{
  "id": "1306.3350",
  "version": "v3",
  "published": "2013-06-14T09:48:04.000Z",
  "updated": "2014-05-30T17:33:11.000Z",
  "title": "Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces",
  "authors": [
    "Michael Brandenbursky"
  ],
  "comment": "30 pages, 5 figures",
  "categories": [
    "math.GT",
    "math.GR",
    "math.SG"
  ],
  "abstract": "Let \\Sigma_g be a closed orientable surface let Diff_0(\\Sigma_g; area) be the identity component of the group of area-preserving diffeomorphisms of \\Sigma_g. In this work we present an extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface \\Sigma_g, i.e. we show that every non-trivial homogeneous quasi-morphism on the braid group on n strings of \\Sigma_g defines a non-trivial homogeneous quasi-morphism on the group Diff_0(\\Sigma_g; area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff_0(\\Sigma_g; area) is infinite dimensional. Let Ham(\\Sigma_g) be the group of Hamiltonian diffeomorphisms of \\Sigma_g. As an application of the above construction we construct two injective homomorphisms from Z^m to Ham(\\Sigma_g), which are bi-Lipschitz with respect to the word metric on Z^m and the autonomous and fragmentation metrics on Ham(\\Sigma_g). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(\\Sigma_g).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-30T17:33:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamiltonian diffeomorphisms",
      "bi-invariant metrics",
      "non-trivial homogeneous quasi-morphism",
      "fragmentation metrics",
      "infinite dimensional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.3350B"
    }
  }
}