Counting loxodromics for hyperbolic actions

Abstract

Let G X be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X-loxodromics, approaches 1 as n → ∞. We also establish several results about the behavior in X of the images of typical geodesic rays in G; for example, we prove that they make linear progress in X and converge to the Gromov boundary ∂X. We discuss various applications, in particular to mapping class groups, Out(FN ), and right-angled Artin groups. and its limit in a suitable compactification (when it exists) is usually called a Patterson-Sullivan (PS) measure. In the second case, for each n one has the convolution measure μ n := μ · · · μ n times , which is precisely the distribution of the nth step of the random walk. In this case, the limit measure in a suitable compactification is the hitting measure for the random walk.The two types of counting, however, need not yield the same notion of genericity; indeed, an underlying theme of much research has been the question: Are Patterson-Sullivan measures also hitting measures for a certain random walk?This is for instance the main question of [64]. As Furstenberg showed [28], for a semisimple Lie group G, the appropriate boundary is B = G/P , where P is a minimal parabolic subgroup. Furthermore, the analogue of the Patterson-Sullivan measure is the unique measure ν on B which is invariant by a maximal compact subgroup. In fact, he proved that for any lattice Λ in G there exists a measure on Λ whose hitting measure is ν. Moreover, for groups acting properly and cocompactly on δ-hyperbolic manifolds, Connell and Muchnik [21] showed that there exists a certain random walk which produces the Patterson-Sullivan measure on the boundary. They also extended this result to arbitrary Gibbs measures in the case the space is CAT(−1) [20].However, for actions of hyperbolic groups G on spaces X, if the harmonic measure and PS measure are in the same measure class, then the metrics on G and on X are quasi-isometric [10]. Interestingly, for hyperbolic groups Gouëzel, Mathéus, and Maucourant [30] recently proved that, when considering a word metric on G, harmonic measures for symmetric random walks of finite support are never in the same measure class as Patterson-Sullivan measures, unless the group is virtually free. One should compare this to Ledrappier's result that for compact hyperbolic surfaces the hitting measure for Brownian motion coincides with the PS measure if and only if the surface has constant curvature [49]. There are many examples of harmonic measures which are singular with respect to the PS-type measure, see for example, [39].In this paper, we will use random walk methods to prove results about genericity with respect to counting in balls in the word metric on G. In particular, we consider the case of G a w...

Genericity of contracting elements in groups

“…This implies, by Lemma 6.1, that (19) d π X (p 1 ∪ p 2 ) ≤ 2A + 8C. On the other hand, applying Proposition 2.2.5, we obtain from (16) and (17) (22) and d(x, X), d(y, X) ≤ C. However, this contradicts to the conclusion of Lemma 6.6. Hence, N C (X) ∩ [gx, go] γ ≠ ∅.…”

“…For instance, partial cases of Theorems 1.7 and 1.9 was obtained there under some automatic hypothesis. Recently, Gehktman, Tylor and Tiozzo [22] estbalished for word metrics the generic elements in a non-elementary hyperbolic group action.…”

A central limit theorem for random closed geodesics: Proof of the Chas–Li–Maskit conjecture

“…where P v is the distribution of the Markov chain starting at vertex v, and x n is the n th step. (Lemma 6.2 of [9] explicitly gives this statement where v is the "initial vertex" of the directed graph, however the same argument gives the more general result for all vertices. )…”

Typical behaviour along geodesic rays in hyperbolic groups

“…Gekhtman, Taylor and Tiozzo asked the above question in a more general setting. They prove the following theorem in [11]. Let ν denote the Patterson-Sullivan measure obtained as the weak star limit…”

“…where δ g denotes the Dirac measure based at g ∈ G and |g| denotes the word length of g. We write [γ ] ∈ ∂G for the element in ∂G that contains γ . Proposition 1.1 (Theorem 1.3 [11]) Suppose a hyperbolic group G has a non-elementary action by isometries on a separable, hyperbolic geodesic metric space X . Then, there is L > 0 such that for every x ∈ X and ν almost every [ γ ] ∈ ∂G,…”