Shimizu Lemma Promotion On Quaternion Hyperbolic Space

In this article, we give some introduction of quaternion, hyperbolic geometry, isometric transformation group, and obtain the necessary condition for subgroup generated by two ele-ments to be discrete. Namely, a generalization of Shimizu’s lemma in quaternion hyperbolic space. Shimizu’ lemma gives a necessary condition for a non-elementary group to be discrete. This thesis is arranged as follows.In Chapter one, we provide some background information about our research and state our main results.In Chapter two, we introduce some basic quaternionic material such as several kinds of Hermitian forms, horoshperical coordinate, screw parabolic elements and some related results.In Chapter three, we establish the special form of Jorgensen’s inequality-Shimizu lemma in PSp(2,1) contains screw parabolic elements.