Counting loxodromics for hyperbolic actions

Giulio Tiozzo (Yale)

Consider 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 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 S.Taylor.