{
  "id": "0803.3841",
  "version": "v5",
  "published": "2008-03-27T00:18:45.000Z",
  "updated": "2016-09-29T15:19:21.000Z",
  "title": "A generalized congruence subgroup property for the hyperelliptic modular group",
  "authors": [
    "Marco Boggi"
  ],
  "comment": "18 pages. A substantial generalisation of the previous results",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "Let $S_{g,n}$, for $2g-2+n>0$, be obtained from a closed Riemann surface of genus $g$ removing $n$ points. The mapping class group $\\Gamma_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. For a fixed hyperelliptic involution $\\iota\\in\\Gamma_g$, the hyperelliptic mapping class group $\\Upsilon_g$ is the centralizer of $\\iota$ in $\\Gamma_g$. For $n> 0$, let $p\\colon\\Gamma_{g,n}\\to\\Gamma_g$ be the natural epimorphism. The hyperelliptic mapping class group $\\Upsilon_{g,n}$ is the inverse image $p^{-1}(\\Upsilon_g)\\leq\\Gamma_{g,n}$. In genus one, we just let $\\Upsilon_{1,n}:=\\Gamma_{1,n}$. As a subgroup of $\\Gamma_{g,n}$, the hyperelliptic modular group admits a natural faithful representation: $\\Upsilon_{g,n}\\to\\operatorname{Out}(\\pi_1(S_{g,n}))$. For $\\mathcal S$ a class of finite groups, let $\\hat{\\pi}_1(S_{g,n})^{\\mathcal S}$ be the pro-$\\mathcal S$ completion of the fundamental group $\\pi_1(S_{g,n})$. There is a natural monomorphism $\\Upsilon_{g,n}\\to\\operatorname{Out}(\\hat{\\pi}_1(S_{g,n})^{\\mathcal S})$ and we let $\\check{\\Upsilon}_{g,n}^{\\mathcal S}$ be the closure of $\\Upsilon_{g,n}$ inside the profinite group $\\operatorname{Out}(\\hat{\\pi}_1(S_{g,n})^{\\mathcal S})$. The main result of the paper is that, if ${\\mathbb Z}/2\\in{\\mathcal S}$, the profinite group $\\check{\\Upsilon}_{g,n}^{\\mathcal S}$ is, virtually, the pro-$\\mathcal S$ completion of the hyperelliptic mapping class group $\\Upsilon_{g,n}$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-01-18T10:00:33.000Z",
      "title": "The congruence subgroup property for the hyperelliptic modular group",
      "abstract": "Let ${\\cal M}_{g,n}$ and ${\\cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $\\Gamma_{g,n}$ and $H_{g,n}$, the so called Teichm{\\\"u}ller modular group and hyperelliptic modular group. A choice of base point on ${\\cal H}_{g,n}$ defines a monomorphism $H_{g,n}\\hookrightarrow \\Gamma_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichm\\\"uller group $\\Gamma_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. As a subgroup of $\\Gamma_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\\hookrightarrow\\out(\\pi_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^\\lambda$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of $\\pi_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}\\rightarrow Out(\\pi_1(S_{g,n})/K)$ is contained in $H^\\lambda$. The main result of the paper is an affermative answer to this question.",
      "comment": "16 pages. Proof of Proposition 2.7 substantially simplified",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2016-09-29T15:19:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30F60",
      "14H10",
      "14H15",
      "32G15",
      "11F80",
      "14H30",
      "14F35"
    ],
    "keywords": [
      "hyperelliptic modular group",
      "congruence subgroup property",
      "finite index characteristic subgroup",
      "finite index subgroup",
      "compact riemann surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.3841B"
    }
  }
}