{
  "id": "1706.07347",
  "version": "v1",
  "published": "2017-06-22T14:50:57.000Z",
  "updated": "2017-06-22T14:50:57.000Z",
  "title": "Volume rigidity at ideal points of the character variety of hyperbolic 3-manifolds",
  "authors": [
    "Stefano Francaviglia",
    "Alessio Savini"
  ],
  "comment": "20 pages",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "Given the fundamental group $\\Gamma$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $\\rho:\\Gamma \\rightarrow \\text{Isom}(\\mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the volume of $\\rho$ is maximal, then $\\rho$ must be conjugated to the holonomy of the hyperbolic structure of $M$. This paper generalizes this rigidity result by showing that if a sequence of representations of $\\Gamma$ into $\\text{Isom}(\\mathbb{H}^3)$ satisfies $\\lim_{n \\to \\infty} \\text{Vol}(\\rho_n) = \\text{Vol}(M)$, then there must exist a sequence of elements $g_n \\in \\text{Isom}(\\mathbb{H}^3)$ such that the representations $g_n \\circ \\rho_n \\circ g_n^{-1}$ converge to the holonomy of $M$. In particular if the sequence $\\rho_n$ converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. We conclude by generalizing the result to the case of $k$-manifolds and representations in $\\text{Isom}(\\mathbb H^m)$, where $m\\geq k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-22T14:50:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M50",
      "53C24",
      "22E40"
    ],
    "keywords": [
      "ideal point",
      "character variety",
      "volume rigidity",
      "representation",
      "finite-volume complete hyperbolic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}