On The Studies Of The Properties Of Hyperbolic Geometry And Kleinian Groups

The hyperbolic geometry and Kleinian groups are important research areas in complex analysis. Due to Ahlfors, Bers, Sullivan's excellent works, the hyper-bolic geometry and Kleinian groups become more and more closely related to the studies of Teichmuller spaces, complex analytic dynamics, hyperbolic man-ifolds and so on. In particular, Thurston's famous works on 3-manifolds enrich the study on the hyperbolic geometry and Kleinian groups, and this study at-tracts more and more attension. The main aim of this dissertation is to disscuss the algebraic and geometric properties of the complex hyperbolic geometry and Kleinian groups. This dissertation consists of 7 chapters. They are arranged as follows.In Chapter 1, we give the background of our research and the statements of our main results.In Chapter 2, we introduce some basic definitions and known results on the complex hyperbolic geometry.In Chapter 3, we generalize Jφrgensen's dicreteness criteria to the setting of complex hyperbolic spaces and find the relations between the discreteness of subgroups G in PU(n,1) and the one of their two-generator subgroups.In Chapter 4, we give a characterization of Fuchsian groups acting on complex hyperbolic spaces which is a generalization of Maskit's result in real Fuchsian groups and construct some examples to show the existence. We also discuss purely elliptic subgroups in PU(n,1) and generalize Katok, Jφrgensen's results to the case of complex hyperbolic spaces.In Chapter 5, we study the volumes of complex hyperbolic orbifolds whose fundamental groups have uniformly bounded and a lower bound is obtained.In Chapter 6, we introduce "F-condition". By using it, we discuss the alge-braic convergence of Kleinian groups. We generalize the corresponding results obtained by Martin etc. In Chapter 7, we complete Kim's discussion on the distance of the points in quotient spaces and their preimages in the corresponding real hyperbolic spaces. Also, by using hyperbolic isosceles right triangles, a characterzation of hyperbolic isometrics in real hyperbolic spaces is obtained.