{
  "id": "math/0502585",
  "version": "v1",
  "published": "2005-02-28T17:45:44.000Z",
  "updated": "2005-02-28T17:45:44.000Z",
  "title": "Non-injective representations of a closed surface group into $PSL(2,\\mathbb R)$",
  "authors": [
    "Louis Funar",
    "Maxime Wolff"
  ],
  "comment": "15 pages, 2 figures",
  "journal": "Math.Proc. Cambridge Phil.Soc. 142(2007), 289-304",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "Let $e$ denote the Euler class on the space $Hom(\\Gamma_g, PSL(2,\\mathbb R))$ of representations of the fundamental group $\\Gamma_g$ of the closed surface $\\Sigma_g$ of genus $g$. Goldman showed that the connected components of $Hom(\\Gamma_g, PSL(2,\\mathbb R))$ are precisely the inverse images $e^{-1}(k)$, for $2-2g\\leq k\\leq 2g-2$, and that the components of Euler class $2-2g$ and $2g-2$ consist of the injective representations whose image is a discrete subgroup of $PSL(2,\\mathbb R)$. We prove that non-faithful representations are dense in all the other components. We show that the image of a discrete representation essentially determines its Euler class. Moreover, we show that for every genus and possible corresponding Euler class, there exist discrete representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-02-28T17:45:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M05",
      "22E40"
    ],
    "keywords": [
      "closed surface group",
      "non-injective representations",
      "discrete representation essentially determines",
      "components",
      "discrete subgroup"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1017/S0305004106009601",
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2007,
      "month": "Apr",
      "volume": 142,
      "number": 2,
      "pages": 289
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007MPCPS.142..289F"
    }
  }
}