Hyperfiniteness of boundary actions of acylindrically hyperbolic groups

TL;DR

This paper demonstrates that for any countable acylindrically hyperbolic group, there exists a generating set making the Cayley graph hyperbolic with a hyperfinite boundary action, expanding the class of groups with such properties.

Contribution

It establishes the existence of generating sets for acylindrically hyperbolic groups that produce hyperbolic Cayley graphs with hyperfinite boundary actions, broadening understanding of boundary dynamics.

Findings

Existence of generating sets with hyperbolic Cayley graphs

Boundary actions are hyperfinite for these groups

The class of groups with such boundary actions is expanded

Abstract

We prove that for any countable acylidrically hyperbolic group $G$, there exists a generating set $S$ of $G$ such that the corresponding Cayley graph $\Gamma(G,S)$ is hyperbolic, $|\partial\Gamma(G,X)|>2$, the natural action of $G$ on $\Gamma(G,S)$ is acylindrical, and the natural action of $G$ on the Gromov boundary $\partial\Gamma(G,S)$ is hyperfinite. This result broadens a class of groups that admit a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action.