Cohomology of $Aut(F_n)$ in the p-rank two case

Abstract

For odd primes p, we examine ${\hat{H}}^{\ast }(Aut(F_{2(p-1)});{\mathbb{Z}}_{(p)})$, the Farrell cohomology of the group of automorphisms of a free group $F_{2(p-1)}$ on $2(p-1)$ generators, with coefficients in the integers localized at the prime $(p)\subset \mathbb{Z}$. This extends results by Glover and Mislin, whose calculations yield ${\hat{H}}^{\ast }(Aut(F_n);{\mathbb{Z}}_{(p)})$ for $n\in \{p-1,p\}$ and is concurrent with work by Chen where he calculates ${\hat{H}}^{\ast }(Aut(F_n);{\mathbb{Z}}_{(p)})$ for $n\in \{p+1,p+2\}$. The main tools used are Ken Brown's ``normalizer spectral sequence'', a modification of Krstic and Vogtmann's proof of the contractibility of fixed point sets for outer space, and a modification of the Degree Theorem of Hatcher and Vogtmann.