{
  "id": "1405.6310",
  "version": "v3",
  "published": "2014-05-24T15:23:01.000Z",
  "updated": "2014-09-21T12:43:33.000Z",
  "title": "Hölder conditions for endomorphisms of hyperbolic groups",
  "authors": [
    "Vítor Araújo",
    "Pedro V. Silva"
  ],
  "comment": "27 pages",
  "categories": [
    "math.GR",
    "math.MG"
  ],
  "abstract": "It is proved that an endomorphism $\\varphi$ of a hyperbolic group $G$ satisfies a H\\\"older condition with respect to a visual metric if and only if $\\varphi$ is virtually injective and $G\\varphi$ is a quasiconvex subgroup of $G$. If $G$ is virtually free or torsion-free co-hopfian, then $\\varphi$ is uniformly continuous if and only if it it satisfies a H\\\"older condition if and only if it is virtually injective. Lipschitz conditions are discussed for free group automorphisms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-31T22:31:00.000Z",
      "title": "Complexity of endomorphisms of hyperbolic groups",
      "abstract": "It is proved that an endomorphism $\\varphi$ of a hyperbolic group $G$ is uniformly continuous with polynomial complexity with respect to a visual metric if and only if $\\varphi$ is virtually injective and $G\\varphi$ is a quasi-convex subgroup of $G$. If $G$ is virtually free, $\\varphi$ is uniformly continuous (with polynomial complexity) if and only if it is virtually injective. Linear uniform continuity is characterized for free group automorphisms.",
      "comment": "24 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-21T12:43:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F67",
      "20E36",
      "51M10"
    ],
    "keywords": [
      "hyperbolic group",
      "endomorphism",
      "polynomial complexity",
      "linear uniform continuity",
      "free group automorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.6310A"
    }
  }
}