{
  "id": "math/0510129",
  "version": "v4",
  "published": "2005-10-06T19:11:45.000Z",
  "updated": "2009-03-03T12:59:52.000Z",
  "title": "A random tunnel number one 3-manifold does not fiber over the circle",
  "authors": [
    "Nathan M Dunfield",
    "Dylan P Thurston"
  ],
  "comment": "This is the version published by Geometry & Topology on 15 December 2006",
  "journal": "Geom. Topol. 10 (2006) 2431-2499",
  "doi": "10.2140/gt.2006.10.2431",
  "categories": [
    "math.GT"
  ],
  "abstract": "We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a \"magic splitting sequence\" and work of Mirzakhani proves the main theorem. The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups \"fiber\" with probability strictly between 0 and 1.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-03-03T12:59:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R22",
      "20F05",
      "57N10"
    ],
    "keywords": [
      "random tunnel number",
      "main theorem",
      "fairly inscrutable virtual fibration conjecture",
      "train tracks/interval exchanges",
      "implement browns algorithm"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10129D"
    }
  }
}