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. Bykovskii gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this presentation 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.

arXiv commentMinor revisions based on referee's comments. Accepted for publication at IMRN

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• Improved homological stability for certain general linear groups
• The Steinberg representation is irreducible
• On the top-dimensional cohomology of arithmetic Chevalley groups