M(?)bius Transformations In Inner Product Spaces And The Discreteness Of M(?)bius Groups

Mobius groups have many close connections with finite or infinite dimensional manifolds endowed with hyperbolic structures, such as Riemann surface, Teichmuller space, Complex dynamic system and so on. However so far, investigations in this direction are restricted to the finite dimensional spaces. A counter example cited by J.Vaisala/15} implied that the definition of Mobius transformations in finite dimensional spaces did not suit for general metric spaces. In the first section of this thesis, we establish the definition of Mobius transformations under the framework of inner product spaces successfully. The main results (Thl, Th2) show the deep meanings and the wide prospects of the research in Mobius transformations in inner product spaces. To prove the results, we use a purely geometric method which does not matter the dimension of a space. It is totally different from the method of seeking Schwarz's derivative used by H.Haruki and T.M.Rassias and the induction priciple used by A.F.Beardon and D.Mindat[3]In the second section of this thesis, we discuss the problem of the discreteness of non-elementary 2-dimensional Mobius groups. We introduce a new approach that one can use a parabolic element as a test element to test the dicreteness of a non-elementary 2-dimerisional Mobius group and it does not matter whether or not the test eleiunet isin the group. Our result(Th3)improves the criteria established by T.J0rgensen(ThD).