Consider a nonelementary action by isometries of a hyperbolic group G on a hyperbolic metric space X. Besides the action of G on its Cayley graph, some examples to bear in mind are actions of G on trees and quasi-trees, actions on nonelementary hyperbolic quotients of G, or examples arising from naturally associated spaces, like subgroups of the mapping class group acting on the curve graph.
We show that the set of elements of G which act as loxodromic isometries of X (i.e those with sink-source dynamics) 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 goes to infinity. 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 boundary of X. Our techniques make use of the automatic structure of G, Patterson-Sullivan measure, and the ergodic theory of randoms walks for groups acting on hyperbolic spaces.
This is joint work with I. Gekhtman and G. Tiozzo.