Random walks and quasi‐convexity in acylindrically hyperbolic groups

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 ∼ = 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.

Markov chains on hyperbolic-like groups and quasi-isometries

“…This is a result of Maher-Tiozzo [MT18] who proved that a (non-elementary) random walk on a group acting on a hyperbolic space X makes linear progress in X. This result feeds into the proof of several others, for example on the translation length [MT18], random subgroups [MS19,AH21], various kinds of projections [ST19], and deviation from quasi-geodesics [MS20], as well as the already mentioned central limit theorem for acylindrically hyperbolic groups, and results on counting measures, see e.g. [GTT18].…”

Some examples of separable convex‐cocompact subgroups

“…The next lemma, which is used in the proof of Proposition 3.1, follows from arguments in [4]. We note that [1, Proposition 4.1] is related to the lemma, though neither implies the other. Lemma Let $$\mathfrak {g}\geqslant 2$$ and let $$H$$ be a convex‐compact subgroup of $$MCG(\Sigma _{\mathfrak {g}})$$.…”