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

  • Dominik Francoeur EMAIL logo
From the journal Journal of Group Theory


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.

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).


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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