The Discreteness Of Isometric Subgroups Acting On Hyperbolic Space And The Moduli Space Of Quadruples Of Points

Hyperbolic geometry and Kleinian groups has many important applications in geometry and topology,physics,dynamical systems,Riemannian geometry.The development of the theory of Kleinian groups was started at the end of the nineteenth-century by Poincar(?),Fricke and Klein.After 1960,as theories of quasi-conformal mapping matured, L.V.Ahlfors and L.Bers brought Kleinian groups theory to active area of complex analysis as a branch of Teichm(u|¨)ller theory.Finally,in about 1980,W.P.Thurston brought a revolution to hyperbolic geometry and Kleinian groups,so that hyperbolic manifolds and Kleinian groups attracted the attention of many topologists.Now,A great deal of advances have been obtained in theory of Kleinian groups.For example,L.V.Ahlfors made Ahlfors' Measure Conjecture for finitely generated Kleinian groups.G.D.Mostow proved a rigidity theorem for hyperbolic n-manifolds of finite volume.D.P.Sullivan studied the dynamics of Kleinian groups acting on the boundary of hyperbolic space.W.P. Thurston provided the classification of geometric 3-manifolds and the foliations structure on surface.Complex hyperbolic Kleinian groups were first studied by Picard at about the same time as Poincar(?) was developing the theory of Fuchsian and Kleinian groups.In spite of work by several other people,including G.Giraud and E.Cartan,the complex hyperbolic theory did not develop as rapidly as the real hyperbolic theory.Later,work of S.Chen and L.Greenberg on symmetric spaces and work of G.D.Mostow on the construction of nonarithmetic lattices led to a resurgence of interest in complex hyperbolic discrete groups. Many famous mathematicians began to investicate the complex hyperbolic geometry and obtained many important results.The main purpose of the present thesis is to disscuss some properties of subgroups of isometries of complex hyperbolic space or real hyperbolic space and describe the moduli space of quadruples of points in complex hyperbolic space together with its boundary. We obtain the following main results.Firstly,the authors obtained a necessary condition for two-generator discrete groups of PU(1,n),and provid a J(?)rgensen's inequality for non-elementary subgroups of isometries of complex hyperbolic space generated by two elements,one of which is regular elliptic element.We give a J(?)rgensen's inequality for subgroups containing regular elliptic with real trace and preserving a Lagrangian plane.Secondly,by using some version of Klein-Maskit combination theorem and the dis- tance between the complex lines or points fixed by two elliptic elements,we explore the conditions for two elliptic elements and two parabolic elements in PU(1,2).to generate discrete free group.Thirdly,we discuss three discreteness criterions of n-dimensional subgroup G of PU(1,n).This result generalize some discreteness criterions established by J.Gilman, S.Yang and A.Fang.The basic idea in proving our results is to note that the sets of loxodromic elements or regular elliptic elements are both open facing the identity element. A dense theorem of S.Chen and L.Greenberg which said that n—dimension subgroup of SU(1,n) is dense or discrete in SU(1,n) also play an important role in our proof.Furthermore,In the late 90s of last century,J.W.Anderson asked whether two finitely generated Kleinian groups G₁,G₂ C Isom(Hⁿ) with the same set of axes are commensurable.We give a very simple example to show the answer of Anderson's question is negative in general.We disscuss the conditons which imply two Kleinian groups are commensurable.Finally,we study the moduli space of quadruples of points with three ideal points in boundary and one point in complex hyperbolic space.We constuct a moduli space of real dimension 6 by using Cartan angular invariant and generalized complex cross ratio.