On The Discreteness And Related Problems Of Hyperbolic Isometric Groups

Discrete groups are located at the junction of various differnet areas of modern mathematics. For instance, they play an important role in complex dynamics, Teichmüller theory, low-dimensional topology, and especially, hyperbolic manifolds. Discrete groups also have applications to quantum physics, chemistry, etc. In this thesis, the discreteness criteria of real and complex hyperbolic groups, the algebraic convergence of discrete hyperbolic groups, and the algebraic, geometric and topological proposition of hyperbolic manifolds are concerned.We use hyperbolic polygons, say Lambert or Sacherri quadrilaterals, as a new characteristic of Mobius transformations; use the relations of dense groups, discrete groups and open sets in Isom(H_n) and Clifford algebra to discuss the discreteness criteria under the assumption that L(I) is finite; use the characterizations, such as the chanis in the complex and Heisenberg geometry, to establish the discreteness criteria, algebaic convergence theorem; prove a finiteness theorem for geometrically finite groups in PU(n,1).This thesis is devided to the following four chapters. Chapter I is devoted to the summary of the dissertation, it recount about the development, the present circumstances and vast background of discrete hyperbolic groups, and then introduce the five problems, related results and main difficulties we shall discuss in the next chapters.In the second chapter, we discuss how to use hyperbolic polygons to give the new characteristic of Mobius transformations. First we prove by a geometric approach that an injection of the hyperbolic plane which preserves Lambert quadrilaterals must be Mobius transformation; then we obtain a class of general hyperbolic n-gons which can also be the characterization of Mobius transformation.In the third chapter, we study the discreteness criteria in Isom(H_n). The novel point is that we discuss the discreteness problem mainly from the topological aspect; i.e., we concern the relation between dense and discrete groups and also give some special open sets in Isom(H_n). The main tools we use includes the generalized J0gensen's inequality established by Cao and Waterman by using Clifford matrix, Chen and Green-berg's theorem about the relation between dense and discrete groups, etc. We also discuss the relation of uniformly bounded torsion, n-dimensional condition, Condition A and that L(I) is finite, and then prove six discreteness criteria in Isom(H_n) under the assumption that L(I) is finite.In the fourth chapter, we study the discreteness criteria and algebaic convergence theorem in PU(2,1). We first give a sufficient condition when the fixed point sets of two boundary elliptic elements are linked by using computations in complex and Heisenberg geometry. Then our main results in this chapter are established by using basmajian and Miner's genralized Jogensen's inequality.In the second chapter, we discuss how to generalize a finiteness theorem og geometrically finite real hyperbolic groups to the case of PU(n, 1). The main tools we use includes: Ratcliffe's results about the discreteness of normalizer; Bowditch's equivalent definitions of geometrically finiteness in the negatively curved isometric groups case; Martin's genralized Jogensen's inequality for the negatively curved manifolds, etc. On the other hand, we construct a cunterexample to show a differnce of real and complex geometry.