Counting loxodromics for hyperbolic actions

Let (Formula presented.) be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X–loxodromics, approaches 1 as n → ∞. 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 ∂X. We discuss various applications, in particular to mapping class groups, Out(F_N), and right–angled Artin groups.