Discreteness Of Hyperbolic Isometric Groups

Hyperbolic geometry and discrete group are an important research direction of the modern complex analysis geometric theory, The results of their research and research methods have important applications in many aspects. As real hyperbolic geometry and M obius group theory, How to judge acting on complex hyperbolic isometry group of the complex hyperbolic space is a discrete group is a very important and fundamental questions. Jorgensen T. established a Jorgensen inequality of two-dimensional M obius group, Which gives a necessary condition for a M obius group is discrete group. As the promotion of high-dimensional of hyperbolic Riemann surface , The research of complex hyperbolic space and complex hyperbolic groups has been attentionnned to many people. This thesis is to establish the special form of Jorgensen inequality—Shimizu lemma of non-elementary subgroup of PU(3,1) with screw parabolic element. Which is also to investigate the properties of the isometric groups acting on real hyperbolic space or complex hyperbolic space. We obtained a necessary condition for the discreteness of non-elementary subgroup of PU(3,1) with screw parabolic element. And make use of cross ratios and Jorgensen inequality of hyperbolic elements to prove the collar inequality about a hyperbolic element of discrete non-elementary subgroup. And educe the radius formular of a precisely invariant strip neighborhood about geodesic.