{
  "id": "1007.2598",
  "version": "v3",
  "published": "2010-07-15T15:56:17.000Z",
  "updated": "2011-07-28T21:01:33.000Z",
  "title": "Abelian covers of graphs and maps between outer automorphism groups of free groups",
  "authors": [
    "Martin R. Bridson",
    "Karen Vogtmann"
  ],
  "comment": "Final version, to appear in Mathematische Annalen. Minor errors and typos corrected, including range of n in Theorem B",
  "categories": [
    "math.GR"
  ],
  "abstract": "We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \\to Out(F_m). We prove that if n > 8 is even and n \\neq m \\leq 2n, or n is odd and n \\neq m \\leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \\to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \\to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-07-28T21:01:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F28",
      "53C24",
      "57S25"
    ],
    "keywords": [
      "outer automorphism groups",
      "free groups",
      "abelian covers",
      "abelian galois group",
      "homomorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.2598B"
    }
  }
}