From isolated subgroups to generic permutation representations

Abstract

Let $G$ be a countable group, $\mathrm{Sub}(G)$ the (compact, metric) space of all subgroups of $G$ with the Chabauty topology and $\mathrm{Is}(G)\subset \mathrm{Sub}(G)$ 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) $\mathrm{Is}(G)$ is dense in $\mathrm{Sub}(G)$, (ii) $G$ admits a "generic permutation representation". Namely there exists some ${\tau }^{\ast }\in \mathrm{Hom}(G,X!)$ such that the collection of permutation representations $\{\varphi \in \mathrm{Hom}(G,X!)|\varphi \text{is permutation isomorphic to}{\tau }^{\ast }\}$ is co-meager in $\mathrm{Hom}(G,X!)$. We call groups satisfying these properties solitary. Examples of solitary groups include finitely generated LERF groups and groups with countably many subgroups.

21 pages