Commensurated subgroups and micro-supported actions (with an appendix by Dominik Francoeur)

  • Pierre-Emmanuel Caprace

    Université Catholique de Louvain, Louvain la Neuve, Belgium
  • Adrien Le Boudec

    École Normale Supérieure de Lyon, France
Commensurated subgroups and micro-supported actions (with an appendix by Dominik Francoeur) cover
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Abstract

Let Γ\Gamma be a finitely generated group and XX be a minimal compact Γ\Gamma-space. We assume that the Γ\Gamma-action is micro-supported, i.e. for every non-empty open subset UXU \subseteq X, there is an element of Γ\Gamma acting non-trivially on UU and trivially on the complement XUX \setminus U. We show that, under suitable assumptions, the existence of certain commensurated subgroups in Γ\Gamma yields strong restrictions on the dynamics of the Γ\Gamma-action: the space XX has compressible open subsets, and it is an almost Γ\Gamma-boundary. Those properties yield in turn restrictions on the structure of Γ\Gamma: Γ\Gamma is neither amenable nor residually finite. Among the applications, we show that the (alternating subgroup of the) topological full group associated to a minimal and expansive Cantor action of a finitely generated amenable group has no commensurated subgroups other than the trivial ones. Similarly, every commensurated subgroup of a finitely generated branch group is commensurate to a normal subgroup; the latter assertion relies on an appendix by Dominik Francoeur, and generalizes a result of Phillip Wesolek on finitely generated just-infinite branch groups. Other applications concern discrete groups acting on the circle, and the centralizer lattice of non-discrete totally disconnected locally compact (tdlc) groups. Our results rely, in an essential way, on recent results on the structure of tdlc groups, on the dynamics of their micro-supported actions, and on the notion of uniformly recurrent subgroups.

Cite this article

Pierre-Emmanuel Caprace, Adrien Le Boudec, Commensurated subgroups and micro-supported actions (with an appendix by Dominik Francoeur). J. Eur. Math. Soc. (2022),

DOI 10.4171/JEMS/1236