Commensurated subgroups and micro-supported actions

Abstract

Let $\Gamma$ be a finitely generated group and $X$ be a minimal compact $\Gamma$-space. We assume that the $\Gamma$-action is micro-supported, i.e. for every non-empty open subset $U \subseteq X$, there is an element of $\Gamma$ acting non-trivially on $U$ and trivially on the complement $X \setminus U$. We show that, under suitable assumptions, the existence of certain commensurated subgroups in $\Gamma$ yields strong restrictions on the dynamics of the $\Gamma$-action: the space $X$ has compressible open subsets, and it is an almost $\Gamma$-boundary. Those properties yield in turn restrictions on the structure of $\Gamma$: $\Gamma$ 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.