  We compute rationally the topological (complex) K-theory of the classifying
space BG of a discrete group provided that G has a cocompact G-CW-model for its
classifying space for proper G-actions. For instance word-hyperbolic groups and
cocompact discrete subgroups of connected Lie groups satisfy this assumption.
The answer is given in terms of the group cohomology of G and of the
centralizers of finite cyclic subgroups of prime power order. We also analyze
the multiplicative structure.
