{
  "id": "math/0306071",
  "version": "v2",
  "published": "2003-06-03T21:54:50.000Z",
  "updated": "2004-12-25T11:18:39.000Z",
  "title": "Distances of Heegaard splittings",
  "authors": [
    "Aaron Abrams",
    "Saul Schleimer"
  ],
  "comment": "Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper2.abs.html",
  "journal": "Geom. Topol. 9(2005) 95-119",
  "categories": [
    "math.GT"
  ],
  "abstract": "J Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S,V, h^n(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V in PML(S). Here h is a pseudo-Anosov homeomorphism of a surface S while V is the set of isotopy classes of simple closed curves in S bounding essential disks in a fixed handlebody. With the same hypothesis we show the distance of the splitting (S,V, h^n(V)) grows linearly with n, answering a question of A Casson. In addition we prove the converse of Hempel's theorem. Our method is to study the action of h on the curve complex associated to S. We rely heavily on the result, due to H Masur and Y Minsky [Invent. Math. 1999], that the curve complex is Gromov hyperbolic.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-12-25T11:18:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M99",
      "51F99"
    ],
    "keywords": [
      "heegaard splittings",
      "curve complex",
      "pseudo-anosov homeomorphism",
      "simple closed curves",
      "hempels theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......6071A"
    }
  }
}