On the genericity of loxodromic actions

TL;DR

This paper investigates the likelihood that a randomly chosen element in a finitely generated group acting on a hyperbolic space acts loxodromically, establishing conditions under which this probability is significantly positive.

Contribution

It proves that under weak automaticity and compatibility conditions, the probability of loxodromic action is bounded away from zero, with applications to pseudo-Anosov braids.

Findings

Probability of loxodromic action is bounded away from zero.

Weak automaticity and compatibility conditions are sufficient.

Application to genericity of pseudo-Anosov braids.

Abstract

One way of picking a "generic" element of a finitely generated group is to pick a random element with uniform probability in a large ball centered on $1$ in the Cayley graph. If the group acts on a $\delta$-hyperbolic space, with at least one element acting loxodromically, then it is plausible that generic elements should act loxodromically with high probability. In this paper we prove that the probability of acting loxodromically is bounded away from 0, provided the group satisfies a very weak automaticity condition, and provided a certain compatibility condition linking the automatic with the $\delta$-hyperbolic structure is satisfied. We present several applications of this result, including the genericity of pseudo-Anosov braids.