{
  "id": "1601.06952",
  "version": "v1",
  "published": "2016-01-26T10:03:54.000Z",
  "updated": "2016-01-26T10:03:54.000Z",
  "title": "Convex real projective orbifolds with radial or totally geodesic ends: a survey of some partial results",
  "authors": [
    "Suhyoung Choi"
  ],
  "comment": "44 pages, 1 figure, This paper surveys the results in \"The convex real projective orbifolds with radial or totally geodesic ends: The closedness and openness of deformations\" (arXiv:1011.1060)",
  "categories": [
    "math.GT"
  ],
  "abstract": "A real projective orbifold is an $n$-dimensional orbifold modeled on ${\\mathbf R}P^n$ with the group $PGL(n+1, {\\mathbf R})$. We concentrate on an orbifold that contains a compact codimension $0$ submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed $(n-1)$-dimensional orbifolds times intervals. A real projective orbifold has a radial end if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a totally geodesic end if the end can be completed to have the totally geodesic boundary. The orbifold is said to be convex if any path can be homotoped to a projective geodesic with endpoints fixed. The purpose of this paper is to announce some partial results. A real projective structure sometimes admits deformations to parameters of real projective structures. We will prove a homeomorphism between the deformation space of convex real projective structures on an orbifold $\\mathcal{O}$ with radial or totally geodesic ends with various conditions with the union of open subspaces of strata of the subset \\[ Hom_{\\mathcal E}(\\pi_{1}(\\mathcal{O}),PGL(n+1, {\\mathbf R}))/PGL(n+1, {\\mathbf R}) \\] of the character variety \\[ Hom(\\pi_{1}(\\mathcal{O}),PGL(n+1, {\\mathbf R}))/PGL(n+1, {\\mathbf R}) \\] given by corresponding end conditions for holonomy representations. Lastly, we will prove the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends, where we need the theory of Crampon-Marquis and Cooper, Long and Tillmann on the Margulis lemma for convex real projective manifolds. The theory here partly generalizes that of Benoist on closed convex real projective orbifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-26T10:03:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M50",
      "53A20",
      "20C99"
    ],
    "keywords": [
      "convex real projective orbifolds",
      "totally geodesic end",
      "partial results",
      "convex real projective structures",
      "dimensional orbifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160106952C"
    }
  }
}