Random walks and quasi-convexity in acylindrically hyperbolic groups

DOI10.1112/TOPO.12205arXiv1909.10876WikidataQ115526844 ScholiaQ115526844MaRDI QIDQ6325881

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Publication date: 24 September 2019

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 $langleH,RanglecongHastR$ 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 $langleH,Rangle$ is convex cocompact and isomorphic to $HastR$.

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