The Geometry Of Mobius Groups In High Dimensions And Mostow Rigid Theorem

in the first part of the thesis, we stiudy the subgroups of MObius translations in high dimension with the Clifford algebra. Let G be a subgroup .of MObius translations which maps H~~? onto itself and contains the translation g0 (x)x+i. If 6 does not contain elliptic elements and fp.f elements, then we derive an upper bound of the radius of the isometric spheres, to all the Ahlfors hyperbolic elements and uniformly elements that do not fix ax The upper bound 1/4 is sharp. Moreover, if 6 is made up of oniy Ahlfors hyperbolic elements and uniformly elements, then we show either 6 is discontinuous in H攡?or 6 is an elementary group which consists of only the elements that fixes cxx In the second part of the thesis, we prove the well-known rigid theorem of Tukia 憇 with a condition that is weaker than that of his. So our theorem is more general. Gu Quan (Morden Complex Analysis) Directed by Doc. Mm Chen...