On closure operations in the space of subgroups and applications

Abstract

We establish some interactions between uniformly recurrent subgroups (URSs) of a group $G$ and cosets topologies $\tau_\mathcal{N}$ on $G$ associated to a family $\mathcal{N}$ of normal subgroups of $G$. We show that when $\mathcal{N}$ consists of finite index subgroups of $G$, there is a natural closure operation $\mathcal{H} \mapsto \mathrm{cl}_\mathcal{N}(\mathcal{H})$ that associates to a URS $\mathcal{H}$ another URS $\mathrm{cl}_\mathcal{N}(\mathcal{H})$, called the $\tau_\mathcal{N}$-closure of $\mathcal{H}$. We give a characterization of the URSs $\mathcal{H}$ that are $\tau_\mathcal{N}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when $G$ belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $\mathcal{A}_G$, and prove that for certain coset topologies on $G$, almost all subgroups $H \in \mathcal{A}_G$ have the same closure. For groups in which amenability is detected by a set of laws, we deduce a criterion for $\mathcal{A}_G$ to be a singleton based on residual properties of $G$.