On The Studies Of Discreteness Of Complex Hyperbolic Isometry Group And Jφrgensen Number Of Triangle Group

The main purpose of this thesis is to investigate the discreteness criteria of subgroupsof complex hyperbolic isometry groups PU （1, n; C）and the Jφrgensen number oftriangle group which is a special discrete group.This dissertation consists of fourchapters. They are arranged as follows.In chapter1, we give some basic knowledge, which is needed in our researth, onthe geometry of discrete group.In chapter2, we mainly discuss the properties of products of isometries. And theother two properties about products of two hyperbolic isometries are derived on thebasis of the existing results.In chapter3, we mainly discuss the discreteness of non-elementary subgroups Gof complex hyperbolic isometry group PU （1,n;C）.There are three criteria,and weacquire two conclusions under the assumption that G satisfies condition A. And thefirst result considers to use a test map to examine the discreteness of G.The nextresult shows that G is discrete,if each two-loxodromic-generator subgroup of G isdiscrete. And the third conclusion strengthens the second,namely,only under theassumpion that some subgroupG₀of G satisfies condition A, G is discrete if eachtwo-loxodromic-generator subgroup of G is discrete.In chapter4, we mainly investigate the Jφrgensen number of triangle groups. Andwe have only found the Jφrgensen number of （p, q,∞） triangle group, but we givesome conjectures on the Jφrgensen number of type （p, q, r）（r≠∞）in the final section.