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

TL;DR

This paper investigates the top cohomology of principal congruence subgroups of special linear groups over Euclidean number rings, establishing conditions for when certain cohomology maps are isomorphisms.

Contribution

It generalizes previous results by analyzing the surjectivity and isomorphism conditions of cohomology maps for Euclidean number rings, extending the understanding of cohomology in this context.

Findings

The natural cohomology map is always surjective for prime $p$ in Euclidean number rings.

Provided sufficient conditions on $p$ to guarantee the map is an isomorphism.

Extended the understanding of cohomology of principal congruence subgroups 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.