{
  "id": "0801.1589",
  "version": "v1",
  "published": "2008-01-10T12:49:23.000Z",
  "updated": "2008-01-10T12:49:23.000Z",
  "title": "Reduction theory for mapping class groups and applications to moduli spaces",
  "authors": [
    "Enrico Leuzinger"
  ],
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "Let $S=S_{g,p}$ be a compact, orientable surface of genus $g$ with $p$ punctures and such that $d(S):=3g-3+p>0$. The mapping class group $\\textup{Mod}_S$ acts properly discontinuously on the Teichm\\\"uller space $\\mathcal T(S)$ of marked hyperbolic structures on $S$. The resulting quotient $\\mathcal M(S)$ is the moduli space of isometry classes of hyperbolic surfaces. We provide a version of precise reduction theory for finite index subgroups of $\\textup{Mod}_S$, i.e., a description of exact fundamental domains. As an application we show that the asymptotic cone of the moduli space $\\mathcal M(S)$ endowed with the Teichm\\\"uller metric is bi-Lipschitz equivalent to the Euclidean cone over the finite simplicial (orbi-) complex $ \\textup{Mod}_S\\backslash\\mathcal C(S)$, where $\\mathcal C(S)$ of $S$ is the complex of curves of $S$. We also show that if $d(S)\\geq 2$, then $\\mathcal M(S)$ does \\emph{not} admit a finite volume Riemannian metric of (uniformly bounded) positive scalar curvature in the bi-Lipschitz class of the Teichm\\\"uller metric. These two applications confirm conjectures of Farb.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-01-10T12:49:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32G15",
      "30F60",
      "20H10",
      "20F67",
      "51K10"
    ],
    "keywords": [
      "mapping class group",
      "moduli space",
      "finite volume riemannian metric",
      "applications confirm conjectures",
      "precise reduction theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0801.1589L"
    }
  }
}