Subgroup dynamics and $C^\ast$-simplicity of groups of homeomorphisms

Adrien Le Boudec, Nicolás Matte Bon

Published 2016-05-05Version 1

We study the uniformly recurrent subgroups of various classes of groups of homeomorphisms, including Thompson's groups and their generalizations, groups of piecewise projective homeomorphisms of the real line, some groups acting on non-rooted trees, the class of branch groups, and some topological full groups. Our results have applications to the study of $C^\ast$-simplicity of these groups. Examples of groups which are shown to be $C^\ast$-simple are Thompson's group $V$, and some groups of piecewise projective homeomorphisms of the real line. This provides in particular examples of finitely presented $C^\ast$-simple groups without free subgroups. We show that if $G$ is a branch group, then $G$ is $C^\ast$-simple if and only if $G$ is non-amenable. We also prove the converse of a result of Haagerup and Olesen, namely that if Thompson's group $F$ is non-amenable, then Thompson's group $T$ is $C^\ast$-simple; and that this is also equivalent to the $C^\ast$-simplicity of other groups of homeomorphisms. We provide a complete classification of the uniformly recurrent subgroups of Thompson's groups $F$, $T$ and $V$. As an application, we deduce rigidity results for their minimal actions on compact spaces.