{
  "id": "math/0504105",
  "version": "v1",
  "published": "2005-04-06T12:34:29.000Z",
  "updated": "2005-04-06T12:34:29.000Z",
  "title": "The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms",
  "authors": [
    "Vadim Kaimanovich",
    "Ilya Kapovich",
    "Paul Schupp"
  ],
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "Given a free group $F_k$ of rank $k\\ge 2$ with a fixed set of free generators we associate to any homomorphism $\\phi$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $\\lambda(\\phi)$, which is a non-commutative generalization of the translation number. We concentrate on the situation when $\\phi:F_k\\to Aut(X)$ corresponds to a free action of $F_k$ on a simplicial tree $X$, in particular, when $\\phi$ corresponds to the action of $F_k$ on its Cayley graph via an automorphism of $F_k$. In this case we are able to obtain some detailed ``arithmetic'' information about the possible values of $\\lambda=\\lambda(\\phi)$. We show that $\\lambda \\ge 1$ and is a rational number with $2k\\lambda\\in \\mathbb Z[ \\frac{1}{2k-1} ]$ for every $\\phi\\in Aut(F_k)$. We also prove that the set of all $\\lambda(\\phi)$, where $\\phi$ varies over $Aut(F_k)$, has a gap between 1 and $1+\\frac{2k-3}{2k^2-k}$, and the value 1 is attained only for ``trivial'' reasons. Furthermore, there is an algorithm which, when given $\\phi$, calculates $\\lambda(\\phi)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-04-06T12:34:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65"
    ],
    "keywords": [
      "generic stretching factor",
      "free group automorphisms",
      "subadditive ergodic theorem",
      "left-invariant semi-norm",
      "translation number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4105K"
    }
  }
}