Counting loxodromics for hyperbolic actions

TL;DR

This paper proves that in hyperbolic groups acting on hyperbolic spaces, most elements are loxodromic, and explores the typical behavior of geodesic rays, with applications to various important groups.

Contribution

It establishes the genericity of loxodromic elements in hyperbolic group actions and analyzes the asymptotic behavior of geodesic rays in such settings.

Findings

Proportion of loxodromic elements approaches 1 as radius increases.

Typical geodesic rays make linear progress and converge to the boundary.

Results apply to groups like Mod(S), Out(F_N), and right-angled Artin groups.

Abstract

Let $G \curvearrowright X$ be a nonelementary action by isometries of a hyperbolic group $G$ on a hyperbolic metric space $X$. We show that the set of elements of $G$ which act as loxodromic isometries of $X$ is generic. That is, for any finite generating set of $G$, the proportion of $X$--loxodromics in the ball of radius $n$ about the identity in $G$ approaches $1$ as $n \to \infty$. 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 $\partial X$. Our techniques make use of the automatic structure of $G$, Patterson--Sullivan measure on $\partial G$, and the ergodic theory of random walks for groups acting on hyperbolic spaces. We discuss various applications, in particular to Mod(S), Out($F_N$), and right--angled Artin groups.