{
  "id": "0907.3317",
  "version": "v4",
  "published": "2009-07-19T19:32:02.000Z",
  "updated": "2009-08-11T14:15:26.000Z",
  "title": "On the arc and curve complex of a surface",
  "authors": [
    "Mustafa Korkmaz",
    "Athanase Papadopoulos"
  ],
  "comment": "Added references, added some results about special surfaces and corrected some misprints",
  "categories": [
    "math.GT"
  ],
  "abstract": "We study the {\\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of $AC(S)$ coincides with the natural image of the extended mapping class group of $S$ in that group. We also show that for any vertex of $AC(S)$, the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in $S$ that represents that vertex. We also give a proof of the fact if $S$ is not a sphere with at most three punctures, then the natural embedding of the curve complex of $S$ in $AC(S)$ is a quasi-isometry. The last result, at least under some slightly more restrictive conditions on $S$, was already known. As a corollary, $AC(S)$ is Gromov-hyperbolic.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-08-11T14:15:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32G15",
      "20F38"
    ],
    "keywords": [
      "curve complex",
      "simplicial automorphism group",
      "finite type",
      "natural image",
      "vertex characterizes"
    ],
    "publication": {
      "doi": "10.1017/S0305004109990387",
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2009,
      "month": "Dec",
      "volume": 148,
      "number": 3,
      "pages": 473
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009MPCPS.148..473K"
    }
  }
}