High-dimensional Quaternion Hyperbolic Space J¦Õrgenson Inequality,

The main aim of this dissertation is to erect the Jφrgensen's inequality in the quaternion hyperbolic n-space.We get a discrete criterion of non-elementary isometry subgroups generated by two elements, one of which is loxodromic.This dissertation is arranged as follows.In Chapter 1, we provide some background information about our research and state our main results.In Chapter 2, we introduce some basic quaternionic material.In Chapter 3, firstly, we introduce the quaternionic vector space and several kinds of Hermitian forms.Accordingly,we discuss several models and isometries of quaternionic hyperbolic n-space. Finally, we deduce some crucial inequalities.In Chapter 4, our focus is the boundary of the quaternionic hyperbolic n-space. The coordinate we use is the horospherical coordinate. We introduce a new metric called Cygan metric on the boundary. Then, we generalize the complex cross-ratio to the quaternion ring.In Chapter 5, we prove the Jφrgensen's inequality in the quaternion hyper-bolic n-space.The means we adopt mainly is algebraic. We construct a sequence of transformations in the isometric subgroup, then we can prove this sequence convergence to an element of this subgroup under some specified condition. Fi-nally, we compare our result with others of this kind.