{
  "id": "1509.04178",
  "version": "v1",
  "published": "2015-09-14T16:10:50.000Z",
  "updated": "2015-09-14T16:10:50.000Z",
  "title": "The Volume of complete anti-de Sitter 3-manifolds",
  "authors": [
    "Nicolas Tholozan"
  ],
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "Up to a finite cover, closed anti-de Sitter $3$-manifolds are quotients of $\\mathrm{SO}_0(2,1)$ by a discrete subgroup of $\\mathrm{SO}_0(2,1) \\times \\mathrm{SO}_0(2,1)$ of the form \\[j\\times \\rho(\\Gamma)~,\\] where $\\Gamma$ is the fundamental group of a closed oriented surface, $j$ a Fuchsian representation and $\\rho$ another representation which is \"strictly dominated\" by $j$. Here we prove that the volume of such a quotient is proportional to the sum of the Euler classes of $j$ and $\\rho$. As a consequence, we obtain that this volume is constant under deformation of the anti-de Sitter structure. Our results extend to (not necessarily compact) quotients of $\\mathrm{SO}_0(n,1)$ by a discrete subgroup of $\\mathrm{SO}_0(n,1) \\times \\mathrm{SO}_0(n,1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-14T16:10:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complete anti-de sitter",
      "discrete subgroup",
      "anti-de sitter structure",
      "finite cover",
      "fundamental group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150904178T"
    }
  }
}