The Farrell Cohomology of $SP(p-1,Z)$

Let $p$ be an odd prime with odd relative class number $h^-$. In this article we compute the Farrell cohomology of $\mathrm{Sp}(p-1,\mathbb Z)$, the first $p$-rank one case. This allows us to determine the $p$-period of the Farrell cohomology of $\mathrm{Sp}(p-1,\mathbb Z)$, which is $2y$, where $p-1=2^r y$, $y$ odd. The $p$-primary part of the Farrell cohomology of $\mathrm{Sp}(p-1,\mathbb Z)$ is given by the Farrell cohomology of the normalizers of the subgroups of order $p$ in $\mathrm{Sp}(p-1,\mathbb Z)$. We use the fact that for odd primes $p$ with $h^-$ odd a relation exists between representations of $\mathbb Z/p\mathbb Z$ in $\mathrm{Sp}(p-1,\mathbb Z)$ and some representations of $\mathbb Z/p\mathbb Z$ in $\mathrm{U}((p-1)/2)$.

2000 Mathematics Subject Classification: 20G10

Keywords and Phrases: Cohomology theory

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