , then | f ( g ) − f ( h ) | < η. The following lemma gives the control we need on the length of paths in the graphstructure.
Lemma 5.3.
Suppose that the graph structure Γ for G is biautomatic. For anyfinite set B ⊆ G there exists a constant B ≥ such that if g and h are finite lengthpaths in Γ , and g = b hb in G , then |(cid:107) g (cid:107) − (cid:107) h (cid:107)| ≤ B . Proof.
Recalling that our graph structures are proper, we note that since there areonly finitely many group elements of a given length, the function n (cid:55)→ min { d G (1 , x ) : (cid:107) x (cid:107) = n } goes to infinity with n . Hence, we choose B sufficiently large so that any path x inΓ of length at least B satisfies d G (1 , x ) ≥ C + 1, where C is as in Definition 2.3 forthe given set B . Now let g, h ∈ Ω ∗ be as given in the statement of the lemma and suppose that (cid:107) h (cid:107) = m ≤ n = (cid:107) g (cid:107) . Since the graph structure of biautomatic, we have that both d G ( g ( n ) , b h ( n )) and d G ( g ( m ) , b h ( m )) are no more than C . But h = h ( m ) = h ( n )and so we see that d G ( g ( m ) , g ( n )) ≤ C . So if p is the path in Γ of length n − m such that g ( m ) · p = g ( n ), then d G (1 , p ) ≤ C . Hence, (cid:107) g (cid:107) − (cid:107) h (cid:107) = n − m = (cid:107) p (cid:107) ≤ B ,by our above choice of B . This completes the proof. (cid:3) We next introduce the primary functions of interest used throughout this section.Define the following functions on Ω ∗ : for any (cid:96) ∈ R , ϕ ( g ) := d ( o, go ) − (cid:96) (cid:107) g (cid:107) (cid:112) (cid:107) g (cid:107) and ψ ( g ) := d ( o, go ) (cid:107) g (cid:107) . Let F : R → R be a continuous, compactly supported function. Given a path g = g . . . g n in the graph, denote g [ a,b ] = g a g a +1 . . . g b its subpath from position a to position b .Finally, we define S n F ( g ) := (cid:107) g (cid:107)− n (cid:88) i =0 F ( ϕ ( g [ i +1 ,i + n ] )) . Lemma 5.4.
If the graph structure and biautomatic (and proper), then the functions ψ and ϕ defined above are uniformly bicontinuous. Moreover, for any continuous,compactly supported F . S n F ( g ) (cid:107) g (cid:107) is also uniformly bicontinuous.Proof. Suppose that h = b gb for some b , b ∈ B . Then by Lemma 5.3 |(cid:107) g (cid:107) − (cid:107) h (cid:107)| ≤ B and by the triangle inequality | d ( o, go ) − d ( o, ho ) | ≤ B where B := 2 max b ∈ B d ( o, bo ). Finally, denote by L the Lipschitz constant so that d ( o, go ) ≤ L(cid:107) g (cid:107) for any g ∈ Ω ∗ .(1) By the above estimates, (cid:12)(cid:12)(cid:12)(cid:12) d ( o, ho ) (cid:107) h (cid:107) − d ( o, go ) (cid:107) g (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) ≤ LB + B (cid:107) g (cid:107) − B , and the right-hand side tends to 0 as (cid:107) g (cid:107) → ∞ . ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 21 (2) We can write | ϕ ( g ) − ϕ ( h ) | = x √ n − x + y √ n + d where x = d ( o, go ) − (cid:96) (cid:107) g (cid:107) , y = d ( o, ho ) − (cid:96) (cid:107) h (cid:107) − d ( o, go ) + (cid:96) (cid:107) g (cid:107) , n = (cid:107) g (cid:107) , and d = (cid:107) h (cid:107) − (cid:107) g (cid:107) .Recall that by the above inequalities | d | ≤ B hence also | y | ≤ B + (cid:96) B and | x | ≤ ( L + (cid:96) ) (cid:107) g (cid:107) . Thus, (cid:12)(cid:12)(cid:12)(cid:12) x √ n − x + y √ n + d (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ( √ n + d − √ n ) (cid:112) n ( n + d ) − y √ n + d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | x | n n (cid:112) n ( n + d ) (cid:12)(cid:12)(cid:12) √ n + d − √ n (cid:12)(cid:12)(cid:12) + | y |√ n + d ≤ ( L + (cid:96) ) n (cid:112) n ( n − B ) ( √ n + B − √ n ) + B + (cid:96) B√ n − B and the right-hand side tends to 0 uniformly in n .(3) Fix (cid:15) >
0. Since F is uniformly continuous, let us pick δ > | F ( x ) − F ( y ) | < (cid:15) whenever | x − y | < δ . By Definition 2.3, there exists a finiteset B (cid:48) ⊆ G such that if h = b gb , then for any i ≤ min {(cid:107) g (cid:107) , (cid:107) h (cid:107)} − n there exist b , b ∈ B (cid:48) such that g . . . g i = b h . . . h i b g . . . g i + n = b h . . . h i + n b Thus, g i +1 . . . g i + n = (cid:0) g [1 ,i ] (cid:1) − g [1 ,i + n ] = b − h i +1 . . . h i + n b . Hence, since ϕ is uniformly bicontinuous, there exists N such that | ϕ ( g i +1 . . . g i + n ) − ϕ ( h i +1 . . . h i + n ) | < δ for any n ≥ N . Thus, by the choice of δ , | F ( ϕ ( g i +1 . . . g i + n )) − F ( ϕ ( h i +1 . . . h i + n )) | < (cid:15). Since there are at most (cid:107) g (cid:107) terms you can compare and the additional terms (ofwhich there are at most 2 |(cid:107) g (cid:107) − (cid:107) h (cid:107)| ≤ B ), are bounded by (cid:107) F (cid:107) ∞ , we have |S n F ( g ) − S n F ( h ) | ≤ (cid:15) (cid:107) g (cid:107) + 2 B(cid:107) F (cid:107) ∞ . Thus (cid:12)(cid:12)(cid:12)(cid:12) S n F ( g ) (cid:107) g (cid:107) − S n F ( h ) (cid:107) h (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) ≤ S n F ( g ) (cid:107) g (cid:107) · B(cid:107) g (cid:107) − B + (cid:15) (cid:107) g (cid:107) + 2 B(cid:107) F (cid:107) ∞ (cid:107) g (cid:107) − B hence, noting that |S n F ( g ) | ≤ (cid:107) F (cid:107) ∞ (cid:107) g (cid:107) , we obtainlim sup (cid:107) g (cid:107)→∞ (cid:12)(cid:12)(cid:12)(cid:12) S n F ( g ) (cid:107) g (cid:107) − S n F ( h ) (cid:107) h (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) and, since this is true for any (cid:15) , the claim follows. (cid:3) Remark 5.5 (Logarithmic perturbations) . As a consequence of the proof that ϕ isuniformly bicontinuous, we observe that for any η > N such that if (cid:107) g (cid:107) ≥ N then for any decomposition g = g g g with (cid:107) g (cid:107) , (cid:107) g (cid:107) ≤ log N we have | ϕ ( g ) − ϕ ( g ) | < η. The main reason why we introduce the bicontinuous functions is the followingproperty. Recall that we denote µ ( i ) n the n th step distribution of the Markov chainassociated to the maximal component C i . Lemma 5.6.
Suppose that the structure is thick and let C i be a maximal component.Let f : Ω ∗ → R be a uniformly bicontinuous function, and suppose that for someconstant a f ( w n ) → a in probability with respect to the Markov chain measure on C i . Then for any othermaximal component C j , we also have f ( w n ) → a in probability with respect to the Markov chain measure on C j .Proof. Let v be a vertex of C i , and let C j be another maximal component. Bythickness, we have S ( j ) n ⊆ (cid:91) b ,b ∈ B (cid:91) | k |≤B b (Γ v ∩ S n + k ) b where S ( j ) n is the set of paths of length n which entirely lie in C j . Hence, for any (cid:15) > n such that for all n ≥ n { g ∈ S ( j ) n : | f ( g ) − a | > (cid:15) } ≤ B (cid:88) | k |≤B { g ∈ Γ v ∩ S n + k : | f ( g ) − a | > (cid:15)/ } . Now, note that there exists
C > C − A ∩ S ( i ) n ) λ n ≤ µ ( i ) n ( A ) ≤ C A ∩ S ( i ) n ) λ n (5)for any i , any n and any set A , hence, by noting that Γ v ∩ S n + k ⊆ S ( i ) n + k , µ ( j ) n ( g : | f ( g ) − a | > (cid:15) ) ≤ B C (cid:88) | k |≤B µ ( i ) n + k ( g : | f ( g ) − a | > (cid:15)/ f ( w n ) → a in probability with respect to µ ( i ) n , the right-hand side tendsto 0, hence also the left-hand side does. (cid:3) ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 23
Uniqueness of drift.
We now show that all maximal components of Γ deter-mine the same drift.
Lemma 5.7.
If the structure is thick, then all (cid:96) i for all maximal components C i arethe same.Proof. By the ergodic theorem, for each maximal component C i we havelim n →∞ d ( o, w n o ) n = (cid:96) i almost surely (and hence in probability) with respect to the Markov chain measure µ ( i ) n . Since ψ ( g ) = d ( o,go ) (cid:107) g (cid:107) is uniformly bicontinuous by Lemma 5.4, the claim thenfollows by Lemma 5.6. (cid:3) By the above lemma, we now define ϕ using (cid:96) = (cid:96) i for any (equivalently all) i .5.3. Uniqueness of variance.
In a similar setting, Calegari–Fujiwara [7] use thenotion of typical path to show that all maximal components have the same variance.Here, we adapt this technique by using the uniform bicontinuity of the functions S n F ( g ) (cid:107) g (cid:107) , plus thickness of our structure. An important difference is that here we useconvergence in probability instead of almost sure convergence, as the first one canbe “transferred” from one component to another using thickness (see Lemma 5.6). Lemma 5.8.
Let C i be a maximal component, and let σ i be the variance of thecorresponding CLT. Then for any compactly supported, continuous function F : R → R and for any n , there exists a constant E ( i ) n ( F ) such that for any (cid:15) > , lim m →∞ µ ( i ) m (cid:18) g : (cid:12)(cid:12)(cid:12)(cid:12) S n F ( g ) (cid:107) g (cid:107) − E ( i ) n ( F ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:19) → and moreover lim n →∞ E ( i ) n ( F ) = (cid:90) R F ( t ) d N σ i ( t ) . Proof.
Recall that for the Markov chain, by the ergodic theorem the limit E ( i ) n ( F ) := lim m →∞ m m − (cid:88) i =0 F ( ϕ ( g i +1 . . . g i + n )) = (cid:90) G F ( ϕ ( g )) dµ ( i ) n ( g )exists almost surely, hence also in probability. Moreover, since the law of ϕ ( w n )converges to N σ i , we havelim n →∞ E ( i ) n ( F ) = lim n →∞ (cid:90) G F ( ϕ ( g )) dµ ( i ) n ( g ) = (cid:90) F ( t ) d N σ i ( t ) . Thus, for any (cid:15) > m →∞ µ ( i ) n + m (cid:32) g : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m m − (cid:88) i =0 F ( ϕ ( g i +1 . . . g i + n )) − E ( i ) n ( F ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:33) → . Since lim m →∞ (cid:107) g (cid:107) m = n + mm = 1 in the above equation, the claim follows. (cid:3) Lemma 5.9.
Suppose that G has a thick graph structure. Then for any two maximalcomponents C i , C j we have σ i = σ j .Proof. Fix F continuous and compactly supported, and let M n F ( g ) := S n F ( g ) (cid:107) g (cid:107) . Fix amaximal component C i . By Lemma 5.8, for any n the function M n F ( w m ) convergesin probability w.r.t. the Markov measure for C i as m → ∞ to some constant E ( i ) n ( F ).Since M n F is uniformly bicontinuous by Lemma 5.4, we have by Lemma 5.6 that M n F ( w m ) converges to the same constant in probability with respect to the Markovchain for C j . Hence, E ( i ) n ( F ) = E ( j ) n ( F ) for any n . Thus, (cid:90) F ( t ) d N σ i ( t ) = lim n →∞ E ( i ) n ( F ) = lim n →∞ E ( j ) n ( F ) = (cid:90) F ( t ) d N σ j ( t ) . Since, this is true for any F , we must have σ i = σ j . (cid:3) The semisimple case
In this section, we prove our main theorem for semisimple graph structures. Thisis completed in Theorem 6.3.6.1.
Convergence to the Markov measure.
So far our work has been for max-imal components of a semisimple graph structure. In this section we consider thewhole graph structure, still in the semisimple case.Let Γ be a semisimple graph structure for G with transition matrix M of spectralradius λ >
1. Let v i be the vertices of the graph, and let v be a vertex of largegrowth, which we take as the initial vertex. Then recall that e Ti M n e j is the numberof paths of length n from v i to v j . Since M is semisimple, the limit M ∞ := lim n →∞ M n λ n exists. In particular, in keeping with notation at the beginning of Section 4, wedefine ρ i := lim n →∞ e Ti M n λ n and u i := lim n →∞ e T M n e i λ n . By construction, ρ = ( ρ i ) satisfies ρ = M ∞ M ρ = λρ , while u = ( u i ) satisfies u T M = λu T . Finally, (cid:80) i u i = ρ and (cid:80) i u i ρ i = ρ .Note that vertices v i for which ρ i > u i > ρ i > P on the space Ωof infinite paths starting at any vertex of Γ. First define the initial distribution ofthe Markov chain to start at vertex v i with probability π i := u i ρ i ρ . Then assign anedge from v i to v j the probability p ij := ρ j λρ i . Obviously, P is supported on pathsthat are entirely contained in components of maximal growth. We denote as P n thedistribution of the n th step of the Markov chain. ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 25
Remark 6.1.
We remark that the induced measure on each maximal component C of Γ rescales to give the Markov measure on C previously considered. This followsimmediately from the construction.The following result relates the Markov measure on the semisimple graph struc-ture to the counting measure. For its statement, let v be any vertex of large growth.For each n , consider the path given by selecting uniformly a path γ starting at v of length n , and take its subpath ˜ γ from position log n to position n − log n . Let ˜ λ n denote the distribution of ˜ γ . Lemma 6.2.
With notation as above, the total variation (cid:107) P n − n ) − ˜ λ n (cid:107) T V → as n → ∞ .Proof. Denote n (cid:48) := n − n . Let γ be a path in the graph, starting at v i andending at v j . Then by definition the proportion of paths of length n , starting at v ,that have γ as “middle subpath” of length n (cid:48) is˜ λ n ( γ ) = ( e T M log n e i )( e Tj M log n e T M n . On the other hand, P n (cid:48) ( γ ) = (cid:26) π i ρ j ρ i λ n (cid:48) If v i has large growth0 otherwise,which is nonzero if both v i and v j belong to a maximal component. In this case, d P n (cid:48) d ˜ λ n ( γ ) = λ log n e T M log n e i · λ log n e Tj M log n · e T M n λ n · π i ρ j ρ i −→ u i · ρ j · ρ · π i ρ j ρ i = 1using that π i = u i ρ i ρ . Moreover, if S i,jn denotes the set of paths of length n (cid:48) from v i to v j , we have ˜ λ n ( S i,jn ) = ( e Ti M n (cid:48) e j )( e T M log n e i )( e Tj M log n e T M n ≤ ( e Ti M n (cid:48) e T M log n e i )( e Tj M log n e T M n → ρ i u i ρ j ρ , hence such a probability tends to 0 unless both v i and v j belong to a maximalcomponent. Finally, if we denote as L n the set of paths of length n (cid:48) which lie entirely in amaximal component, we have for any set A (cid:12)(cid:12)(cid:12) P n (cid:48) ( A ) − ˜ λ n ( A ) (cid:12)(cid:12)(cid:12) ≤ (cid:88) x ∈ A ∩L n (cid:12)(cid:12)(cid:12)(cid:12) P n (cid:48) ( x )˜ λ n ( x ) ˜ λ n ( x ) − ˜ λ n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + ˜ λ n ( A \ L n ) ≤ sup x ∈L n (cid:12)(cid:12)(cid:12)(cid:12) P n (cid:48) ( x )˜ λ n ( x ) − (cid:12)(cid:12)(cid:12)(cid:12) + ˜ λ n ( L cn )and both terms tend to 0 as n → ∞ , independently of A . (cid:3) Central limit theorem for the counting measure in the semisimplecase.
We are now ready to prove the following. For its statement, let S n denotethe set of length n paths beginning at the initial vertex v . Theorem 6.3.
Let Γ be a semisimple, thick, biautomatic graph structure for anonelementary group G of isometries of a δ -hyperbolic space ( X, d ) , and let o ∈ X be a base point. Then there exists (cid:96) ≥ , σ ≥ such that for any a < b we have lim n →∞ S n (cid:26) g ∈ S n : d ( o, go ) − n(cid:96) √ n ∈ [ a, b ] (cid:27) = (cid:90) ba d N σ ( t ) . In the following proof and later on, we will use the notation N σ ( x ) := (cid:82) x −∞ d N σ ( t ). Remark 6.4.
Note that if, additionally, the graph structure Γ is semisimple andhas a unique maximal component, Theorem 6.3 holds even without assuming thatthe structure is biautomatic.
Proof.
Let C , . . . , C k be the maximal components, and let µ ( i ) n be the n th stepdistribution for the Markov chain associated to that component, as in Section 4.Theorem 4.1 shows a CLT for all such measures, and by Lemmas 5.7 and 5.9 allsuch measures have the same drift and variance, that we denote by (cid:96), σ .Now, since the starting probability ( π i ) in the above construction is nonzeroprecisely on the set of vertices which belong to a maximal component, there existweights c i ≥ (cid:80) i c i = 1 such that P n = k (cid:88) i =1 c i µ ( i ) n for any n . Thus, for any x ∈ R ,(6) P n ( g : ϕ ( g ) ≤ x ) = k (cid:88) i =1 c i µ ( i ) n ( g : ϕ ( g ) ≤ x ) → N σ ( x ) , where we recall that ϕ ( g ) = d ( o, go ) − (cid:96) (cid:107) g (cid:107) (cid:112) (cid:107) g (cid:107) . Now, we use that the counting measure can be approximated by the Markov chainmeasure. If g is a path of length n , we denote as g = g g g where g is the prefix ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 27 of length log n , g is the middle part of length n − n and g is the final part oflength log n . By Remark 5.5, there exists n such that(7) | ϕ ( g ) − ϕ ( g ) | ≤ (cid:15) for any n ≥ n and g with (cid:107) g (cid:107) = n .Fix x ∈ R and (cid:15) >
0. Then we have λ n ( g : ϕ ( g ) ≤ x ) = λ n ( g = g g g : ϕ ( g ) ≤ x )and, by eq. (7), for n large = λ n ( g = g g g : ϕ ( g ) ≤ x + (cid:15) )then by definition of ˜ λ n = ˜ λ n ( g : ϕ ( g ) ≤ x + (cid:15) )and by Lemma 6.2, for n large, = P n − n ( g : ϕ ( g ) ≤ x + (cid:15) ) + (cid:15). Hence, by eq. (6) we obtainlim sup n →∞ λ n ( g : ϕ ( g ) ≤ x ) ≤ N σ ( x + (cid:15) ) + (cid:15) and, by taking (cid:15) smaller and smaller and using the continuity of N σ ,lim sup n →∞ λ n ( g : ϕ ( g ) ≤ x ) ≤ N σ ( x ) . The lower bound follows analogously. (cid:3)
Indeed, the same proof shows the following stronger statement. Let λ ( i ) n denotethe counting measure on the set of paths of length n starting at v i . Corollary 6.5.
Let Γ be a semisimple, thick, biautomatic graph structure for anonelementary group G of isometries of a δ -hyperbolic space ( X, d ) , and let o ∈ X be a base point. Then there exists (cid:96) ≥ , σ ≥ such that for any vertex v i of largegrowth for Γ and any a < b we have lim n →∞ λ ( i ) n (cid:32) g : d ( o, go ) − (cid:96) (cid:107) g (cid:107) (cid:112) (cid:107) g (cid:107) ∈ [ a, b ] (cid:33) = (cid:90) ba dN σ ( t ) . Proof.
Let us fix a vertex v (cid:96) of large growth for M . Then we can define a Markovchain measure P ( (cid:96) ) on the space of infinite paths as follows. The transition probabil-ities will always be the same p ij = ρ j λ p ρ i , while for each vertex v (cid:96) one finds a differentset of starting probabilities π ( (cid:96) ) i given by π ( (cid:96) ) i := u ( (cid:96) ) i ρ i ρ (cid:96) , where u ( (cid:96) ) i := lim n →∞ e T(cid:96) M n e i λ n . Just as before, there exist constants c ( (cid:96) ) i ≥ (cid:80) i c ( (cid:96) ) i = 1 and P ( (cid:96) ) n = k (cid:88) i =1 c ( (cid:96) ) i µ ( i ) n . The proof then proceeds exactly as for Theorem 6.3. (cid:3) The CLT for displacement and translation length
Now suppose that Γ is an almost semisimple graph structure for G with transitionmatrix M . Then M has some period p ≥ M p is semisimple. We denoteby Γ p the corresponding p step graph structure on G . That is, Γ p is the graph withthe same vertex set as Γ and an edge joining v i to v j for each directed path from v i to v j of length p , whose label is the word in G spelled by the corresponding path.The transition matrix for Γ p is M p , hence Γ is a semisimple graph structure for G .Since the previous results require this structure to be thick, we need the followinglemma. Lemma 7.1.
The following properties pass to the p step graph structure: • If v is a large growth vertex of Γ , then its also a large growth vertex of Γ p . • If Γ is a thick structure, then Γ p is also thick. • If Γ is biautomatic, then so is Γ p .Proof. The first statement holds because any path from v that ends in a componentof maximal growth can be extended to a path whose length is a multiple of p byadding on a path in that component of length less than p .Now suppose that Γ is thick. Let v be a vertex in a maximal component of Γ p .Then v is also a vertex in a maximal component of Γ. Let Γ v,p be the semigroup ofloops based at v of lengths multiple of p . Consider the semigroup homomorphism f : Γ v → N → N /p N given by taking the length and reducing it mod p . Clearly, the image of f is asubsemigroup of N /p N , which is a finite group, hence the image is also a group. Let γ i , . . . , γ k ⊆ Γ v be a set of representatives for each remainder class in the image of f . Now, let γ ∈ Γ v . Then (cid:107) γ (cid:107) belongs to the image of f , hence there exists γ i (the representative of the inverse modulo p ), such that γγ i has length multiple of p ,hence it belongs to Γ v,p . Hence, by setting B (cid:48) the set { γ − i : 1 ≤ i ≤ k } , we haveΓ v ⊆ Γ v,p B (cid:48) in the group. Since Γ is thick, there exists B (cid:48)(cid:48) such that G = B (cid:48)(cid:48) Γ v B (cid:48)(cid:48) ,hence also G = B (cid:48)(cid:48) Γ v,p B (cid:48) B (cid:48)(cid:48) , hence Γ p is also thick.Finally, note that any path g in Γ p of length k can be naturally thought of as apath g † in Γ of length pk such that for all i ≥ g † ( pi ) = g ( i ). Hence, if B and C ≥ g, h are paths in Γ p with g = b hb for b , b ∈ B , then we also have that g † = b h † b . Then biautomaticiy ofΓ implies that d G ( g † ( i ) , b h † ( i )) ≤ C, for all i ≥
0. Hence, restricting to multiples of p completes the proof. (cid:3) ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 29
Now, let us consider the semisimple matrix M p . Note that irreducible componentsof M p may be proper subsets of irreducible components of M . Given a vertex v i ,let us denote by λ ( i ) k the counting measure on paths starting at v i of length k for Γ.Note that if k = np , then this also counts paths of length n in Γ p starting at v i .By applying Theorem 6.5 to Γ p , we immediately obtain: Corollary 7.2.
Let Γ be a thick, biautomatic structure of period p for a nonele-mentary group G of isometries of a δ -hyperbolic space ( X, d ) , and let o ∈ X be abase point. Then there exists (cid:96), σ such that the following holds. For any vertex v i oflarge growth for Γ and for any x , we have λ ( i ) pn (cid:32) g : d ( o, go ) − (cid:96) (cid:107) g (cid:107) (cid:112) (cid:107) g (cid:107) ≤ x (cid:33) → (cid:90) x −∞ d N σ ( t ) as n → ∞ . We are now ready to prove the following. Recall that S n denotes the set of length n paths beginning at the initial vertex v . Theorem 7.3.
Let Γ be a thick, biautomatic graph structure for a nonelementarygroup G of isometries of a δ -hyperbolic space ( X, d ) , and let o ∈ X be a base point.Then there exists (cid:96) ≥ , σ ≥ such that for any a < b we have lim n →∞ S n (cid:26) g ∈ S n : d ( o, go ) − (cid:96)n √ n ∈ [ a, b ] (cid:27) = (cid:90) ba d N σ ( t ) . Proof.
Let v be the initial vertex, let S n be the set of paths of length n based at v , and let λ n be the uniform measure on S n .Let us fix 0 ≤ r ≤ p −
1. Then we can write the counting measure on S pn + r ,starting at the initial vertex v , by first picking randomly a path g of length r from v with a certain probability µ , and then picking a random path starting at v i = t ( g ) with respect to the counting measure on the set of paths of length n starting at v i .To compute µ , let us consider a path g of length r starting at v and ending at v i . Then, if v i is of large growth for Γ p , { paths from v i of length pn } { paths from v of length pn + r } = e i M pn e M pn + r → e i M ∞ e M r M ∞ . Thus, we define µ ( g ) := e i M ∞ e M r M ∞ . Note that µ ( g ) = 0 if the end vertex of g has small growth and moreover (cid:88) (cid:107) g (cid:107) = r µ ( g ) = (cid:88) i µ ( g ) { g ∈ S r : t ( g ) = v i } = (cid:88) i e M r e i e i M ∞ e M r M ∞ . Let λ (cid:48) pn + r be the measure on S pn + r given by first taking randomly a path g oflength r from v with distribution µ and then taking uniformly a path of length pn starting from t ( g ). Now we show that the CLT holds for λ (cid:48) pn + r . Let (cid:96), σ be given by Theorem 7.2,and let ϕ ( g ) := d ( o,go ) − (cid:96) (cid:107) g (cid:107) √ (cid:107) g (cid:107) . By Theorem 7.2, for any vertex v i of large growth, wehave λ ( i ) pn ( g : ϕ ( g ) ≤ x ) → N σ ( x ) . Then if g = g g , and t ( g ) denotes the (index of the) end vertex of g , λ (cid:48) pn + r ( g : ϕ ( g ) ≤ x ) = (cid:88) g ∈ S r µ ( g ) λ ( t ( g )) pn ( g : ϕ ( g g ) ≤ x ) −→ (cid:88) g µ ( g ) N σ ( x ) = N σ ( x ) , where we used that ϕ is uniformly bicontinuous as in the proof of Theorem 6.3.Now we prove that (cid:107) λ (cid:48) pn + r − λ pn + r (cid:107) T V → n → ∞ . Indeed, if γ = g g is a path from v of length pn + r and g is its prefixof length r ending at a vertex v i of large growth, then λ (cid:48) pn + r ( γ ) λ pn + r ( γ ) = µ ( g ) · e i M pn e M pn + r → . On the other hand, if the end vertex of g is of small growth, then λ (cid:48) pn + r ( g ) = 0,and also λ pn + r ( g = g g : g ends at a small growth vertex) → n → ∞ . Now, let A x := { g : ϕ ( g ) ≤ x } and L r be the set of paths starting at v whose prefix of length r ends in a vertex of large growth. Then λ pn + r ( g : ϕ ( g ) ≤ x ) = λ pn + r ( g ∈ L r : ϕ ( g ) ≤ x ) + λ pn + r ( g / ∈ L r : ϕ ( g ) ≤ x )= λ pn + r ( A x ∩ L r ) λ (cid:48) pn + r ( A x ∩ L r ) λ (cid:48) pn + r ( A x ∩ L r ) + λ pn + r ( A x \ L r ) → · N σ ( x ) + 0 = N σ ( x ) . We have thus obtained a CLT for λ pn + r , for any 0 ≤ r ≤ p −
1, always with thesame (cid:96), σ . Since there are only finitely many values r , the claim follows. (cid:3) A CLT for translation length.
We now prove a more general version of oursecond main result, Theorem 1.1 (2).
Theorem 7.4.
Let Γ be a thick, biautomatic graph structure for a nonelementarygroup G of isometries of a δ -hyperbolic space ( X, d ) , let o ∈ X be a base point, andlet (cid:96), σ be as in Theorem 7.3. Then for any a < b we have lim n →∞ S n (cid:26) g ∈ S n : τ ( g ) − (cid:96)n √ n ∈ [ a, b ] (cid:27) = (cid:90) ba d N σ ( t ) . ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 31
Proof.
Let us recall that the translation length of an isometry g of a δ -hyperbolicspace can be computed by (see e.g. [28, Proposition 5.8])(8) τ ( g ) = d ( o, go ) − go, g − o ) o + O ( δ )where O ( δ ) is a constant which only depends on the hyperbolicity constant of X .Now, by choosing f ( n ) = (cid:15) √ n in [17, Proposition 5.8], for any (cid:15) we have λ n ( g : ( go, g − o ) o ≤ (cid:15) √ n ) → n → ∞ . The claim then follows by combining this statement and the statementof Theorem 7.3 into formula (8). (cid:3) Zero variance.
We finally complete our main theorem by characterizing thecase where σ = 0. First, we give a general criterion. Proposition 7.5.
In the hypotheses of Theorem 7.3 we have σ = 0 if and only ifthere is C ≥ such that for all finite length paths g in Γ , | d ( o, go ) − (cid:96) (cid:107) g (cid:107)| ≤ C. We note that the proposition implies that σ > G (cid:121) X isnonproper. Proof.
Suppose that σ = 0 for the CLT for the counting measure. Then by ourprevious discussion, we have σ = 0 also for the Markov chain on any maximalcomponents. Then by Theorem 4.2, we also have σ = 0 for the random walk on theloop semigroup driven by ˇ µ v . Hence, as in [4, Proof of Theorem 4.7 (b)], for any n n (cid:90) ( η ( g, ξ )) d ˇ µ ∗ nv ( g ) dν v ( ξ ) = 0where η is the centering of η . This implies η ( g, ξ ) = 0for any g ∈ Γ − v and ν v -a.e. ξ ∈ X h . Thus, since | η − η | ≤ (cid:107) ψ (cid:107) ∞ is bounded, wehave | β ξ ( o, g − o ) − (cid:96) (cid:107) g (cid:107)| = | η ( g, ξ ) | ≤ (cid:107) ψ (cid:107) ∞ hence by [23, Corollary 2.3] there exists a constant C for which | d ( o, go ) − (cid:96) (cid:107) g (cid:107)| ≤ C for any g in the support of ˇ µ ∗ nv ( g ).Hence, by thickness we have for any g ∈ Ω ∗ there exists b , b ∈ B and h ∈ Γ − v such that h = b gb , thus by Lemma 5.3 and the triangle inequality | d ( o, go ) − (cid:96) (cid:107) g (cid:107)| ≤ | d ( o, ho ) − (cid:96) (cid:107) h (cid:107)| + B + (cid:96) B thus there exists a constant C (cid:48) such that | d ( o, go ) − (cid:96) (cid:107) g (cid:107)| ≤ C (cid:48) for any g ∈ Ω ∗ . This completes the proof. (cid:3) We conclude with a corollary that applies when the graph structure is geodesic.For the action G (cid:121) X , denote the translation length of h by τ X ( h ). We use thenotation τ G ( h ) to denote the translation length of h with respect to the word metric d G induced by the graph structure Γ: τ G ( h ) = lim n →∞ n d G (1 , h n ) . Corollary 7.6.
Suppose that Γ is a thick bicombing of G . If σ = 0 in the CLT,then for all h ∈ G τ X ( h ) = (cid:96) τ G ( h ) , where (cid:96) is the corresponding drift.Proof. Let g n be a path in Γ representing h n for h ∈ G . That is h n = g n . Since thestructure is geodesic, || g n || = d G (1 , h n ). Applying Proposition 7.5, we get that | d ( o, h n o ) − (cid:96)d G (1 , h n ) | = O (1) . The corollary follows after dividing by n and taking a limit. (cid:3) Applications
The main theorem of Section 1 now follows easily from the results in Section 7.
Proof of Theorem 1.1.
Since G has a thick bicombing with respect to S , the length (cid:107) g (cid:107) of a path in the graph equals the word length with respect to S of its evaluation g ∈ G , and the sphere of radius n in the Cayley graph of G is in bijection with theset of paths of length n in the graph. Then (1) follows immediately from Theorem7.3, (2) follows from 7.4 and (3) from Corollary 7.6. (cid:3) We now give proofs of the applications in the introduction. We first recall someexamples of groups which admit thick bicombings; for further details, see also [17].
Lemma 8.1.
The following groups admit thick bicomings: (1)
A (word) hyperbolic group G admits a thick bicombing with respect to anygenerating set. (2) If G is relatively hyperbolic with virtually abelian peripheral subgroups, thenevery finite generating set S (cid:48) can be extended to a finite generating set S for G which admits a thick bicombing. (3) If G is a right-angled Artin group or right-angled Coxeter group that does notdecompose as a product and S is the vertex generating set, then G admits athick combing for S whose graph structure has only one maximal component,which is aperiodic.Proof. (1) By [9], a hyperbolic G has a bicombing with respect to any generatingset. Such a structure is thick by [19, Lemma 4.6].(2) By [2, Corollary 1.9], the generating set S (cid:48) of G can be enlarged to a generatingset S , so that the pair ( G, S ) admits a geodesic graph structure. By [22, Theorem5.2.7], this can be turned into a bicombing for the same generating set S . ENTRAL LIMIT THEOREMS FOR COUNTING MEASURES 33
Hence, it remains to show that this bicombing is thick. Yang [45] proves thatany relatively hyperbolic group has the growth quasitightness property (see [17,Definition 1.2], inspired by [3]) with respect to any finite generating set. Since growthquasitightness implies thickness by [17, Proposition 7.2], the proof is complete.(3) In [17, Corollary 10.4], building on Hermiller–Meier [21], we proved that thelanguage of lexicographically first geodesics in the vertex generators is parameterizedby a thick graph structure. In fact, the graph structure we construct has only onemaximal component, which is aperiodic. (cid:3)
Proof of Theorem 1.2.
Note that π ( M ) is hyperbolic relative to its parabolic sub-groups, which are virtually abelian since M has constant curvature. Hence, byLemma 8.1 (2) the given generating set S (cid:48) can be enlarged to a finite generatingset S that is associated to a thick bicombing on π ( M ). The theorem then followsfrom Theorem 1.1. Finally, σ > (cid:3) Proof of Theorem 1.3.
In the case where M has no rank 2 cusps, we have that π ( M ) is hyperbolic. Indeed, by the Tameness Theorem ([8], [1]), M is the interiorof a compact manifold M , which by assumption does not have tori as boundarycomponents. Then Thurston’s Hyperbolization Theorem (see [25]), M admits aconvex cocompact hyperbolic structure on its interior. Hence, π ( M ) is hyperbolic.The result now follows from Lemma 8.1 and Theorem 1.1.For the moreover statement, the argument above gives that M admits a geomet-rically finite hyperbolic structure. Hence, π ( M ) is hyperbolic relative to its rank2 parabolic subgroups, which are virtually Z × Z . The proof then proceeds as inTheorem 1.2 (cid:3) Proof of Theorem 1.4.
First, since π ( M ) is word hyperbolic, by Lemma 8.1 (1) ithas a thick bicombing with respect to any generating set.Second, let T = T Σ be the dual tree associated to Σ ⊂ M . For details of thisstandard construction and the properties we need, see [42, Section 1.4]. Alterna-tively, T is the Bass–Serre tree associated to the splitting of π ( M ) induced byΣ. Since Σ is not fiber-like, T is not the real line, and since the quotient G ofthe action π ( M ) (cid:121) T is compact (it is the underlying graph of the associatedgraph-of-groups), the action is nonelementary.Finally, the intersection number i ( γ, Σ) equals the translation length of γ withrespect to the action π ( M ) (cid:121) T . To see this, note that the translation length of γ for this action is equal to the number of edges e γ crossed by the shortest represen-tative of γ in G . If we embed G in M dual to Σ, this shows that i ( γ, Σ) ≤ e γ . Forthe opposite inequality, recall that there is a retraction r : M → G mapping eachcomponent of Σ to the midpoint of some edge. Thus by taking a representative of γ intersecting Σ minimally, considering its image under the retraction, and homo-toping it off edges that it does not fully cross, we obtain that e γ ≤ i ( γ, Σ) . Hence, i ( γ, Σ) = (cid:96) ( γ ) for the action on T . We now obtain the CLT by applying Theorem 1.1 to this action. If σ = 0, thenTheorem 1.1 (3) implies that the action π ( M ) (cid:121) T is proper. However, this isimpossible since only virtually free groups admit proper actions on trees. (cid:3) For the following application, let us assume G is a hyperbolic group, let ∂G beits Gromov boundary, and let d be a metric on G . We define the growth rate of themetric d as v := lim sup n →∞ n log (cid:8) g ∈ G : d (1 , g ) ≤ n (cid:9) and for each s ≥ v let us consider the measure on G ∪ ∂G : ν s := (cid:80) g ∈ G e − sd (1 ,g ) δ g (cid:80) g ∈ G e − sd (1 ,g ) . Then any limit point of ( ν s ) as s → v is supported on ∂G and is called a Patterson–Sullivan (PS) measure . By Coornaert [13], any two limit measures are absolutelycontinuous with respect to each other, with bounded Radon–Nikodym derivative,so the Patterson-Sullivan measure class is well-defined.
Proof of Theorem 1.5.
Since G is word hyperbolic, it has a thick bicombing byLemma 8.1 (1). The first statement then follows immediately from Theorem 1.1, byconsidering the action of G on the Cayley graph of G (cid:48) .For the moreover statement, if σ = 0, Theorem 1.1 (3) implies that |(cid:107) φ ( g ) (cid:107) S (cid:48) − (cid:96) (cid:107) g (cid:107) S | is bounded independently of g ∈ G , hence φ has finite kernel.Now, consider the factorization G π → G := G ker φ φ → G (cid:48) , and define S := π ( S ).Then the Cayley graph of G carries the two metrics d ( g, h ) := (cid:107) h − g (cid:107) S d ( g, h ) := (cid:107) φ ( h − g ) (cid:107) S (cid:48) and they satisfy(9) | d ( g, h ) − (cid:96)d ( g, h ) | ≤ C for any g, h ∈ G . Now, by [15, Theorem 2], eq. (9) holds if and only if the Bowen–Margulis measures on the double boundary ∂G × ∂G associated to d , d are thesame, which by [15, Proposition 1] holds if and only if the Patterson–Sullivan mea-sure classes for d , d on ∂G are the same. Finally, if φ has finite kernel, there exist C > | d ( π ( g ) , π ( h )) − d S ( g, h ) | ≤ C for any g, h ∈ G . Hence, the PS measure class for ( G, d S ) on ∂G pushes forward tothe PS measure class for ( φ ( G ) , d S (cid:48) ) if and only if σ = 0. (cid:3) Proof of Theorem 1.6.
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E-mail address : [email protected] Department of Mathematics, Temple University, 1805 North Broad Street Philadel-phia, PA 19122, U.S.A,
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