On the top dimensional cohomology groups of congruence subgroups of ${\text{SL}}_n(\mathbb{Z})$

Abstract

Let ${\Gamma }_n(p)$ be the level-$p$ principal congruence subgroup of ${\text{SL}}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of ${\Gamma }_n(p)$ vanishes above degree $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. We study the cohomology in this top degree $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. Let ${\mathcal{T}}_n(\mathbb{Q})$ denote the Tits building of ${\text{SL}}_n(\mathbb{Q})$. Lee-Szczarba conjectured that $H^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}({\Gamma }_n(p))$ is isomorphic to ${\tilde{H}}_{n-2}({\mathcal{T}}_n(\mathbb{Q})/{\Gamma }_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}({\Gamma }_n(p))\to {\tilde{H}}_{n-2}({\mathcal{T}}_n(\mathbb{Q})/{\Gamma }_n(p))$ is always surjective, but is only injective for $p\le 5$. In particular, we completely calculate $H^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}({\Gamma }_n(5))$ and improve known lower bounds for the ranks of $H^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}({\Gamma }_n(p))$ for $p\ge 5$.

51 pages, 6 figures; minor corrections; to appear in Geom. Topol

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