The Discreteness Of Mobius Transformation Group

Since the theory of Klein group were proposed, they have advanced with excep-tional speed. For their great effectiveness on lower dimension topology, dynamic system, Riemann geometry, etc., the qualitative analysis of Klein group fixe a large number of experts’s attention. The paper mainly study the discreteness of Mobius transformation group, answering the conjecture of S. Yang. By means of test map, we get some criterias of discreteness of Mobius transformation group for lower dimension. Moreover, the paper also study Mobius transformation group on higher dimension. The main structure of the paper is as follows:In the first part, we briefly introduce the history and the current research situation of Klein group, and give a clean reins of discreteness for Mobius trans-formation group. Through citing and analyzing numerous works in this field, we introduce what would be studied in our subsequent sections.In the second part, we discuss the discreteness criteria of Mobius transforma-tion group on lower dimension. In this part we proof the conjecture of S. Yang, and we get three theorems about discreteness by test map and certain element (elliptic element or loxodromic element or parabolic element).In the third part, we discuss the discreteness criteria for Mobius transformation group on higher dimension. In this part we use the discreteness of WY(G) with the discreteness of regular elliptic element and parabolic element (or parabolic element and loxodromic element, or regular elliptic element and parabolic element) getting three new theorems on discreteness when dim(σ(L(G))) is even.