{
  "id": "0711.2704",
  "version": "v4",
  "published": "2007-11-16T23:49:17.000Z",
  "updated": "2011-05-10T19:04:00.000Z",
  "title": "The fundamental group of random 2-complexes",
  "authors": [
    "Eric Babson",
    "Christopher Hoffman",
    "Matthew Kahle"
  ],
  "comment": "28 pages, 3 figures; major revisions",
  "journal": "J. Amer. Math. Soc. 24 (2011), 1-28",
  "categories": [
    "math.CO",
    "math.GR",
    "math.GT",
    "math.PR"
  ],
  "abstract": "We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\\H{o}s-R\\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(n)/n. We use a variant of Gromov's local-to-global theorem for linear isoperimetric inequalities to show that when p = O(n^{-1/2 -\\epsilon}) the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2-dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-05-10T19:04:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "05C80"
    ],
    "keywords": [
      "fundamental group",
      "study linial-meshulam random",
      "first homology group",
      "linear isoperimetric inequalities",
      "gromovs local-to-global theorem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "J. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.2704B"
    }
  }
}