On the genericity of loxodromic actions (Q2408020)

Let \(G\) be a finitely generated group acting by isometries on a \(\delta\)-hyperbolic space, with at least one element acting loxodromically, and assume that the elements of \(G\) have a normal form with respect to some finite state automaton. In the paper under review, the author proves that under a certain compatibility condition between the automatic and the hyperbolic structure, in the ball consisting of elements of \(G\) whose normal form is of length at most \(l\), the proportion of elements acting loxodromically is bounded away from zero as \(l\to\infty\). The author also obtains some corollaries, including the fact that pseudo-Anosov braids are generic in the sense defined in this paper.