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The Farrell cohomology of \(\text{Sp}(p-1,\mathbb{Z})\). (English) Zbl 1025.20033

Let \(\text{Sp}(p-1,\mathbb{Z})\) be the symplectic group over the integers, where \(p\) is an odd prime number with odd relative class number \(h^-\). The author computes the Farrell cohomology (\(\widehat H^*\)) and the period of these groups.
The main theorems are Theorem 3.7. Let \(p\) an odd prime for which \(h^-\) is odd. Then \[ \widehat H^*((\text{Sp}(p-1),\mathbb{Z}),\mathbb{Z})_{(p)}\cong\prod_{\substack{ k\mid p-1\\ k\text{ odd}}}(\prod_1^{\widetilde{\mathcal K}_k}\mathbb{Z}/p\mathbb{Z}[x^k,x^{-k}]), \] where \(\widetilde{\mathcal K}_k\) denotes the number of conjugacy classes of subgroups of order \(p\) of \(\text{Sp}(p-1,\mathbb{Z})\) for which \(|N/C|=k\), \(N\) and \(C\) denotes the normalizer and centralizer of the corresponding subgroups.
As these groups have periodic cohomology, the following is of interest. Theorem 3.8. Let \(p\) be an odd prime for which \(h^-\) is odd and let \(y\) be such that \(p-1=2^ry\) and \(y\) is odd. Then the period of \(\widehat H^*(\text{Sp}(p-1),\mathbb{Z})_{(p)}\) is \(2y\). – Several examples are provided.

MSC:

20G10 Cohomology theory for linear algebraic groups
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