{
  "id": "2407.10222",
  "version": "v1",
  "published": "2024-07-14T14:26:07.000Z",
  "updated": "2024-07-14T14:26:07.000Z",
  "title": "On closure operations in the space of subgroups and applications",
  "authors": [
    "Dominik Francoeur",
    "Adrien Le Boudec"
  ],
  "categories": [
    "math.GR",
    "math.OA"
  ],
  "abstract": "We establish some interactions between uniformly recurrent subgroups (URSs) of a group $G$ and cosets topologies $\\tau_\\mathcal{N}$ on $G$ associated to a family $\\mathcal{N}$ of normal subgroups of $G$. We show that when $\\mathcal{N}$ consists of finite index subgroups of $G$, there is a natural closure operation $\\mathcal{H} \\mapsto \\mathrm{cl}_\\mathcal{N}(\\mathcal{H})$ that associates to a URS $\\mathcal{H}$ another URS $\\mathrm{cl}_\\mathcal{N}(\\mathcal{H})$, called the $\\tau_\\mathcal{N}$-closure of $\\mathcal{H}$. We give a characterization of the URSs $\\mathcal{H}$ that are $\\tau_\\mathcal{N}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when $G$ belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $\\mathcal{A}_G$, and prove that for certain coset topologies on $G$, almost all subgroups $H \\in \\mathcal{A}_G$ have the same closure. For groups in which amenability is detected by a set of laws, we deduce a criterion for $\\mathcal{A}_G$ to be a singleton based on residual properties of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-14T14:26:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "applications",
      "natural closure operation",
      "finite index subgroups",
      "faithful minimal profinite action",
      "largest amenable urs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}