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Article Dans Une Revue Bulletin de la société mathématique de France Année : 2022

Piecewise strongly proximal actions, free boundaries and the Neretin groups

Résumé

A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally disconnected locally compact group $G$ on a compact space $X$ ensuring that no relatively amenable subgroup of $G$ can be confined. This property is equivalent to the fact that the action of $G$ on its Furstenberg boundary is free. Our criterion applies to the Neretin groups. We deduce that each Neretin group has two inequivalent irreducible unitary representations that are weakly equivalent. This implies that the Neretin groups are not of type I, thereby answering a question of Y.~Neretin.
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Dates et versions

hal-03357491 , version 1 (05-10-2021)
hal-03357491 , version 2 (05-07-2023)

Identifiants

Citer

Pierre-Emmanuel Caprace, Adrien Le Boudec, Nicolás Matte Bon. Piecewise strongly proximal actions, free boundaries and the Neretin groups. Bulletin de la société mathématique de France, 2022, 4 (150), pp.773-795. ⟨10.24033/bsmf.2861⟩. ⟨hal-03357491v2⟩
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