A group has Kazhdan's property (T) if every representation which has almost invariant vectors has in fact a nonzero invariant vector. The thesis investigates this property for various classes of groups and develops techniques for the explicit computation of Kazhdan constants. The latter provide a quantitative version of property (T). The first part is concerned with discrete groups. If such groups have property (T) a family of expanding graphs can be constructed from a family of normal subgroups. A theorem of A. Zuk gives a criterion for the restriction of the Cayley graph to a set of generators. For less than six nodes all such graphs have been determined. The second part deals with semi-simple Lie groups. A lower bound for the infinitesimal Kazhdan constant of the Lie groups Sp(n, R) for n > 1 is determined as well as for SL(n, R) (n > 3). The used methods of proof permit the determination of explicit asymptotics of the matrix coefficients of Sp(n, R) for n > 1. From that a Kazhdan constant of Sp(n, R) can be derived. The last section is concerned with loop groups in particular the loop group of Sp(n, C). There explicit Kazhdan constants are determined. Loop groups are examples of Kazhdan groups which are not locally compact and not finite dimensional. The last part considers compact groups. Here the computation of lower bounds of explicit Kazhdan constants is particularly interesting. This is done relative to a conjugacy class. The lower bound only depends on the nontrivial irreducible characters of the compact group. Explicit lower bounds are computed for the symmetric groups and SU(n). It is also shown that the lower bound is nonzero if the conjugacy class generates the group. The used methods of proof yield also other lower bounds for Kazhdan constants.     «

A group has Kazhdan's property (T) if every representation which has almost invariant vectors has in fact a nonzero invariant vector. The thesis investigates this property for various classes of groups and develops techniques for the explicit computation of Kazhdan constants. The latter provide a quantitative version of property (T). The first part is concerned with discrete groups. If such groups have property (T) a family of expanding graphs can be constructed from a family of normal subgroups...     »