{
  "id": "1405.1979",
  "version": "v2",
  "published": "2014-05-08T15:44:21.000Z",
  "updated": "2015-02-06T11:17:17.000Z",
  "title": "On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups",
  "authors": [
    "Mattia Mecchia",
    "Bruno P. Zimmermann"
  ],
  "comment": "13 pages; this is the final version to appear in Fund. Math",
  "categories": [
    "math.GT"
  ],
  "abstract": "It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group, is bounded by a quadratic polynomial in g (but not by a linear one). In the present paper we prove a generalization for handlebodies of arbitrary dimension d, uniformizing handlebodies by Schottky groups and considering finite groups of isometries of such handlebodies. We prove that the order of a finite group of isometries of a handlebody of dimension d acting faithfully on the fundamental group is bounded by a polynomial of degree d/2 in g if d is even, and of degree (d+1)/2 if d is odd, and that the degree d/2 for even d is best possible. This implies then analogous polynomial Jordan-type bounds for arbitrary finite groups of isometries of handlebodies (since a handlebody of dimension d > 3 admits S^1-actions, there does not exist an upper bound for the order of the group itself ).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-08T15:44:21.000Z",
      "title": "On finite groups of isometries of handlebodies of dimension d \\ge 3 and finite extensions of Schottky groups",
      "comment": "12 pages",
      "journal": null,
      "doi": null,
      "authors": [
        "Bruno P. Zimmermann"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-02-06T11:17:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S17",
      "57S25",
      "57M60",
      "20H10"
    ],
    "keywords": [
      "handlebody",
      "schottky groups",
      "finite extensions",
      "isometries",
      "fundamental group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1979M"
    }
  }
}