{
  "id": "math/0209191",
  "version": "v1",
  "published": "2002-09-16T14:22:58.000Z",
  "updated": "2002-09-16T14:22:58.000Z",
  "title": "Homomorphisms from automorphism groups of free groups",
  "authors": [
    "Martin R Bridson",
    "Karen Vogtmann"
  ],
  "comment": "10 Pages, to appear in J. London Math. Soc",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m$ is less than $n$ then a homomorphism $Aut(F_n)\\to Aut(F_m)$ can have cardinality at most 2. More generally, this is true of homomorphisms from $\\Aut(F_n)$ to any group that does not contain an isomorphic copy of the symmetric group $S_{n+1}$. Strong constraints are also obtained on maps to groups that do not contain a copy of $W_n= (\\Bbb Z/2)^n\\rtimes S_n$, or of $\\Bbb Z^{n-1}$. These results place constraints on how $\\Aut(F_n)$ can act. For example, if $n\\ge 3$ then any action of $\\Aut(F_n)$ on the circle (by homeomorphisms) factors through $\\text{\\rm{det}}:Aut(F_n) \\to \\Bbb Z_2$ .",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-09-16T14:22:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F28"
    ],
    "keywords": [
      "automorphism group",
      "homomorphism",
      "results place constraints",
      "single element",
      "finitely generated free group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......9191B"
    }
  }
}