## Abstract

Let ${\mathrm{\Gamma}}_{\phantom{\rule{-0.17em}{0ex}}n}\left(p\right)$ be the level-$p$ principal congruence subgroup of ${SL}_{n}\left(\mathbb{Z}\right)$. Borel and Serre proved that the cohomology of ${\mathrm{\Gamma}}_{\phantom{\rule{-0.17em}{0ex}}n}\left(p\right)$ 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 ${\mathcal{\mathcal{T}}}_{n}\left(\mathbb{Q}\right)$ denote the Tits building of ${SL}_{n}\left(\mathbb{Q}\right)$. Lee and Szczarba conjectured that ${H}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\mathrm{\Gamma}}_{\phantom{\rule{-0.17em}{0ex}}n}\left(p\right)\right)$ is isomorphic to ${\stackrel{\u0303}{H}}_{n-2}\left({\mathcal{\mathcal{T}}}_{n}\left(\mathbb{Q}\right)\u2215{\mathrm{\Gamma}}_{\phantom{\rule{-0.17em}{0ex}}n}\left(p\right)\right)$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map ${H}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\mathrm{\Gamma}}_{\phantom{\rule{-0.17em}{0ex}}n}\left(p\right)\right)\to {\stackrel{\u0303}{H}}_{n-2}\left({\mathcal{\mathcal{T}}}_{n}\left(\mathbb{Q}\right)\u2215{\mathrm{\Gamma}}_{\phantom{\rule{-0.17em}{0ex}}n}\left(p\right)\right)$ is always surjective, but is only injective for $p\le 5$. In particular, we completely calculate ${H}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\mathrm{\Gamma}}_{\phantom{\rule{-0.17em}{0ex}}n}\left(5\right)\right)$ and improve known lower bounds for the ranks of ${H}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\mathrm{\Gamma}}_{\phantom{\rule{-0.17em}{0ex}}n}\left(p\right)\right)$ for $p\ge 5$.

## Citation

Jeremy Miller. Peter Patzt. Andrew Putman. "On the top-dimensional cohomology groups of congruence subgroups of $\mathrm{SL}(n,\mathbb{Z})$." Geom. Topol. 25 (2) 999 - 1058, 2021. https://doi.org/10.2140/gt.2021.25.999

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