Commensurated subgroups and micro-supported actions
(2023) Journal of the European Mathematical Society — Vol. 25, n° 6, p. 2251-2294 (2023)
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- Abstract
- Let Γ be a finitely generated group and X be a minimal compact Γ-space. We assume that the Γ-action is micro-supported, i.e. for every non-empty open subset U⊆X, there is an element of Γ acting non-trivially on U and trivially on the complement X∖U. We show that, under suitable assumptions, the existence of certain commensurated subgroups in Γ yields strong restrictions on the dynamics of the Γ-action: the space X has compressible open subsets, and it is an almost Γ-boundary. Those properties yield in turn restrictions on the structure of Γ: Γ 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.
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Caprace, P.-E., & Adrien Le Boudec. (2023). Commensurated subgroups and micro-supported actions. Journal of the European Mathematical Society, 25(6), 2251-2294. https://doi.org/10.4171/JEMS/1236 (Original work published 2023)