Hyperfiniteness of boundary actions of acylindrically hyperbolic groups

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.

arXiv commentSection 5 "Application to topologically amenable actions" was added. Appeared in Forum of Mathematics, Sigma

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