Counting loxodromics for hyperbolic actions

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.