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

This is the authors' abstract. We don't add key points for this paper.

Let K be a number field with ring of integers R = OK. We show that if R is not a principal ideal domain, then the symplectic group Sp₂ₙ(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 Sp₂ₙ(K). We show that this module is not generated by integral apartment classes.