Let ‫ވ‬ n be the n-dimensional hyperbolic space with n ≥ 2. Suppose that G is a discrete, sense-preserving subgroup of Isom‫ވ‬ n , the isometry group of ‫ވ‬ n . Let p be the projection map from ‫ވ‬ n to the quotient space M = ‫ވ‬ n /G. The first goal of this paper is to prove that for any a ∈ ‫ވ∂‬ n (the sphere at infinity of ‫ވ‬ n ), there exists an open neighbourhood U of a in ‫ވ‬ n