{
  "id": "math/9712250",
  "version": "v1",
  "published": "1997-12-13T20:53:19.000Z",
  "updated": "1997-12-13T20:53:19.000Z",
  "title": "Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds",
  "authors": [
    "Sungbok Hong",
    "Darryl McCullough"
  ],
  "comment": "33 pages",
  "journal": "Pacific J. Math. 188 (1999), 275-301",
  "categories": [
    "math.GT"
  ],
  "abstract": "Mapping class groups of Haken 3-manifolds enjoy many of the homological finiteness properties of mapping class groups of 2-manifolds of finite type. For example, H(M) has a torsionfree subgroup of finite index, which is geometrically finite (i. e. is the fundamental group of a finite aspherical complex). This was proven by J. Harer for 2-manifolds and by the second author for Haken 3-manifolds. In this paper we prove that H(M) acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in H(M) is geometrically finite. Also, a simplified proof of the fact that torsionfree subgroups of finite index in H(M) exist is given. All results are proven for mapping class groups that preserve a boundary pattern in the sense of K. Johannson. As an application, we show that if F is a nonempty compact 2-manifold in the boundary of M, then the classifying space BDiff(M rel F) of the diffeomorphism group of M relative to F has the homotopy type of a finite aspherical complex.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1997-12-13T20:53:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M99"
    ],
    "keywords": [
      "mapping class groups",
      "geometric finiteness",
      "torsionfree subgroup",
      "finite index",
      "finite aspherical complex"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1997math.....12250H"
    }
  }
}