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

2023/09/11 by Brück, Benjamin, Himes, Zachary
#11F75 #20E42 #55U10 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2309.05456

Abstract

Let K be a number field with ring of integers R = O_K. We show that if R is not a principal ideal domain, then the symplectic group 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 Sp_2n(K). We show that this module is not generated by integral apartment classes.

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