The Discreteness And Extension Of M(?)bius Groups

This paper mainly discusses about the discreteness criteria and the extension of Mobius groups.Firstly, by using Clifford algebra, we get the general expression of n-dimensional hyperbolic transformations. Then according to the properties of Clifford cross-ratios we classify the type of elements which have the half-turn decomposition and that of the commutators.Secondly, we discuss the discreteness of Mobius groups. Our main results are: (1) Let G be a non-elementary subgroup of SL(2,C) and go is a loxodromic (or parabolic , elliptic) element of SL(2,C). If every non-elementary subgroup < /, 90 > where / € G, is discrete, then G is discrete. (2) Let G C SL(2,C) be non-elementary. Then G is discrete if and only if each subgroup generated by two different elements of Gh (or Gp (if G contains some parabolic element)) is discrete.Finally, the extension of Mobius group is discussed. We get a necessary and sufficient condition for that G C 5L(2, Tn) is the extension of G' C SL(2, C) (or SL(2,R) respectively). They are: (1) G is conjugate in SL(2,Fn) to a group G' C SL(2, C) if and only if G' satisfies the following properties: (A) There exist loxodromic elements g0, h 6 G' such that fix(g0) = {0, }, fix(g0) fix(h) = 0 and fix(h) n C 0; (B) tr(g) € C for each loxodromic element g € G'. (2) G is conjugate in 5L(2, Fn) to a group G' C 5L(2, R) if and only if G' satisfies the following properties: (A) There exists a loxodromic element g0 G' such that fix(g0) {0, } 0; (B) tr(g) R for each loxodromic element g e G'.