{
  "id": "2407.19440",
  "version": "v1",
  "published": "2024-07-28T09:37:11.000Z",
  "updated": "2024-07-28T09:37:11.000Z",
  "title": "Computably locally compact groups and their closed subgroups",
  "authors": [
    "Alexander G. Melnikov",
    "Andre Nies"
  ],
  "categories": [
    "math.GR",
    "math.GN",
    "math.LO"
  ],
  "abstract": "Given a computably locally compact Polish space $M$, we show that its 1-point compactification $M^*$ is computably compact. Then, for a computably locally compact group $G$, we show that the Chabauty space $\\mathcal S(G)$ of closed subgroups of $G$ has a canonical effectively-closed (i.e., $\\Pi^0_1$) presentation as a subspace of the hyperspace $\\mathcal K(G^*)$ of closed sets of $G^*$. We construct a computable discrete abelian group $H$ such that $\\mathcal S(H)$ is not computably closed in $\\mathcal K(H^*)$; in fact, the only computable points of $\\mathcal S(H)$ are the trivial group and $H$ itself, while $\\mathcal S(H)$ is uncountable. In the case that a computably locally compact group $G$ is also totally disconnected, we provide a further algorithmic characterization of $\\mathcal S(G)$ in terms of the countable meet groupoid of $G$ introduced recently by the authors (arXiv: 2204.09878). We apply our results and techniques to show that the index set of the computable locally compact abelian groups that contain a closed subgroup isomorphic to $(\\mathbb{R},+)$ is arithmetical.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-28T09:37:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "computably locally compact group",
      "closed subgroup",
      "locally compact polish space",
      "computable discrete abelian group",
      "computable locally compact abelian groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}