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

Le Boudec, Adrien; Matte Bon, Nicolás

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 557-602.

Nous étudions les sous-groupes uniformément récurrents de groupes agissant par homéomorphismes sur un espace topologique. Nous prouvons un résultat général reliant les sous-groupes uniformément récurrents aux stabilisateurs rigides de l'action, et en déduisons un critère de $C^{*}$-simplicité basé sur la non moyennabilité des stabilisateurs rigides. Comme application, nous prouvons que le groupe de Thompson $V$ est $C^{*}$-simple, de même que certains groupes d'homéomorphismes projectifs par morceaux de la droite réelle. Cela fournit des exemples de groupes finiment présentés qui sont $C^{*}$-simples et sans sous-groupes libres. Nous prouvons qu'un groupe branché est soit moyennable, soit $C^{*}$-simple. Nous prouvons également la réciproque d'un résultat de Haagerup et Olesen: si le groupe de Thompson $F$ n'est pas moyennable alors le groupe de Thompson $T$ est $C^{*}$-simple. Nos résultats fournissent de plus des conditions suffisantes sur un groupe d'homéomorphismes sous lesquelles les sous-groupes uniformément récurrents sont complètement compris. Cela s'applique aux groupes de Thompson, pour lesquels nous déduisons également des résultats de rigidité sur leurs actions sur des espaces compacts.

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.

Classification : 37B05, 54H20, 37B20, 20E08, 20F65,
Keywords: Chabauty space, Uniformly recurrent subgroups, Minimal, strongly and extremely proximal group actions, C*-simple groups
Mot clés : Espace de Chabauty, sous-groupes uniformément récurrents, actions de groupes minimales, fortement et extrêmement proximales, groupes C*-simples.

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