Stabilizer Subgroups and the Simplicity of Reduced Crossed Products

TL;DR

This paper characterizes when reduced crossed products are simple for minimal group actions on compact spaces, linking simplicity to stabilizer subgroups with trivial amenable radicals, and answers a question of Ozawa.

Contribution

It provides a complete characterization of simplicity for reduced crossed products in various classes of groups, including linear and hyperbolic groups, based on stabilizer subgroup properties.

Findings

Simplicity of reduced crossed products implies existence of points with stabilizers having trivial amenable radical.

Characterization of simplicity for minimal actions of linear and hyperbolic groups.

Almost every subgroup has a trivial amenable radical in certain recurrent subgroup scenarios.

Abstract

Given a minimal action $G\curvearrowright X$ of a countable group $G$ on a compact space $X$, we prove that if the reduced crossed product $G\ltimes_rC(X)$ is simple, then there exists a point whose stabilizer subgroup has trivial amenable radical. As a consequence, we give a complete characterization of the simplicity of the reduced crossed product of minimal actions of countable linear groups, hyperbolic groups, and, more generally, for groups with countably many amenable subgroups. This answers a question of Ozawa (2014) for these classes of groups. Furthermore, in the case of an infinite uniformly recurrent subgroup of a $C^*$-simple group, we prove that almost every subgroup has a trivial amenable radical, with respect to a fully supported, atomless probability measure.