Discrete Groups In Complex Hyperbolic Space And The Problems On The Volumes Of The Manifold

M(?)bius group has been quite important in the complex analysis for over a hundred years, and it is always a main branch and studied by a lot of famous mathematicians, who also applied m(?)bius groups to the complex hyperbolic manifold. The criterion of the discrete condition for the mobius group is one of the main subjects in the m(?)bius group study, and also, it has great influence on the manifold and the algebra property of the discrete groups. Recently, the researchof the discreteness of the mobius groups in the complex hyperbolic space attracts many mathematicians at home and abroad. One important result is given by Shigeyasu Kamiya and John R.Parker in PU(2,1) whose subgroup is generated by a screw parabolic motion and another mobius transformation. Based on the research by Shigeyasu Kamiya and John R.Parker, this paper extends the results in PU(2,1) to that in PU(n,1) and gives further research on the criterion of the discrete condition for groups of complex hyperbolic isometriesone of whose generators is a screw parabolic motion in PU(n,1). Then we use this result to give a sub-horospherical region precisely invariant under the stabliser of the fixed point of the screw parabolic motion in G. Besides, we also show that, given a discrete subgroup of PU(n,1), if its subgroup stablising a point on the boundary of complex hyperbolic space consists only of screw parabolic motions (and the identity), the precisely invariant horoball or sub-horosphericalregion contains a ball of a uniform size with Bergman radius of 0.2589 which doesn't intersect any of its images.We get the three main results in the paper:Theroem 1. Let g be a positively oriented screw parabolic element of PU(n,1) fixing q_∞. Let A∈PU(n - 1) denote the rotational part of g and suppose that ||A - I|| < 1/4. Let h be any element of PU(n,1) not projectively fixing q_∞and let r_h denote the radius of the isometric sphere of h. Ifthen < g,h > is not discrete.Theroem 2. Let g be a positively oriented screw parabolic map, g : (ξ, v, u)→ (Aξ,v + t,u), ||A - I|| < 2/9 , where ||A - I|| < 2/9. Let G be a discrete subgroup of PU(n,1) for which any element of G_∞has the same axis as g. Then the sub-horospherical region U defined byis precisely invariant under G_∞in G.Let z₁ = (ξ₁,v₁,u₁) be any point of Hⁿ(C). As G is discrete, there is an element of G_∞- {I} with the shortest Bergman translation length at Z₁. That is, if f is any other element of G_∞- {I}, thenρ(z₁,g(z₁))≤ρ(z₁, f(z₁)). This means that the open ball centered at z₁ with radiusρ(z-1,g(z₁))/2 does't intersect any of its images under G_∞- {I}. In this condition, we can get the result as following:Theroem 3. Let g be a positively oriented scew parabolic map as above, g : (ξ,v,u)→(Aξ,v+t,u),||A-I|| < 2/9 , where ||A-I|| < 2/9. Let G be a discrete subgroup of PU(n,1) for which any element of G_∞has the same axis as g. Then the orbifold Hⁿ(C)/G contains an embeded open ball with Bergman radiusρwhere e^ρ≈1.29551, that isρ≈0.2589.