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arXiv · 2309.05456

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

Benjamin Brück·Zachary Himes
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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.

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Benjamin Brück, Zachary Himes. 2023-09-11. Top-degree rational cohomology in the symplectic group of a number ring. https://doi.org/10.1007/s00029-025-01051-8

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