Random walks and quasi-convexity in acylindrically hyperbolic groups

TL;DR

This paper extends known properties of quasi-convex subgroups in hyperbolic groups to a probabilistic setting, showing that random walks generate free products with high probability, preserving quasi-convexity.

Contribution

It introduces a probabilistic generalization of quasi-convex subgroup properties in hyperbolic groups, applicable to groups acting on hyperbolic spaces and involving random walks.

Findings

Random walks generate free products with high probability as length increases.

The resulting subgroup remains quasi-convex in the ambient group.

Results apply to mapping class groups and convex cocompact subgroups.

Abstract

It is known that every infinite index quasi-convex subgroup $H$ of a non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that when $R$ is a subgroup generated by independent random walks in $G$, then $\langle H, R\rangle\cong H\ast R$ with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in $G$. Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits. In particular, when $G$ is the mapping class group of a surface and $H$ is a convex cocompact subgroup we show that $\langle H, R\rangle$ is convex cocompact and isomorphic to $ H\ast R$.