{
  "id": "1202.3536",
  "version": "v2",
  "published": "2012-02-16T09:06:10.000Z",
  "updated": "2012-03-14T17:05:57.000Z",
  "title": "Distortion for diffeomorphisms of surfaces with boundary",
  "authors": [
    "Kiran Parkhe"
  ],
  "comment": "13 pages. arXiv admin note: text overlap with arXiv:math/0404532",
  "categories": [
    "math.DS"
  ],
  "abstract": "If $G$ is a finitely generated group with generators $\\{g_1,..., g_s\\}$, we say an infinite-order element $f \\in G$ is a distortion element of $G$ provided that $\\displaystyle \\liminf_{n \\to \\infty} \\frac{|f^n|}{n} = 0$, where $|f^n|$ is the word length of $f^n$ with respect to the given generators. Let $S$ be a compact orientable surface, possibly with boundary, and let $\\Diff(S)_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result is that if $S$ has genus at least two, and $f$ is a distortion element in some finitely generated subgroup of $\\Diff(S)_0$, then $\\supp(\\mu) \\subseteq \\Fix(f)$ for every $f$-invariant Borel probability measure $\\mu$. Under a small additional hypothesis the same holds in lower genus. For $\\mu$ a Borel probability measure on $S$, denote the group of $C^1$ diffeomorphisms that preserve $\\mu$ by $\\Diff_\\mu(S)$. Our main result implies that a large class of higher-rank lattices admit no homomorphisms to $\\Diff_{\\mu}(S)$ with infinite image. These results generalize those of Franks and Handel to surfaces with boundary.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-03-14T17:05:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C85",
      "57M60",
      "22F10"
    ],
    "keywords": [
      "diffeomorphisms",
      "invariant borel probability measure",
      "distortion element",
      "small additional hypothesis",
      "main result implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.3536P"
    }
  }
}