A theorem on discrete, torsion free subgroups of Isom $H^n$

Abstract

Let $H^n$ be the hyperbolic n-space with $n\ge 2$. Suppose that $\Gamma <Isom{H}^n$ is a discrete, torsion free subgroup and $a$ is a point in the domain of discontinuity $\Omega (\Gamma )$. Let $p$ be the projection map from $H^n$ to the quotient manifold $M=H^n/\Gamma$. In this paper we prove that there exists an open neighborhood $U$ of $a$ in $H^n\cup \Omega (\Gamma )$ such that $p$ is an isometry on $U\cap H^n$.

arXiv comment7 pages, 1 figures, To be appear in Geom. Dedicata