Typical behaviour along geodesic rays in hyperbolic groups

Abstract

In this note we study the limiting behaviour of real valued functions on hyperbolic groups as we travel along typical geodesic rays in the Gromov boundary of the group. Our results apply to group homomorphisms, certain quasimorphisms and to the displacement functions associated to convex cocompact group actions on CAT$$(-1)$$ ( - 1 ) metric spaces.

Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

“…The following result establishes exponential counting large deviation estimates in another setting, that of isometric actions on Gromov-hyperbolic spaces. This setting has recently attracted much attention both from probabilistic [2,4,7,14,40,53] and counting [18,22,23,24,34,35,68] perspectives. To state our result, recall that the action of a group Γ on a Gromov-hyperbolic space H by isometries is said to be nonelementary if it there exists γ 1 , γ 2 ∈ Γ acting as loxodromic elements (see §3.5) with disjoint pairs of fixed points on the Gromov boundary of H.…”

“…The proof makes use of Bougerol's results [12] which are translated to the group theoretic setting using techniques due to Calegari-Fujiwara [18] and extensions of these techniques due to Cantrell [23]. The scheme of proof, somewhat common to the next Theorem 1.11, is expounded in §1.4 below.…”

“…We summarize here a construction of a Markov chain on the strongly Markov structure of a Gromov-hyperbolic group and its connection with the Patterson-Sullivan measure (both due to Calegari-Fujiwara [18] in this setting). We also include some further related observations from Cantrell [23]; other more specific ones will be included/proven in later sections where they are needed.…”

“…A key property of the Parry-like measure µ is that by [18,Lemma 4.19] (and more precisely [23,Proposition 4.6]), it is closely related to the measure ν on Y ⊆ Σ A constructed above using the Patterson-Sullivan measure ν: we have…”

“…On the other hand, by [23,Proposition 4.2] the Parry-like measure µ is nothing but a linear combination of the Parry measures of maximal irreducible components of Σ A : for each maximal component (B j ) j=1,...,m , there exists α j > 0 such that m j=1 α j = 1 and…”