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 $Γ(G,S)$ is hyperbolic, $|\partialΓ(G,X)|>2$, the natural action of $G$ on $Γ(G,S)$ is acylindrical, and the natural action of $G$ on the Gromov boundary $\partialΓ(G,S)$ is hyperfinite. This result broadens a class of groups that admit a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action.