From isolated subgroups to generic permutation representations

Y. Glasner, D. Kitroser, J. Melleray

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 {∈ Hom(G,X!) | 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.

Original languageEnglish
Pages (from-to)688-708
Number of pages21
JournalJournal of the London Mathematical Society
Issue number3
StatePublished - 1 Jan 2016

ASJC Scopus subject areas

  • Mathematics (all)


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