An extreme boundary of acylindrically hyperbolic groups

TL;DR

This paper proves that acylindrically hyperbolic groups have a minimal, extremely proximal action on a compact space, with topologically free actions under certain conditions, impacting the study of $C^*$-algebras.

Contribution

It establishes the existence of a minimal, extremely proximal action for all acylindrically hyperbolic groups, answering a question of Ozawa and exploring applications to $C^*$-algebras.

Findings

Every acylindrically hyperbolic group admits a minimal, extremely proximal action.

If no nontrivial finite normal subgroups exist, the action is topologically free.

Applications to $C^*$-algebras are discussed.

Abstract

We prove that every acylindrically hyperbolic group admits a minimal and extremely proximal action on a compact metrizable space. If there are no nontrivial finite normal subgroups, then the action is topologically free. This answers positively a question of Ozawa and the applications to $C^\ast$-algebras are discussed.