Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings

TL;DR

This paper demonstrates that in acylindrically hyperbolic groups, random subgroups generated by independent random walks are almost surely free and hyperbolically embedded, revealing typical subgroup structures in such groups.

Contribution

It introduces the concept that random subgroups in acylindrically hyperbolic groups are almost surely free and hyperbolically embedded, advancing understanding of subgroup dynamics.

Findings

Random subgroups are almost surely free.

Such subgroups are hyperbolically embedded.

Results hold with asymptotic probability one.

Abstract

Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup H of G is a free group, and the semidirect product of H acting on E(G) is hyperbolically embedded in G, where E(G) is the unique maximal finite normal subgroup of G.