{
  "id": "math/0109045",
  "version": "v1",
  "published": "2001-09-06T19:44:52.000Z",
  "updated": "2001-09-06T19:44:52.000Z",
  "title": "Curvature and rank of Teichmüller space",
  "authors": [
    "Jeffrey Brock",
    "Benson Farb"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.GT",
    "math.DG",
    "math.GR"
  ],
  "abstract": "Let S be a surface with genus g and n boundary components and let d(S) = 3g-3+n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmuller space Teich(S) is Gromov-hyperbolic if and only if d(S) <= 2. When d(S) >= 3 the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of R^k, k >= 2); when d(S) <= 2 we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of CP(S) prove Gromov-hyperbolicity. We prove moreover that Teich(S) admits no geodesically complete Gromov-hyperbolic metric of finite covolume when d(S) >= 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) >= 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-09-06T19:44:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "teichmüller space",
      "weil-petersson metric",
      "employ metric properties",
      "teichmuller space teich",
      "complete riemannian metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......9045B"
    }
  }
}