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

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