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Random walks and quasi-convexity in acylindrically hyperbolic groups
Journal of Topology  (IF1.582),  Pub Date : 2021-09-29, DOI: 10.1112/topo.12205
Carolyn Abbott, Michael Hull

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 $⟨ H , R ⟩ ≅ H * R$ 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 $H * R$.