On the cohomology of linear groups over imaginary quadratic fields

Let Γ be the group GLN(OD)GLN(OD), where ODOD is the ring of integers in the imaginary quadratic field with discriminant D<0D<0. In this paper we investigate the cohomology of Γ for N=3,4N=3,4 and for a selection of discriminants: D≥−24D≥−24 when N=3N=3, and D=−3,−4D=−3,−4 when N=4N=4. In particular we compute the integral cohomology of Γ up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for Γ developed by Ash [4, Ch. II] and Koecher [24]. Our results extend work of Staffeldt [40], who treated the case N=3N=3, D=−4D=−4. In a sequel [15] to this paper, we will apply some of these results to computations with the K  -groups K4(OD)K4(OD), when D=−3,−4D=−3,−4.