Abstract
Arzhantseva proved that every infinite-index quasi-convex subgroup H of a torsion-free 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⟩ approximately equal to 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⟩ is convex cocompact and isomorphic to H*R.