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

Abstract

We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^{\ast }$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^{\ast }$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^{\ast }$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^{\ast }$-simple. We also prove the converse of a result of Haagerup and Olesen: if Thompson's group $F$ is non-amenable, then Thompson's group $T$ must be $C^{\ast }$-simple. Our results further provide sufficient conditions on a group of homeomorphisms under which uniformly recurrent subgroups can be completely classified. This applies to Thompson's groups $F$, $T$ and $V$, for which we also deduce rigidity results for their minimal actions on compact spaces.

45 pages. Minor changes

References in corpus (6)

• Boundaries of reduced C*-algebras of discrete groups
• An intrinsic characterization of C*-simplicity
• Rigidity and flexibility of group actions on the circle
• A new look at C*-simplicity and the unique trace property of a group
• Ideal structure of the C*-algebra of Thompson group T
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Cited by in corpus (5)

• Uniformly recurrent subgroups and the ideal structure of reduced crossed products
• C*-simplicity of free products with amalgamation and radical classes of groups
• $C^{\ast }$-simplicity and representations of topological full groups of groupoids
• On minimal actions of countable groups
• Uniformly recurrent subgroups and simple $C^{\ast }$-algebras