Normalizer Of Kleinian Subgroup Of PU(2,1) And Discreteness Of M(?)bius Groups In High Dimensions

Mobius group theory has been studied for over a hundred years. It has lots of applications in many fields. Nowadays, as an extension of hyperbolic manifolds, complex hyperbolic manifolds is concerned more and more.The discreteness of the isometry group of complex hyperbolic space, especially, complex 2-dimension, is a remarkable field people studied in. There are significant connections between nonnalizer of the Kleinian subgroups of complex hyperbolic model and the isometry group on complex hyperbolic manifolds. The discreteness of normal-izer is an significant object.There are remarkable connections among discrete groups, Riemann surfaces and hyperbolic manifolds. As an indispensable part of Mobius groups theory, the discreteness of Mobius groups is an important problem many workers concerned. We try to discuss this problem by using new method.In section 1, we consider the discreteness of the normalizer of a Kleinian subgroup of PU(2,1). We give "the condition of dimension" for the normalizer of a Kleinian subgroup in complex hyperbolic space of complex 2-dimension to be discrete. That is, if G is a non-elementary, discrete subgroup of PU(2,1) and dim L(G) = 3, then the normalizer N of G is discrete in PU(2,1). Finally, we give an example to show that "the condition of dimension" can not be weaken.In section 2, we try to find the necessary condition for a non-elementary two...