The Matrix Group Embedded With New J¦Õrgensen Of Inequality

The main aim of this dissertation is obtain J(?)rgensen's inequality by embedding S L(2, C) into PSO(1,3). We get a discrete criterion of non-elementary isometry subgroups generated by two elements, one of which is loxodromic.The dissertation is arranged as follows.In Chapter 1. we provide some background information about J(?)rgensen's inequality and state our main results.In Chapter 2. we introduce some basic facts about quaternions, the models of quaternionic hyperbolic space, the classification of isometries of hyperbolic space and the cross-radio.In Chapter 3, we introduce the horospherical coordinate of quaternionic hyperbolic n-space and the Heisenberg group.In Chapter 4, we prove the J(?)rgensen's inequality by embedding SL(2.C) into PSO(1,3). At last, we compare our result with the classical result.