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Annales scientifiques de l'ENS - Parutions - série 4, 51 (2018)

Parutions < série 4, 51

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 51, fascicule 3 (2018)

Adrien Le Boudec, Nicolás Matte Bon
Subgroup dynamics and C^-simplicity of groups of homeomorphisms
Annales scientifiques de l'ENS 51, fascicule 3 (2018), 557-602

Télécharger cet article : Fichier PDF

Résumé :
Dynamique dans l'espace des sous-groupes et C^-simplicité de groupes d'homéomorphismes
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.

Mots-clefs : Espace de Chabauty; sous-groupes uniformément récurrents; actions de groupes minimales, fortement et extrêmement proximales; groupes C*-simples.

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.

Keywords: Chabauty space; Uniformly recurrent subgroups; Minimal, strongly and extremely proximal group actions; C*-simple groups

Class. math. : 37B05, 54H20, 37B20, 20E08, 20F65,


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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