{
  "id": "1010.5987",
  "version": "v2",
  "published": "2010-10-28T15:08:00.000Z",
  "updated": "2011-06-06T20:12:30.000Z",
  "title": "Notes on non-archimedean topological groups",
  "authors": [
    "Michael Megrelishvili",
    "Menachem Shlossberg"
  ],
  "comment": "17 pages, revised version",
  "categories": [
    "math.GN",
    "math.GM",
    "math.GR"
  ],
  "abstract": "We show that the Heisenberg type group $H_X=(\\Bbb{Z}_2 \\oplus V) \\leftthreetimes V^{\\ast}$, with the discrete Boolean group $V:=C(X,\\Z_2)$, canonically defined by any Stone space $X$, is always minimal. That is, $H_X$ does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean $G$ there exists a (resp., locally compact) non-archimedean minimal group $M$ such that $G$ is a group retract of $M.$ For discrete groups $G$ the latter was proved by S. Dierolf and U. Schwanengel. We unify some old and new characterization results for non-archimedean groups. Among others we show that every continuous group action of $G$ on a Stone space $X$ is a restriction of a continuous group action by automorphisms of $G$ on a topological (even, compact) group $K$. We show also that any epimorphism $f: H \\to G$ (in the category of Hausdorff topological groups) into a non-archimedean group $G$ must be dense.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-06T20:12:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-archimedean topological groups",
      "continuous group action",
      "stone space",
      "non-archimedean group",
      "strictly coarser hausdorff group topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.5987M"
    }
  }
}