From isolated subgroups to generic permutation representations

Let G be a countable group, Sub(G) be the (compact, metric) space of all subgroups of G with the Chabauty topology and Is(G) ⊆ Sub(G) be 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) Is(G) is dense in Sub(G); (ii) G admits a ‘generic permutation representation’. Namely, there exists some τ∗ ∈ Hom(G,X!) such that the collection of permutation representations {φ ∈ Hom(G,X!) |φ is permutation isomorphic to τ∗} is co-meager in Hom(G,X!). We call groups satisfying these properties solitary . Examples of solitary groups include finitely generated locally extended residually finite groups and groups with countably many subgroups.