  In this paper we provide descriptions of the Whitehead groups with
coefficients in a ring of the Hilbert modular group and its reduced version, as
well as for the topological K-theory of $C^*$-algebras, after tensoring with
$\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones
and the Baum-Connes conjecture respectively. We also construct a model for the
classifying space of the Hilbert modular group for the family of virtually
cyclic subgroups.
