On the top-degree cohomology groups of congruence subgroups of $\text{Sp}_{2n}(\mathbb{Z})$

Abstract

Let $Γ_{2n}^ω(p)$ be the level-$p$ principal congruence subgroup of $\text{Sp}_{2n}(\mathbb{Z})$ for all prime $p$. Borel--Serre demonstrated that the cohomology of $Γ_{2n}^ω(p)$ vanishes above degree $n^2$. We prove that $\text{H}^{n^2}(Γ_{2n}^ω(p); \mathbb{Q})$ surjects onto the homology of the quotient of the symplectic Tits building for $\mathbb{Q}$ by $Γ_{2n}^ω(p)$ and we compute the homology of this quotient. We conclude that $\text{H}^{n^2}(Γ_{2n}^ω(p);\mathbb{Q})$ is nontrivial and provide a lower bound of its rank.