{
  "id": "math/0604271",
  "version": "v2",
  "published": "2006-04-12T12:33:49.000Z",
  "updated": "2007-01-11T14:49:27.000Z",
  "title": "Fundamental groups of moduli stacks of stable curves of compact type",
  "authors": [
    "Marco Boggi"
  ],
  "comment": "25 pages; minor corrections in some proofs; typos corrected; theorem numbering changed",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $\\widetilde{\\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, stable complex curves of compact type. Various characterizations and properties are obtained of both the algebraic and topological fundamental groups of the stack $\\widetilde{\\cal M}_{g,n}$. Let $\\Gamma_{g,n}$, for $2g-2+n>0$, be the Teichm\\\"uller group associated with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$, i.e. the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $K_{g,n}$ be the normal subgroup of $\\Gamma_{g,n}$ generated by Dehn twists along separating circles on $S_{g,n}$. As a first application of the above theory, a characterization of $K_{g,n}$ is given for all $n\\geq 0$ (for $n=0,1$, this was done by Johnson). Let then ${\\cal T}_{g,n}$ be the Torelli group, i.e. the kernel of the natural representation $\\Gamma_{g,n}\\ra Sp_{2g}(Z)$. The abelianization of ${\\cal T}_{g,n}$ is determined for all $g\\geq 1$ and $n\\geq 1$, thus completing classical results by Johnson and Mess.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-01-11T14:49:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H10",
      "30F60",
      "14F35",
      "14H15",
      "32G15"
    ],
    "keywords": [
      "fundamental groups",
      "moduli stack",
      "compact type",
      "stable curves",
      "compact riemann surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......4271B"
    }
  }
}