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- Title
Commensurated subgroups and micro-supported actions.
- Authors
Caprace, Pierre-Emmanuel; Le Boudec, Adrien
- Abstract
Let Г be a finitely generated group and X be a minimal compact Г-space. We assume that the Г-action is micro-supported, i.e. for every non-empty open subset U ⊆ X, there is an element of Г acting non-trivially on U and trivially on the complement X n U. We show that, under suitable assumptions, the existence of certain commensurated subgroups in Г yields strong restrictions on the dynamics of the Г-action: the space X has compressible open subsets, and it is an almost Г-boundary. Those properties yield in turn restrictions on the structure of Г: Г 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.
- Subjects
COMPACT groups; DISCRETE groups; SUBSET selection; GENERALIZATION; MATHEMATICAL models
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2023, Vol 25, Issue 6, p2251
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/1236