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.
Communicated by: Benjamin Klopsch
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