The top cohomology of principal congruence subgroups of special linear groups over Euclidean number rings

Abstract

For $R$ a Euclidean number ring, and let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(R)$. Borel--Serre showed that the cohomology of $\Gamma_n(p)$ vanishes above a degree $\nu$ that is quadratic in $n$. Let $K$ be the fraction field of $R$, and $\mathcal{T}_n(K)$ the Tits building of $\text{SL}_n(K)$. For $R=\mathbb{Z}$, Lee--Szczarba asked when $\text{H}^\nu(\Gamma_n(p))$ is isomorphic to $\widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/\Gamma_n(p))$, which was answered by Miller--Patzt--Putman. We study a generalized version of Lee--Szczarba's question. We prove that for a prime $p$ in a Euclidean number ring $R$ with fraction field $K$, that a natural map $\text{H}^\nu(\Gamma_n(p)) \to \widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/\Gamma_n(p))$ is always surjective, and give a sufficent set of conditions on $p \in R$ that guarantee when this map is an isomorphism.