{
  "id": "math/0511621",
  "version": "v1",
  "published": "2005-11-25T05:29:09.000Z",
  "updated": "2005-11-25T05:29:09.000Z",
  "title": "End Invariants for $\\SL(2,C)$ characters of the one-holed torus",
  "authors": [
    "Ser Peow Tan",
    "Yan Loi Wong",
    "Ying Zhang"
  ],
  "comment": "24 pages, 6 figures",
  "journal": "American Journal of Mathemaics 130 (2008), 385-412",
  "categories": [
    "math.GT",
    "math.DG",
    "math.DS"
  ],
  "abstract": "We define and study the set ${\\mathcal E}(\\rho)$ of end invariants of a $\\SL(2,C)$ character $\\rho$ of the one-holed torus $T$. We show that the set ${\\mathcal E}(\\rho)$ is the entire projective lamination space $\\mathscr{PL}$ of $T$ if and only if (i) $\\rho$ corresponds to the dihedral representation, or (ii) $\\rho$ is real and corresponds to a SU(2) representation; and that otherwise, ${\\mathcal E}(\\rho)$ is closed and has empty interior in $\\mathscr{PL}$. For real characters $\\rho$, we give a complete classification of ${\\mathcal E}(\\rho)$, and show that ${\\mathcal E}(\\rho)$ has either 0, 1 or infinitely many elements, and in the last case, ${\\mathcal E}(\\rho)$ is either a Cantor subset of $\\mathscr{PL}$ or is $\\mathscr{PL}$ itself. We also give a similar classification for \"imaginary\" characters where the trace of the commutator is less than 2. Finally, we show that for discrete characters (not corresponding to dihedral or SU(2) representations), ${\\mathcal E}(\\rho)$ is a Cantor subset of $\\mathscr{PL}$ if it contains at least three elements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-25T05:29:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M05",
      "30F60",
      "20H10",
      "37F30"
    ],
    "keywords": [
      "end invariants",
      "one-holed torus",
      "cantor subset",
      "entire projective lamination space",
      "discrete characters"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11621P"
    }
  }
}