On The Discreteness Of Isometric Subgroups Acting On Hyperbolic Space And The Estimation Of The Volume Of Hyperbolic Manifold

Kleinian groups and hyperbolic geometry have many important applicationsin topology, Riemannian geometry and dynamical systems. And the developmentof the theory of Kleinian groups was started at the end of the nineteenth-century.After the80’s of the twenty-century, as theories of quasi-conformal mapping ma-tured, L. V. Ahlfors etc. brought the theory of Kleinian groups to active the area ofcomplex analysis and made the the theory of Kleinian groups become an importantbranch of complex analysis. Finally, in about1980, Thurston brought a revolu-tion on hyperbolic geometry and Kleinian groups so that hyperbolic geometry andKleinian groups played an important role in topology.In spite of,the complex hyperbolic theory did not develop as rapidly as thereal hyperbolic theory. Until S. Chen and L. Greenberg studied the symmetricspaces, and G. D. Mostow made the construction of non-arithmetic latices of thecomplex hyperbolic spaces, many famous mathematicians began to investigate thecomplex hyperbolic geometry, for example, R.Schwarz, W. M. Goldman and J. R.Parker.The main purpose of the present thesis is to discuss the discreteness of iso-metric subgroups acting on hyperbolic space and the estimation of the volume ofhyperbolic manifold.This thesis is divieded to the following seven chapters.In the first chapter, we devoted to the summary of the dissertation, it recountabout the development, the present circumstances and vast background of discretehyperbolic isometric groups, then introduce the four problems, related results andmain difculties we shall discuss in the next chapters. And the main work of thispaper is also simply introduced. A catalogue of notation is also given there.In the second chapter, we introduce the complex hyperbolic space, and someof the basic properties needed later are developed.In the third chapter, by using the classified according to their set of fixedpoints of the isometric element and the rank of matric, we explore the conditionfor the commutator of two elements inU(1, n; C) be a parabolic element.In the fourth chapter, we consider the necessary condition for a non-elementarytwo generator subgroup of SL(2, C) to be discrete. By embedding SL(2, C) intoU(1,1; H), and we obtain a new type of J rgensen’s inequality which is in termsof the coefcients of involved isometries. We provide an example to show that thisresult gives an improvement over the classical J rgensen’s inequality. As applica- tions, we construct some solid tubes around short geodesics and some propertiesof them are obtained.In the fifth chapter, we study the discreteness criteria in P U(1, n; C). Theseresults generalize some discreteness criterions. We first establish a new set GLinG, then our main results in this chapter are obtained by using Dai B, Fang A andNai B’s generalized J rgensen’s inequality together with the set GL.In the sixth chapter, we discuss how to estimate the Margulis number of anon-elementary discrete subgroup of P U(1,2; C).In the seventh chapter, we expand J rgensen’s inequality by using of Clifordmatrix of n-dimensional Mo¨bius transformation, and by the relation between theinequality of the norm and hyperbolic geometry, prove that one kind of hyperbolicn+1dimensional manifold correspondent and Dirichlet basic polyhedrom includesa sphere which does not correlate to n.