Discrete Groups Of M(?)bius Transformations In Higher Dimensions

The main aim of this dissertation is to disscuss the properties of MSbius transformations,discrete MSbius gronps in higher dimensions from the viewpoints of algebra and geometry.This dissertation is arranged in the following way.In Chapter 1,we provide the background of our research and the statements of our main results.In Chapter 2,we introduce the representations of M(o|¨)bius transformations in(?)~n by using Cartesian coordinates and Clifford matrices.In Chapter 3,we study the discreteness of the group G(?)M((?)~n)by using the discreteness of non-elementary subgroups generated by two loxodromic elements,or two parabolic elements,or two g-elliptic elements of G.Correspondingly, we obtain three discreteness criteria.In Chapter 4,we generalize Martin's result concerning the algebraic convergence theorem in(?)~n([56]).When n=2,our result completely coincides with Jφrgensen and Klein's classic one in[19].Moreover,we make a further discussion about Apanasov's result in[65]and get its generalization.In Chapter 5,we study the discreteness of the group G(?)M((?)~n)by using non-elementary subgroups generated by a loxodromic element and a g-elliptic element, or a loxodromic element and a parabolic element,or a parabolic element and a g-elliptic element of G.Correspondingly,we obtain three discreteness criteria.In Chapter 6,the first Klein-Maskit combination theorem in SL(2,C)([22]) is generalized to M((?)~n).And some simple applications of our obtained results are given.In Chapter 7,we generalize the second Klein-Maskit combination theorem in SL(2,C)([22])to M((?)~n).As simple applications,several examples are constructed.