Discreteness Of Isometric Subgroups In Complex Hyperbolic Space

Complex hyperbolic geometry keeps close relation with Riemannian geometry, contact geometry, theory of Lie groups, harmonic analysis and algebraic geometry, etc, which is an important research object in the field of complex analysis. The development of the theory of complex hyperbolic geometry was began at the end of the nineteenth-century. Although it was born at the same time as real hyperbolic theory, complex hyperbolic space did not develop as rapidly as the real hyperbolic theory due of its rich structure. Later, symmetric space was studied by Chen and Greenberg, and Mostow constructed non-arithmetic lattices, so more and more scholars began to study the complex hyperbolic theory. Since Epstein, Toledo, Goldman, Schwartz, Parker, Falbel, Zocca, Deraux, etc., have done a lot of work and promoted the development of this field, the enthusiasm of more young scholars has been inspired.The main purpose of the thesis is to discuss discrete necessary condition of isometric subgroups of PU(2,1), discrete sufficient condition of two-generator sub-groups in complex hyperbolic plane, and the C-decomposability of isometric ele-ments.In the first chapter, we survey the history and development of the theory of complex hyperbolic geometry, and then we give a summary outlining of the main results of the thesis. At last, we introduce basic notations in this chapter that will be used throughout the text.In the chapter 2, the basic knowledge about complex hyperbolic geometry are reviewed. Firstly, we introduce two standard models and Cygan metric of complex hyperbolic plane, in which the affinely convex of all Cygan balls plays an impor-tant role in the chapter 3. Secondly, we investigate four types of totally geodesic subspaces, which are points, geodesics, complex lines and R-planes. Finally, we also recall the definition of topology groups and discrete groups, and classify the holomorphic isometries into certain types.The problem of discreteness of isometric groups have been receiving more and more attention from many researchers. In the chapter 3, we mainly study the discreteness criterion for a subgroup of PU(2,1) with a screw parabolic element, which can be considered as a extension of Shimizu’s lemma. The nature of screw parabolic elements are recalled, and then we define the boundary function Bg(r) relying upon the geometry of Margulis region. With the help of the theory associ- ated with continued fraction, we give an universal upper bound for the boundary function Bg(r), so we can also get an explicit region precisely invariant under Γ∞. The main theorem in this chapter is proved by using the property of Margulis region and the convexity of Cygan ball.Discrete groups are related to triangle groups, that is, groups generated by three involutions. There are three types of involutions in the isometric group of complex hyperbolic plane. One is the reflection of order 2 about a complex line, we call it complex symmetry; Another is the reflection of order 2 about a point in complex hyperbolic plane; The last one is the reflection of order 2 about a R-plane named Lagrangian reflection. In the chapter 4, we study the C-strongly reversibility of a parabolic and an elliptic element respectively, and the C-decomposability is introduced. A pair of isometries (A, B) is said to be C-decomposable if there are three complex symmetries I1, I2 and I3 such that A=I1I2 and B=I2I3 holds. In this chapter, the C-decomposability criterions for a pair of parabolic elements and elliptic elements are established respectively, and moreover, we also obtain the sufficient condition of C-decomposability when one is loxodromic element and the other one is parabolic element.In the chapter 5. we study a new sufficient condition for a two-generator subgroup to be discrete, where the pair of generators can be C-decomposable. In particular, we extend the result of two-generator groups to groups generated by n elements, where n> 2. We first turn the problem of discreteness for two-generator subgroups to judgment of the discreteness of the corresponding triangle groups. After that, we recall the definitions and basic properties of bisectors and introduce the definition of NSD/TB group. By combining with Klein’s combination theorem, we obtain the discreteness criterion. Finally, we illustrate the feasibility of this discreteness criterion.