  Let Gamma be the group GL_N (OO_D), where OO_D is the ring of integers in the
imaginary quadratic field with discriminant D<0. In this paper we investigate
the cohomology of Gamma for N=3,4 and for a selection of discriminants: D >=
-24 when N=3, and D=-3,-4 when N=4. In particular we compute the integral
cohomology of Gamma up to p-power torsion for small primes p. Our main tool is
the polyhedral reduction theory for Gamma developed by Ash and Koecher. Our
results extend work of Staffeldt, who treated the case n=3, D=-4. In a sequel
to this paper, we will apply some of these results to the computations with the
K-groups K_4 (OO_{D}), when D=-3,-4.
