# The Farrell cohomology of SP (𝑝−1, ℤ)

### Cornelia Busch

Katholische Universität Eichstätt, Germany

## Abstract

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

## Cite this article

Cornelia Busch, The Farrell cohomology of $SP(p−1,Z)$. Doc. Math. 7 (2002), pp. 239–254

DOI 10.4171/DM/126