Finitely approximable groups and actions Part I: The Ribes--Zalesski\u\i{} property

TL;DR

This paper establishes a precise link between finitely approximable group actions by isometries and the closure properties of subgroup products in the profinite topology, extending Solecki's theorem.

Contribution

It provides a new characterization of finitely approximable actions of discrete groups via subgroup closure properties in the profinite topology.

Findings

Finitely approximable actions correspond to closed subgroup products in the profinite topology.

Extension of Solecki's theorem to a broader class of group actions.

Equivalence between action approximation and subgroup closure conditions.

Abstract

We investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces \cite{solecki1} and obtain the following exact equivalence: any action of a discrete group $\Gamma$ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of $\Gamma$ is closed in the profinite topology on $\Gamma$.