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

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