On closure operations in the space of subgroups and applications

TL;DR

This paper explores closure operations on uniformly recurrent subgroups (URSs) within groups, linking topological and algebraic properties, and provides criteria for the structure and uniqueness of certain URSs.

Contribution

It introduces a natural closure operation for URSs related to coset topologies and characterizes when URSs are closed, with implications for amenability and residual properties.

Findings

Defined a closure operation for URSs using coset topologies.

Characterized $ au_ ext{N}$-closed URSs in terms of stabilizer URSs.

Provided criteria for the singleton property of the largest amenable URS.

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 (a property that is variant of the Tits alternative), we deduce a criterion for $\mathcal{A}_G$ to be a singleton based on residual properties of $G$.