{
  "id": "math/0509036",
  "version": "v3",
  "published": "2005-09-02T09:40:44.000Z",
  "updated": "2005-12-01T10:31:15.000Z",
  "title": "Large groups, Property (tau) and the homology growth of subgroups",
  "authors": [
    "Marc Lackenby"
  ],
  "comment": "35 pages, 2 figures; minor corrections and improvements from version 2",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homology if the infimum of d_p(G_i)/[G:G_i] is positive. We show that if this holds and each G_i is normal in its predecessor and has index a power of p, then one of the following possibilities must be true: G is large (that is, some finite index subgroup admits a surjective homomorphism onto a non-abelian free group) or G has Property (tau) with respect to {G_i}. The arguments are based on the geometry and topology of finite 2-complexes. This has several consequences. It implies that if the pro-p completion of a finitely presented group G has exponential subgroup growth, then G has Property (tau) with respect to some nested sequence of finite index subgroups. It also has applications to low-dimensional topology. We use it to prove that a group-theoretic conjecture of Lubotzky-Zelmanov would imply the following: any lattice in PSL(2,C) with torsion is large. We also relate linear growth of mod p homology to the existence of certain important error-correcting codes: those that are `asymptotically good', which means that they have large rate and large Hamming distance.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-12-01T10:31:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E07",
      "20F65",
      "94B05",
      "57N10"
    ],
    "keywords": [
      "homology growth",
      "large groups",
      "finite index subgroup admits",
      "exponential subgroup growth",
      "relate linear growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9036L"
    }
  }
}