{
  "id": "0903.2419",
  "version": "v1",
  "published": "2009-03-13T16:10:58.000Z",
  "updated": "2009-03-13T16:10:58.000Z",
  "title": "Flows, Fixed Points and Rigidity for Kleinian Groups",
  "authors": [
    "Kingshook Biswas"
  ],
  "categories": [
    "math.GT",
    "math.DS"
  ],
  "abstract": "We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group $G_1$ and a quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show that if the conjugacy $h$ is not conformal then this group contains a non-trivial one parameter subgroup. This leads to rigidity results; for example, Mostow rigidity is an immediate consequence. We are also able to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group $G$, by a $G$-invariant pattern we mean a $G$-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of $G$. We prove that (in dimension at least three) for $G_1, G_2$ cocompact Kleinian groups, any quasiconformal map pairing a $G_1$-invariant pattern to a $G_2$-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas-Mj who proved rigidity for Poincare Duality subgroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-13T16:10:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M50"
    ],
    "keywords": [
      "fixed points",
      "cocompact kleinian group",
      "invariant pattern",
      "compact subsets minus singletons",
      "cocompact group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.2419B"
    }
  }
}