On the cohomology of linear groups over imaginary quadratic fields

Abstract

Abstract. Let Γ be the group GL N (O D ), where O D is the ring of integers in the imaginary quadratic field with discriminant D < 0. In this paper we investigate the cohomology of Γ 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 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 [18]. Our results extend work of Staffeldt [29], who treated the case n = 3, D = −4. In a sequel [11] to this paper, we will apply some of these results to the computations with the K-groups K 4 (O D ), when D = −3, −4.

The Farrell–Tate and Bredon homology for PSL4(Z) via cell subdivisions

“…While for Coxeter groups with a small system of generators [18] and arithmetic groups of rank 2 [14], general formulae for the equivariant K-homology have been established, the only known higherrank case to date is the example SL 3 (Z) in [17]. Although there are by now considerably more arithmetic groups for which cell complexes have been worked out [6,7,9], no further computations of Bredon homology H Fin n (EG; R C ) have been done since 2008 because the relevant cell complexes fail to have the rigidity property required for Sanchez-Garcia's method. We discuss an explicit example, cf.…”

“…e.g. [7,9]) do not provide complexes with this rigidity property, and both for the computation of Farrell-Tate cohomology (resp. the torsion at small prime numbers in group cohomology) of arithmetic groups as well as for the computation of Bredon homology, this lack of rigid cell complexes constitutes a significant bottleneck.…”