{
  "id": "math/0010046",
  "version": "v1",
  "published": "2000-10-04T19:05:17.000Z",
  "updated": "2000-10-04T19:05:17.000Z",
  "title": "Hall invariants, homology of subgroups, and characteristic varieties",
  "authors": [
    "Daniel Matei",
    "Alexander I. Suciu"
  ],
  "comment": "34 pages, 3 figures",
  "journal": "International Math. Research Notices 2002:9 (2002), 465-503",
  "doi": "10.1155/S107379280210907X",
  "categories": [
    "math.GR",
    "math.CO",
    "math.GT"
  ],
  "abstract": "Given a finitely-generated group G, and a finite group \\Gamma, Philip Hall defined \\delta_\\Gamma to be the number of factor groups of G that are isomorphic to \\Gamma. We show how to compute the Hall invariants by cohomological and combinatorial methods, when G is finitely-presented, and \\Gamma belongs to a certain class of metabelian groups. Key to this approach is the stratification of the character variety by the jumping loci of the cohomology of G, with coefficients in rank 1 local systems over a suitably chosen field \\K. Counting relevant torsion points on these \"characteristic\" subvarieties gives \\delta_\\Gamma(G). In the process, we compute the distribution of prime-index, normal subgroups K of G according to the dimension of the the first homology group of K with \\K coefficients, provided \\char\\K does not divide the index of K in G. In turn, we use this distribution to count low-index subgroups of G. We illustrate these techniques in the case when G is the fundamental group of the complement of an arrangement of either affine lines in \\C^2, or transverse planes in \\R^4.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-10-04T19:05:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J05",
      "57M05",
      "20E07",
      "52C35"
    ],
    "keywords": [
      "hall invariants",
      "characteristic varieties",
      "counting relevant torsion points",
      "first homology group",
      "count low-index subgroups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....10046M"
    }
  }
}