Random walks and quasi-convexity in acylindrically hyperbolic groups

Abstract

Arzhantseva proved that every infinite-index quasi-convex subgroup (Formula presented.) of a torsion-free hyperbolic group (Formula presented.) is a free factor in a larger quasi-convex subgroup of (Formula presented.). We give a probabilistic generalization of this result. That is, we show that when (Formula presented.) is a subgroup generated by independent random walks in (Formula presented.), then (Formula presented.) with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in (Formula presented.). Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits. In particular, when (Formula presented.) is the mapping class group of a surface and (Formula presented.) is a convex cocompact subgroup we show that (Formula presented.) is convex cocompact and isomorphic to (Formula presented.).