On the top-dimensional cohomology groups of congruence subgroups of \(\operatorname{SL}(n,\mathbb{Z})\) (Q2024752)

The authors pursue the computation of the top-dimensional cohomology of the level \(p\) principal congruence subgroups \(\Gamma_n(p)\) in the classical modular groups \(\mathrm{SL}(n,\mathbb{Z})\), refining the approach of \textit{R. Lee} and \textit{R. H. Szczarba} [Invent. Math. 33, 15--53 (1976; Zbl 0332.18015)]. This approach is based on Borel-Serre duality \[ \mathrm{H}^N(\Gamma_n(p); \mathbb{Q}) \cong \mathrm{H}_0(\Gamma_n(p);\mathrm{St}(n)\otimes \mathbb{Q}), \] where \(N = \frac{n(n-1)}{2}\) is the virtual cohomological dimension, and \(\mathrm{St}(n)\) is the Steinberg module. The right hand side of this isomorphism are the co-invariants \((\mathrm{St}(n)\otimes \mathbb{Q})_{\Gamma_n(p)}\), and the approach focuses on computing them on the Tits building \(T_n\), making use of that the Steinberg module \(\mathrm{St}(n)\) is ismorphic to the reduced homology \(\widetilde{\mathrm{H}}_{n-2}(T_n)\) (the authors are tacitly using the convention that the reduced homology of the empty set is concentrated as a rank \(1\) module in degree \(-1\), as well as the fact that the Tits building for \(\mathrm{SL}(1)\) is the empty set). A conjecture stated by Lee and Szczarba, and proved for the prime \(p=3\), is that for \(n\geq 2\), the map \[ (\mathrm{St}(n)\otimes \mathbb{Q})_{\Gamma_n(p)} \mapsto \widetilde{\mathrm{H}}_{n-2}(T_n/\Gamma_n(p)) \] induced by the quotient map \(T_n \mapsto T_n/\Gamma_n(p)\) is an isomorphism. \textit{A. Ash} [Bull. Am. Math. Soc. 83, 367--368 (1977; Zbl 0349.22009)] did disprove this conjecture for \(n=3\) and \(p\geq 7\), using a method that apparently does not extend to other matrix ranks \(n\). Furthermore, \textit{A. Paraschivescu} [Duke Math. J. 89, No. 1, 1--8 (1997; Zbl 0895.20044)] provided a lower bound for the rank of \(\mathrm{H}^N(\Gamma_n(p))\). For all \(n \geq 2\), the authors establish a complete answer for the validity of Lee and Szczarba's conjecture, proving it for \(p \leq 5\), and showing that it fails for \(p > 5\), the map being proven surjective but not injective. Consequently, they substantially improve Paraschivescu's lower bound, and, for \(p=5\), provide a beautiful explicit formula for the rank of \(\mathrm{H}^N(\Gamma_n(5))\), which involves only addition, multiplication and recursion, so it can easily be evaluated from low to very high matrix ranks \(n\).