#### Volume 25, issue 2 (2021)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
On the top-dimensional cohomology groups of congruence subgroups of $\mathrm{SL}(n,\mathbb{Z})$

### Jeremy Miller, Peter Patzt and Andrew Putman

Geometry & Topology 25 (2021) 999–1058
##### Abstract

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

##### Keywords
congruence subgroups, Steinberg module, cohomology of arithmetic groups
Primary: 11F75