Top-degree rational cohomology in the symplectic group of a number ring

TL;DR

This paper proves that symplectic groups over non-principal ideal rings of number fields have non-trivial top-degree rational cohomology, contrasting with Euclidean cases, by analyzing the symplectic Steinberg module.

Contribution

It establishes the non-triviality of top-degree rational cohomology for symplectic groups over certain rings, using properties of the symplectic Steinberg module.

Findings

Non-trivial rational cohomology in the virtual cohomological dimension for non-principal ideal rings.

The symplectic Steinberg module is not generated by integral apartment classes.

Contrast with Euclidean rings where such cohomology vanishes.

Abstract

Let $K$ be a number field with ring of integers $R = \mathcal{O}_K$. We show that if $R$ is not a principal ideal domain, then the symplectic group $\operatorname{Sp}_{2n}(R)$ has non-trivial rational cohomology in its virtual cohomological dimension. This demonstrates a sharp contrast to the situation where $R$ is Euclidean. To prove our result, we study the symplectic Steinberg module, i.e. the top-dimensional homology group of the spherical building associated to $\operatorname{Sp}_{2n}(K)$. We show that this module is not generated by integral apartment classes.