## Abstract

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 language | English |
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Pages (from-to) | 688-708 |

Number of pages | 21 |

Journal | Journal of the London Mathematical Society |

Volume | 94 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 2016 |

## ASJC Scopus subject areas

- General Mathematics