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 ${\tilde{\text{H}}}_{n-2}({\mathcal{T}}_n(K)/{\Gamma }_n(p))$, 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 ${\text{H}}^{\nu }({\Gamma }_n(p))\to {\tilde{\text{H}}}_{n-2}({\mathcal{T}}_n(K)/{\Gamma }_n(p))$ is always surjective, and give a sufficent set of conditions on $p\in R$ that guarantee when this map is an isomorphism.

44 pages, 9 figures. Comments welcome! v2: Major revision - significant improvement to the main result of Theorem 1.5