Subgroup mixing and random walks in groups acting on hyperbolic spaces

TL;DR

This paper analyzes the topological dynamics of the conjugation action of a countable group $G$ on its subgroup space ${\rm Sub}(G)$ for groups acting on hyperbolic spaces, focusing on acylindrically hyperbolic groups with trivial finite radical. It develops a random-walk framework using permissible measures $\mu$ to produce elements with strong mixing properties, then shows that the action on the space of infinite-index convex cocompact subgroups ${\rm Sub}_{\infty}^{cc}(G\curvearrowright S)$ is topologically $\mu$-mixing, hence highly topologically transitive. A key technical advance is the existence of transverse loxodromic elements to convex cocompact subgroups (and their relative-hyperbolic analogues), which enables the random-generation arguments establishing that random $k$-generated subgroups are free and combine with a convex cocompact subgroup as a free product, preserving geometric niceties. The results extend to relatively hyperbolic groups and groups with infinitely many ends, yielding mu-mixing on closures of spaces of relatively quasi-convex subgroups and yielding strong subgroup-dynamics consequences with potential model-theoretic applications. Overall, the work provides a robust mechanism to obtain mixing and high transitivity phenomena for subgroup actions via random walks in broad hyperbolic-geometry contexts, strengthening connections between random walks, convex cocompactness, and subgroup dynamics.

Abstract

We study the topological dynamics of the action of an acylindrically hyperbolic group on the space of its infinite index convex cocompact subgroups by conjugation. We show that, for any suitable probability measure $μ$, random walks with respect to $μ$ will produce elements with strong mixing properties for this action asymptotically almost surely. In particular, when the group has no finite normal subgroups this implies that the action is highly topologically transitive. Along the way, we prove technical results about convex cocompact subgroups which allow us to extend some results on random walks of Abbott and the first author.

Figures (1)

• Figure 1: Illustration of the proof of Proposition \ref{['prop:existence-transversal']}

Theorems & Definitions (98)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: AMS