{
  "id": "math/0512261",
  "version": "v3",
  "published": "2005-12-13T11:04:39.000Z",
  "updated": "2008-05-27T13:23:13.000Z",
  "title": "New lower bounds on subgroup growth and homology growth",
  "authors": [
    "Marc Lackenby"
  ],
  "comment": "39 pages, 2 figures; v3 has minor changes from v2, incorporating referee's comments; v2 has minor changes from v1; to appear in the Proceedings of the London Mathematical Society",
  "doi": "10.1112/plms/pdn032",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subgroup growth of any finitely generated group. This is achieved by free non-abelian groups and is slightly faster than exponential. As a consequence, we obtain good estimates on the number of covering spaces of a hyperbolic 3-manifold with given covering degree. We also obtain slightly weaker information on the number of covering spaces of closed 4-manifolds with non-positive Euler characteristic. The results on subgroup growth follow from a new theorem which places lower bounds on the rank of the first homology (with mod p coefficients) of certain subgroups of a group. This is proved using a topological argument.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-05-27T13:23:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E07",
      "57M10",
      "57N10",
      "57N13"
    ],
    "keywords": [
      "subgroup growth",
      "homology growth",
      "strong lower bounds",
      "free non-abelian groups",
      "covering spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}