Geodesic
Subgroup dynamics and C*-simplicity of groups of homeomorphisms
[Dynamique dans l'espace des sous-groupes et C*-simplicité de groupes d'homéomorphismes]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 557-602.

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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.

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.

Publié le :
DOI : 10.24033/asens.2361
Classification : 37B05, 54H20, 37B20, 20E08, 20F65,
Keywords: Chabauty space, Uniformly recurrent subgroups, Minimal, strongly and extremely proximal group actions, C*-simple groups
Mots-clés : Espace de Chabauty, sous-groupes uniformément récurrents, actions de groupes minimales, fortement et extrêmement proximales, groupes C*-simples. 
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     author = {Le Boudec, Adrien and Matte Bon, Nicol\'as},
     title = {Subgroup dynamics and $C^\ast $-simplicity  of groups of homeomorphisms},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {557--602},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
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Le Boudec, Adrien; Matte Bon, Nicolás. Subgroup dynamics and $C^\ast $-simplicity  of groups of homeomorphisms. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 557-602. doi : 10.24033/asens.2361. http://geodesic.mathdoc.fr/articles/10.24033/asens.2361/

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