On the top-dimensional cohomology groups of congruence subgroups of $\mathrm{SL}(n,\mathbb{Z})$

Abstract

Let ${\mathrm{\Gamma }}_n\left(p\right)$ be the level-$p$ principal congruence subgroup of ${SL}_n\left(\mathbb{Z}\right)$. Borel and Serre proved that the cohomology of ${\mathrm{\Gamma }}_n\left(p\right)$ vanishes above degree $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$. We study the cohomology in this top degree $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$. Let ${\mathcal{𝒯}}_n\left(\mathbb{Q}\right)$ denote the Tits building of ${SL}_n\left(\mathbb{Q}\right)$. Lee and Szczarba conjectured that $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\mathrm{\Gamma }}_n\left(p\right)\right)$ is isomorphic to ${\tilde{H}}_{n-2}\left({\mathcal{𝒯}}_n\left(\mathbb{Q}\right)∕{\mathrm{\Gamma }}_n\left(p\right)\right)$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\mathrm{\Gamma }}_n\left(p\right)\right)\to {\tilde{H}}_{n-2}\left({\mathcal{𝒯}}_n\left(\mathbb{Q}\right)∕{\mathrm{\Gamma }}_n\left(p\right)\right)$ is always surjective, but is only injective for $p\le 5$. In particular, we completely calculate $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\mathrm{\Gamma }}_n\left(5\right)\right)$ and improve known lower bounds for the ranks of $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\mathrm{\Gamma }}_n\left(p\right)\right)$ for $p\ge 5$.