Subgroup dynamics and C*-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 * -simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group V is C * -simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented C * -simple groups without free subgroups. We prove that a branch group is either amenable or C * -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 * -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.

Higman–Thompson‐like groups of higher rank graph C*‐algebras

“…Similar to the proof of [16, Proposition 4.12] one has Theorem Let $$\Lambda$$ be an aperiodic single‐vertex $$k$$‐graph. Then the action $$\phi$$ of $$\operatorname{\Lambda _{ht}}$$ on $$\Lambda ^\infty$$ has Hausdorff germs.…”

“…This section provides some necessary backgrounds on groupoids, topological full groups of groupoids, higher rank graph algebras, and URS. The main resources are from [8, 13, 16, 20, 22–24].…”

“…Let $$G$$ be a faithful and minimal action by homeomorphisms on a compact space $$X$$. Then this action induces a URS known as the stabilizer URS of the action $$G$$ on $$X$$, which is denoted by $${\mathcal {S}}_G(X)$$ (see [10, 16] for more information). For $$x\in X$$, we write $$G_x$$ as the stabilizer of $$x$$, and $$G_x^0$$ as the normal subgroup of $$G_x$$ which consists of all elements $$g\in G_x$$ such that $$g$$ fixes pointwise a neighborhood of $$x$$.…”

“…In what follows, we show that $$\phi :G\curvearrowright \Lambda ^\infty$$ is an extreme boundary action, that is, $$\Lambda ^\infty$$ is compact, $$\phi$$ is minimal, and extremely proximal (see [16, Subsection 2.1] for all relevant definitions). Lemma Let $$\Lambda$$ be an aperiodic single‐vertex $$k$$‐graph.…”

“…In Section 5, we first prove that the commutator group $$\operatorname{\Lambda _{ht}^\prime }$$ of $$\operatorname{\Lambda _{ht}}$$ is simple and that the stabilizer uniformly recurrent subgroup (URS) is the only non‐trivial URS of $$\operatorname{\Lambda _{ht}^\prime }$$ if $$\Lambda$$ is strongly connected. This is heavily facilitated by some recent results in [16, 18]. When $$\Lambda$$ is single‐vertex, we obtain more properties of $$\operatorname{\Lambda _{ht}}$$: (1) The action $$\phi$$ of $$\operatorname{\Lambda _{ht}}$$ on $$\Lambda ^\infty$$ is an extreme boundary action and its stabilizer URS is also explicitly described; (2) $$\operatorname{\Lambda _{ht}}$$ is C*‐simple and so has the unique trace property; and (3) the C*‐simplicity of $$\operatorname{\Lambda _{ht}}$$ provides another characterization of the aperiodicity of $$\Lambda$$.…”