The Farrell cohomology of SP (𝑝−1, ℤ)

Abstract

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 $\mathrm{U}((p-1)/2)$.