| Hyperbolic geometry and discrete group are an important research direction of the modern complex analysis geometric theory, The results and methods of their research have important applications in many aspects. At the beginning of the twentieth century, the research of real hyperbolic geometry reached its peak, with the perfect of real hyperbolic theory, more and more researchers are interested in complex hyperbolic space and complex hyperbolic group and have obtained many well-known research outcomes. In recent years, quaternion hyperbolic geometry is concerned by many mathematicians and constantly enriched under the influence of Riemann geometry, Complex analysis, Symplectic geometry and many other areas of Mathematics. As real hyperbolic geometry and complex hyperbolic geometry, how to judge acting on quaternion hyperbolic isometry group of the quaternion hyperbolic space is a discrete group is a very important and fundamental question.In this paper, we mainly study necessary condition for subgroups of quaternion hyperbolic isometry with screw-parabolic elements. The arrangement is as follows:In chapter1, we introduce the research background of hyperbolic geometry and briefly introduce the main work of this paper.In chapter2, we introduce some basic propositions of quaternion hyperbolic geometry in n-dimension, which contain definition of quaternion, quaternion hy-perbolic space, Hermitian form, Heisenberg group, Cygan measure, Ford isometric spherical and so on.In chapter3, using a similar structure and method of proof in [25], Shimizu lemma will be extended to groups of quaternion hyperbolic isometries with screw-parabolic elements in three dimension.In chapter4, we give a necessary condition for PSp(3,1) with screw-parabolic elements to be discrete.In chapter5, basing on the work of Kamiya and Parker, we go on to study a necessary condition for subgroups of quaternion hyperbolic isometries with screw-parabolic elements in three dimension to be discrete. |
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