Properties And Discreteness Of M(?)bius Groups

The main purpose of this thesis is to investigate the discreteness criteria of the M(o|¨)bius groups and the inequalities of Cygan metric in complex hyperbolic geometry. It is arranged as follows.In Chapter 1,we mainly provides some background information about M(o|¨)biusgroups and complex hyperbolic geometry, and also state the main results.In Chapter 2, we introduce some basic concepts and properties of M(o|¨)bius groups, including the Poincare extension, elementary subgroups, non-elementary subgroups. And also introduces some basic concepts and properties of complex hyperbolic geometry, state three models and three Hermitian form, Cygan metric,etc.In Chapter 3, we mainly discuss the discreteness of non-elementary groups of M(o|¨)bius groups. Normalize the non-elementary groups, we get the discreteness of its subgroups.In Chapter 4, we mainly consider the classification of isometries in complex hyperbolic geometry and some properties. We also consider the inequality of Cygan metric. Finally, we give the inequality of Cygan metric in view of the regular elliptic and boundary elliptic. Therefore generalize the conclusion related.