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On the stabilisers of points in groups with micro-supported actions

Dominik Francoeur
From the journal Journal of Group Theory

Abstract

Given a group 𝐺 of homeomorphisms of a first-countable Hausdorff space 𝒳, we prove that if the action of 𝐺 on 𝒳 is minimal and has rigid stabilisers that act locally minimally, then the neighbourhood stabilisers of any two points in 𝒳 are conjugated by a homeomorphism of 𝒳. This allows us to study stabilisers of points in many classes of groups, such as topological full groups of Cantor minimal systems, Thompson groups, branch groups, and groups acting on trees with almost prescribed local actions.

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR-10-LABX-0070

Award Identifier / Grant number: ANR-11-IDEX-0007

Funding statement: This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Acknowledgements

The author would like to thank Laurent Bartholdi, Adrien Le Boudec, Tatiana Nagnibeda, Volodymyr Nekrashevych and Aitor Pérez for useful discussions regarding this work.

  1. Communicated by: Benjamin Klopsch

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Received: 2020-09-15
Revised: 2020-12-02
Published Online: 2020-12-22
Published in Print: 2021-05-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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