$C^*$-simplicity for groups with non-elementary convergence group actions

TL;DR

This paper proves that countable groups with certain types of convergence group actions are Powers groups, leading to their subgroups being $C^*$-simple, which advances understanding of their algebraic and operator algebraic properties.

Contribution

It establishes that groups with effective minimal non-elementary convergence actions are Powers groups and their subgroups are $C^*$-simple, a significant extension in group theory and operator algebras.

Findings

Countable groups with non-elementary convergence actions are Powers groups.

Such groups are strongly Powers groups, ensuring $C^*$-simplicity of subgroups.

Provides new criteria linking convergence actions to $C^*$-simplicity.

Abstract

We prove that a countable group with an effective minimal non-elementary convergence group action is a Powers group. More strongly we prove that it is a strongly Powers group and thus its non-trivial subnormal subgroups are $C^*$-simple.