{
  "id": "0903.2306",
  "version": "v2",
  "published": "2009-03-13T02:35:08.000Z",
  "updated": "2010-02-24T18:21:15.000Z",
  "title": "On endomorphisms of torsion-free hyperbolic groups",
  "authors": [
    "O. Bogopolski",
    "E. Ventura"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $H$ be a torsion-free $\\delta$-hyperbolic group with respect to a finite generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of $H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there is a uniform conjugator if and only if $W(a_{1*},..., a_{n*})$ is conjugate to $W(a_1,..., a_n)$ for every word $W$ in $n$ variables and length up to a computable constant depending only on $\\delta$, $\\sharp{S}$ and $\\sum_{i=1}^n |a_i|$. As a corollary, we deduce that there exists a computable constant $\\mathcal{C}=\\mathcal{C}(\\delta, \\sharp S)$ such that, for any endomorphism $\\phi$ of $H$, if $\\phi(h)$ is conjugate to $h$ for every element $h\\in H$ of length up to $\\mathcal {C}$, then $\\phi$ is an inner automorphism. Another corollary is the following: if $H$ is a torsion-free conjugacy separable hyperbolic group, then $\\text{\\rm Out}(H)$ is residually finite. When particularizing the main result to the case of free groups, we obtain a solution for a mixed version of the classical Whitehead's algorithm.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-02-24T18:21:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F67"
    ],
    "keywords": [
      "torsion-free hyperbolic groups",
      "endomorphism",
      "torsion-free conjugacy separable hyperbolic group",
      "computable constant",
      "whiteheads algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.2306B"
    }
  }
}