{
  "id": "math/0210316",
  "version": "v4",
  "published": "2002-10-21T14:02:00.000Z",
  "updated": "2003-11-26T09:53:27.000Z",
  "title": "The asymptotic behaviour of Heegaard genus",
  "authors": [
    "Marc Lackenby"
  ],
  "comment": "14 pages. Final version, including a new expository section. To appear in Mathematical Research Letters",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let M be a closed orientable 3-manifold with a negatively curved Riemannian metric. Let {M_i} be a collection of finite regular covers with degree d_i. (1) If the Heegaard genus of M_i grows more slowly than the square root of d_i, then M_i has positive first Betti number for all sufficiently large i. (2) The strong Heegaard genus of M_i cannot grow more slowly than the square root of d_i. (3) If the Heegaard genus of M_i grows more slowly than the fourth root of d_i, then M_i fibres over the circle for all sufficiently large i. These results provide supporting evidence for the Heegaard gradient conjecture and the strong Heegaard gradient conjecture. As a corollary to (3), we give a necessary and sufficient condition for M to be virtually fibred in terms of the Heegaard genus of its finite covers.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2003-11-26T09:53:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10",
      "57M10"
    ],
    "keywords": [
      "asymptotic behaviour",
      "strong heegaard gradient conjecture",
      "square root",
      "strong heegaard genus",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....10316L"
    }
  }
}