A note on LERF groups and generic group actions

TL;DR

This paper explores the properties of LERF and A-separable groups, establishing their equivalences and introducing A-separable groups as a broader class that includes LERF and amenable groups.

Contribution

It introduces A-separable groups, generalizing LERF groups by replacing finite index with co-amenable subgroups, and investigates their properties.

Findings

LERF property is equivalent to generic homomorphisms having finite orbits.

A-separable groups include all LERF and amenable groups.

A-separable groups form a broader class with distinct properties.

Abstract

Let $G$ be a finitely generated group, $\mathrm{Sub}(G)$ the (compact, metric) space of all subgroups of $G$ with the Chaubuty topology and $X!$ the (Polish) group of all permutations of a countable set $X$. We show that the following properties are equivalent: (i) Every finitely generated subgroup is closed in the profinite topology, (ii) the finite index subgroups are dense in $\mathrm{Sub}(G)$, (iii) A Baire generic homomorphism $\phi: G \rightarrow X!$ admits only finite orbits. Property (i) is known as the LERF property. We introduce a new family of groups which we call {\it{A-separable}} groups. These are defined by replacing, in (ii) above, the word "finite index" by the word "co-amenalbe". The class of A-separable groups contains all LERF groups, all amenable groups and more. We investigate some properties of these groups.