The \(\text{mod-}p\) cohomology of \(\text{GL}(2p-2,\mathbb{Z})\) (Q1913977)

The author studies the Farrell cohomology with \(\mathbb{Z}/p\mathbb{Z}\) coefficients of the group \(\Gamma = \text{GL}(2p - 2,\mathbb{Z})\) when \(p\) is an odd prime. This is the first matrix size where rank two elementary abelian \(p\)-groups occur. He uses the spectral sequence of Brown [see last chapter in \textit{K. S. Brown}, Cohomology of Groups (Graduate Texts Math. 87, Springer 1982; Zbl 0584.20036)] which expresses the Farrell cohomology of \(\Gamma\) in terms of a \(\Gamma\)-equivariant system of coefficients \(\mathcal F\) of Farrell cohomology groups on the simplicial complex \(\mathcal A\) of elementary abelian \(p\)-subgroups. The description of \(\mathcal A\) and \(\mathcal F\) requires a classification of the elementary abelian \(p\)-subgroups of \(\Gamma\) and their normalizers. This occupies the first half of the paper. Then \(\mathcal F\) is partially computed, through a computation of ordinary cohomology of the normalizers. He gets enough of the spectral sequence to obtain large lower bounds on the Farrell cohomology of \(\Gamma\).