On the top dimensional cohomology groups of congruence subgroups of $\text{SL}_n(\mathbb{Z})$

TL;DR

This paper investigates the top degree cohomology of principal congruence subgroups of SL(n,Z), providing partial proofs of a conjecture, explicit calculations for p=5, and improved bounds for p≥5.

Contribution

It partially proves and disproves Lee-Szczarba's conjecture, computes cohomology for p=5, and enhances bounds on cohomology ranks for larger primes.

Findings

The natural map is always surjective.

Injectivity holds only for p ≤ 5.

Complete calculation of H^{n(n-1)/2}(Γ_n(5)).

Abstract

Let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of $\Gamma_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let $\mathcal{T}_n(\mathbb{Q})$ denote the Tits building of $\text{SL}_n(\mathbb{Q})$. Lee-Szczarba conjectured that $H^{\binom{n}{2}}(\Gamma_n(p))$ is isomorphic to $\widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/\Gamma_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\binom{n}{2}}(\Gamma_n(p)) \rightarrow \widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/\Gamma_n(p))$ is always surjective, but is only injective for $p \leq 5$. In particular, we completely calculate $H^{\binom{n}{2}}(\Gamma_n(5))$ and improve known lower bounds for the ranks of $H^{\binom{n}{2}}(\Gamma_n(p))$ for $p \geq 5$.