{
  "id": "1405.1000",
  "version": "v2",
  "published": "2014-05-05T19:14:26.000Z",
  "updated": "2017-10-10T15:06:21.000Z",
  "title": "Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms",
  "authors": [
    "Emmanuel Militon"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let S be a compact connected surface and let f be an element of the group Homeo\\_0(S) of homeomorphisms of S isotopic to the identity. Denote by \\tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal cover. The homeomorphism f is said to be non-spreading if the sequence (d\\_{n}/n) converges to 0, where d\\_{n} is the diameter of \\tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a nonempty boundary. We prove that, if S is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in Homeo\\_{0}(S) arbitrarily close to the identity. In the case where the surface S is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in Homeo\\_{0}(S) arbitrarily close to a rotation (this was already known in most cases by a theorem by B{\\'e}guin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces S, an element of Homeo\\_{0}(S) is distorted if and only if it is non-spreading.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-05T19:14:26.000Z",
      "abstract": "Let S be a compact connected surface and let f be an element of the group Homeo_0(S) of homeomorphisms of S isotopic to the identity. Denote by \\tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal cover. The homeomorphism f is said to be non-spreading if the sequence (d_{n}/n) converges to 0, where d_{n} is the diameter of \\tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a nonempty boundary. We prove that, if S is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in Homeo_{0}(S) arbitrarily close to the identity. In the case where the surface S is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in Homeo_{0}(S) arbitrarily close to a rotation (this was already known in most cases by a theorem by B\\'eguin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces S, an element of Homeo_{0}(S) is distorted if and only if it is non-spreading.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-10-10T15:06:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homeomorphism",
      "distortion elements",
      "conjugacy class",
      "universal cover",
      "arbitrarily close"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1000M"
    }
  }
}