Random walks and quasi-convexity in acylindrically hyperbolic groups

Abstract

Arzhantseva proved that every infinite index quasi-convex subgroup H of a 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 H, R ∼ = H * 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 H, R ∼ = H * R is convex cocompact.