On the generalized Bykovskii presentation of Steinberg modules

Abstract

We study presentations of the virtual dualizing modules of special linear groups of number rings, the Steinberg modules. Bykovskiȋ gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this to the Gaussian integers and the Eisenstein integers. We also show that this generalization does not give a presentation for the Steinberg modules of several Euclidean number rings.

The Steinberg representation is irreducible

“…This gave rise to a large literature using St(G; F) to study the cohomology of such arithmetic subgroups. Some representative papers include [2,3,7,8,9,14,17,19,21,22,23,24].…”