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

Abstract

Let $H^n$ be the hyperbolic n-space with $n\geq 2$. 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 $U\cap H^n$.