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

Abstract

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