Explicit Kazhdan constants for representations of semisimple and arithmetic groups

Consider a simple non-compact algebraic group, over any locally compact nondiscrete field, which has Kazhdan's property (T). For ang such group, G, we present a Kazhdan set of two elements, and compute its best Kazhdan constant. Then, settling a question raised by Serre and by de la Harpe and Valette, explicit Kazhdan constants for every lattice Gamma in G are obtained, for a "geometric" generating set of the form Gamma boolean AND B-r, where B-r subset of G is a ball of radius r, and the dependence of r on Gamma is described explicitly. Furthermore, for all rank one Lie groups we derive explicit Kazhdan constants, for any family of representations which admits a spectral gap. Several applications of our methods are discussed as well, among them, an extension of Howe-Moore's theorem.