Kazhdan's Property(T) and Structure of Exotic Hyperbolic Manifold

This thesis considers complex and exotic hyperbolic manifolds. So called exotic manifolds are those manifolds that can be viewed as quotient spaces of Lie groups whose base number fields are quaternions and octonions. It is well known that a complex hyperbolic manifold can be viewed as a quotient of complex unit ball Bn modulo an action of discrete subgroup of the whole Mobius group on Bn. It is proved in the first part of this thesis that there exits a positive Green's function on complex hyperbolic manifold provided certain condition on the limit set of the discrete group action. In the more general point of view, exotic hyperbolic manifolds can be viewed as double coset space Gamma G/K, where G is a semisimple Lie group, K the maximum compact subgroup of G and Gamma a discrete subgroup of G. The structure at infinity of these hyperbolic manifolds were studied in many articles. This thesis mainly focuses on the relation between the Kazhdan's Property (T) of G and the structure at infinity. In the second part of thesis, it is proved that Kazhdan's Property (T) implies simple structure of exotic hyperbolic manifolds at infinity.