{
  "id": "2011.15075",
  "version": "v1",
  "published": "2020-11-30T18:14:38.000Z",
  "updated": "2020-11-30T18:14:38.000Z",
  "title": "Automorphisms of procongruence mapping class groups",
  "authors": [
    "Marco Boggi"
  ],
  "comment": "55 pages",
  "categories": [
    "math.GT",
    "math.AG",
    "math.NT"
  ],
  "abstract": "For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, with $\\chi(S)=2-2g-n<0$, let $\\Gamma(S)$ be the mapping class group of $S$ and $\\hat{\\Gamma}(S)$ and $\\check{\\Gamma}(S)$, respectively, its profinite and its congruence completion. The latter is the image of the natural representation $\\hat{\\Gamma}(S)\\to\\operatorname{Out}(\\hat{\\pi}_1(S))$, where $\\hat{\\pi}_1(S)$ is the profinite completion of the fundamental group of the surface $S$. Let $\\operatorname{Out}^{\\mathbb{I}_0}(\\check{\\Gamma}(S))$ be the group of outer automorphisms of $\\check{\\Gamma}(S)$ which preserve the conjugacy class of a procyclic subgroup generated by a nonseparating Dehn twist and put $d(S)=3g-3+n$. The main result of the paper is that, for $d(S)>1$, there is a natural isomorphism: \\[\\operatorname{Out}^{\\mathbb{I}_0}(\\check{\\Gamma}(S))\\cong\\widehat{\\operatorname{GT}},\\] where $\\widehat{\\operatorname{GT}}$ is the profinite Grothendieck-Teichm\\\"uller group. We will actually prove a slightly stronger result which implies that the automorphism group of the procongruence Grothendieck-Teichm\\\"uller tower is also isomorphic to $\\widehat{\\operatorname{GT}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-30T18:14:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G32"
    ],
    "keywords": [
      "procongruence mapping class groups",
      "completion",
      "natural representation",
      "automorphism group",
      "fundamental group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}