On the mod-p cohomology of Out(F2(p−1))

We study the mod- p cohomology of the group O u t ( F n ) of outer automorphisms of the free group F n in the case n = 2 ( p − 1 ) which is the smallest n for which the p -rank of this group is 2 . For p = 3 we give a complete computation, at least above the virtual cohomological dimension of O u t ( F 4 ) (which is 5). More precisely, we calculate the equivariant cohomology of the p -singular part of outer space for p = 3 . For a general prime p > 3 we give a recursive description in terms of the mod- p cohomology of A u t ( F k ) for k ≤ p − 1 . In this case we use the O u t ( F 2 ( p − 1 ) ) -equivariant cohomology of the poset of elementary abelian p -subgroups of O u t ( F n ) .