Shimizu's Lemma For Screw-Parabolic Elements

With the perfect of real hyperbolic theory, complex Hyperbolic Geometry is concerned by many international mathematician. It plays an important role in theoretical physics, quantum mechanics and systems sciences. Discrete group theory also plays an important role in complex hyperbolic space.In this paper, we mainly study a non-discrete group of complex hyperbolic isometries that contain a parabolic screw motion. Thus Shimizu′s Lemma. Arranges as follows.In chapter 1, the background of our interested problems is introduced and our main results are brief stated.In chapter 2, We give the basic propositions of complex hyperbolic in n+1-dimension. Which contain the proposition and the classification of isometry groups in n+1-dimension,Hermitian form,Cygan metric,isometric sphere and so on.In chapter 3, we study a non-discrete group of complex hyperbolic isometries that contain a parabolic screw motion. We use character value ,Hilbert-Schmidt norm proof the theorem 3.1.In chapter 4, we discuss the non-discreteness of complex hyperbolic isomerties that a parabolic screw motion with a small rotation angle in three dimension.