A Theorem on Discrete,Torsion Free Subgroups of Isom H n

Let H ⁿ be the hyperbolic n-space with n⩾ 2. Suppose that Γ H ⁿ is a discrete, torsion free subgroup and a is a point in the domain of discontinuity Ω (Γ). Let p be the projection map from H ⁿ to the quotient manifold M=H ⁿ/Γ. In this paper we prove that there exists an open neighborhood U of a in H ⁿ∪ Ω (Γ) such that p is an isometry on U∩ H ⁿ.Mathematics Subject Classification (2000). primary: 51M10; secondary: 22E40, 57M50.