{
  "id": "2106.12547",
  "version": "v1",
  "published": "2021-06-23T17:16:33.000Z",
  "updated": "2021-06-23T17:16:33.000Z",
  "title": "Automatic continuity for groups whose torsion subgroups are small",
  "authors": [
    "Daniel Keppeler",
    "Philip Möller",
    "Olga Varghese"
  ],
  "comment": "18 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We prove that a group homomorphism $\\varphi\\colon L\\to G$ from a locally compact Hausdorff group $L$ into a discrete group $G$ either is continuous, or there exists a normal open subgroup $N\\subseteq L$ such that $\\varphi(N)$ is a torsion group provided that $G$ does not include $\\mathbb{Q}$ or the $p$-adic integers $\\mathbb{Z}_p$ or the Pr\\\"ufer $p$-group $\\mathbb{Z}(p^\\infty)$ for any prime $p$ as a subgroup, and if the torsion subgroups of $G$ are small in the sense that any torsion subgroup of $G$ is artinian. In particular, if $\\varphi$ is surjective and $G$ additionaly does not have non-trivial normal torsion subgroups, then $\\varphi$ is continuous. As an application we obtain results concerning the continuity of group homomorphisms from locally compact Hausdorff groups to many groups from geometric group theory, in particular to automorphism groups of right-angled Artin groups and to Helly groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-23T17:16:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22D05",
      "20F65"
    ],
    "keywords": [
      "automatic continuity",
      "locally compact hausdorff group",
      "group homomorphism",
      "non-trivial normal torsion subgroups",
      "normal open subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}