From isolated subgroups to generic permutation representations

TL;DR

This paper characterizes groups with dense isolated subgroups as those admitting a generic permutation representation, linking topological subgroup properties to permutation dynamics.

Contribution

It establishes the equivalence between the density of isolated subgroups and the existence of a generic permutation representation for countable groups.

Findings

Solitary groups include finitely generated LERF groups.

Groups with countably many subgroups are solitary.

The paper provides a topological characterization of permutation representations.

Abstract

Let $G$ be a countable group, $\operatorname{Sub}(G)$ the (compact, metric) space of all subgroups of $G$ with the Chabauty topology and $\operatorname{Is}(G) \subset \operatorname{Sub}(G)$ the collection of isolated points. We denote by $X!$ the (Polish) group of all permutations of a countable set $X$. Then the following properties are equivalent: (i) $\operatorname{Is}(G)$ is dense in $\operatorname{Sub}(G)$, (ii) $G$ admits a "generic permutation representation". Namely there exists some $\tau^* \in \operatorname{Hom}(G,X!)$ such that the collection of permutation representations $\{\phi \in \operatorname{Hom}(G,X!) \ | \ \phi {\text{is permutation isomorphic to}} \tau^*\}$ is co-meager in $\operatorname{Hom}(G,X!)$. We call groups satisfying these properties solitary. Examples of solitary groups include finitely generated LERF groups and groups with countably many subgroups.