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A theorem on torsion free Kleinian group

Let H be the hyperbolic 3-space and IsomH be the group of isometries of H. Let G be a discrete subgroup of IsomH. The action of G on H extends to a continuous action on the compactification of H by the sphere at infinity S ∞. The limit set Λ(G) is the set of all accumulation points of the orbit of a point in H. The limit set does not depend on the choice of the point in H and the limit set is a subset of S ∞ because G is discrete. The domain of discontinuity Ω(G) is the complement of Λ(G) in S ∞. A Kleinian group is a discrete subgroup of IsomH 3 with nonempty domain of discontinuity. An element of IsomH is called elliptic if it has a fixed point in H. An element of IsomH is called parabolic if it has no fixed point in H and it has unique fixed point on S ∞. An element of IsomH 3 is called hyperbolic if it has no fixed point in H and it has exactly two fixed point on S ∞. Any element of IsomH 3 is an elliptic, parabolic or hyperbolic isometry and torsion free Kleinian group does not contain any elliptic