{
  "id": "math/0506396",
  "version": "v4",
  "published": "2005-06-20T18:00:53.000Z",
  "updated": "2008-02-03T19:18:55.000Z",
  "title": "Singular surfaces, mod 2 homology, and hyperbolic volume, I",
  "authors": [
    "Ian Agol",
    "Marc Culler",
    "Peter B. Shalen"
  ],
  "comment": "Ian Agol has been added as a co-author. The main topological result has been considerably strengthened. It now applies to singular surfaces of any genus, and in the genus 2 case considered in the earlier version the homology bound has been lowered from 11 to 7. In addition, the arguments have been substantially simplified",
  "categories": [
    "math.GT"
  ],
  "abstract": "This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that \\pi_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at least 4g-1 then M contains a closed incompressible surface of genus at most g. This result should be viewed as an analogue of Dehn's Lemma for \\pi_1-injective singular surfaces. The geometric application states that if M is a closed orientable hyperbolic 3-manifold with volume less than 3.08 then the rank of H_1(M;Z/2Z) is at most 6. The proof of the geometric theorem combines the topological theorem with several deep geometric results, including the Marden tameness conjecture,recently established by Agol and by Calegari-Gabai; a co-volume estimate for 3-tame, 3-free Kleinian groups due to Anderson, Canary, Culler and Shalen; and a volume estimate for hyperbolic Haken manifolds recently proved by Agol, Storm and W. Thurston.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2008-02-03T19:18:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M50"
    ],
    "keywords": [
      "singular surfaces",
      "hyperbolic volume",
      "hyperbolic haken manifolds",
      "marden tameness conjecture",
      "deep geometric results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6396C"
    }
  }
}