{
  "id": "1903.08436",
  "version": "v1",
  "published": "2019-03-20T11:02:34.000Z",
  "updated": "2019-03-20T11:02:34.000Z",
  "title": "Oligomorphic groups are essentially countable",
  "authors": [
    "Andre Nies",
    "Philipp Schlicht",
    "Katrin Tent"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We study the complexity of the isomorphism relation on classes of closed subgroups of $S_\\infty$, the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. A closed subgroup $G$ of $S_\\infty$ is called $\\mathit{oligomorphic}$ if for each $n$, its natural action on $n$-tuples of natural numbers has only finitely many orbits. We show that the isomorphism relation for oligomorphic subgroups of $S_\\infty$ is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of groups topologically isomorphic to oligomorphic subgroups of $S_\\infty$. Given a closed subgroup $G$ of $S_\\infty$, the coarse group $\\mathcal{M}(G)$ is the structure whose domain consists of cosets of some open subgroups of $G$ and a single ternary relation $AB \\sqsubseteq C$. If $G$ has only countably many open subgroups, this translates $G$ into a countable coarse group structure $\\mathcal{M}(G)$ coding $G$. Coarse groups form our main tool in studying closed subgroups of $S_\\infty$ with only countably many open subgroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-20T11:02:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22A05",
      "20B27",
      "54H05",
      "03D80"
    ],
    "keywords": [
      "oligomorphic groups",
      "closed subgroup",
      "open subgroups",
      "essentially countable",
      "natural numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}