On a generalization of the double coset formula
The double coset formula is a very useful tool in computations of group homology for finite groups. In this paper we prove a generalization to the case of infinite index subgroups, using locally finite homology. In this context a transfer can be constructed, and a double coset formula, which is very similar to the usual one, holds. As an application, we prove an estimate for the rank of the homology at the \(vcd\) of a congruence subgroup of \(\text{Sl}_n(\mathbb{Z})\). More precisely, let \(\Gamma(n,p)\) be the subgroup of \(\text{Sl}_n(\mathbb{Z})\) consisting of matrices congruent to identity mod \(p\), \(n\geq 2\), \(p\geq 3\), \(p\) a prime. Then it is known [see \textit{A. Borel} and \textit{J.-P. Serre}, Comment. Math. Helv. 48, 436-491 (1973; Zbl 0274.22011)]\ that the group \(H_{n(n-1)/2}(\Gamma(n,p),\mathbb{Z})\) is a finitely generated free abelian group. The result is the following Theorem 1. The rank of \(H_{n(n-1)/2}(\Gamma(n,p),\mathbb{Z})\) is at least \(\geq p^{{n\choose 2}}((p-1)/2)^{n-1}\). This improves the result of \textit{R. Lee} and \textit{R. H. Szczarba} [Invent. Math. 33, 15-53 (1976; Zbl 0332.18015)]\ by the factor \(((p-1)/2)^{n-1}\). While the above estimate can be derived without using our double coset formula, we believe this setting makes the main idea more apparent.
scientific article; zbMATH DE number 1054072