On the cohomology of linear groups over imaginary quadratic fields

TL;DR

This paper investigates the cohomology of linear groups over imaginary quadratic fields, computing integral cohomology for specific cases using polyhedral reduction theory, extending prior work and setting the stage for K-group applications.

Contribution

It provides new computations of the cohomology of GL_N over imaginary quadratic integer rings for N=3,4 and specific discriminants, using polyhedral reduction theory.

Findings

Computed integral cohomology up to p-power torsion for small primes

Extended previous work on cohomology of these groups

Laid groundwork for future K-group computations

Abstract

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.