The Discreteness Of The Normalizers Of High Dimensional Kleinian Groups

..Mobius group theory has important applications in many fields, such as Riemann surface, Teichmu..ller spaces, hyperbolic manifolds, potential theory, complex analytic dynamics and so on.People have been making wide and deep studies in Kleinian groups since 40 date of last century. The close connections among Kleinian groups or discrete Mo..bius groups and hyperbolic manifolds make it contain plentiful contents. The criteria of discreteness and the rigidity theorems are ones of its important contents.Maskit B. has proved that: Let LetΓin M( B 2) be a Kleinian group, then the normalizer N(Γ) ofΓin M( B 2) is also a Kleinian group. After that, Ratcliffe G. used a sufficient condition about the discreteness of normalizers of Kleinian groups when he studied the finiteness of the isometry group of the hyperbolic manifold. Also, Ratcliffe G. showed that: LetΓin M( B n) be a Kleinian group, then the isometry group I( B nΓ) of the hyperbolic manifold B nΓis isomorphic to N (Γ)Γ.From above, we see that normalizers play a very important role in the study of Kleinian groups.We extend Kleinian groups from the algebraic view, then we extend it to its normalizer. Obviously, Kleinian group is the normal subgroup of its normalizer. The...