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

We study the mod-pp cohomology of the group Out(Fn)Out(Fn) of outer automorphisms of the free group FnFn in the case n=2(p−1)n=2(p−1) which is the smallest nn for which the pp-rank of this group is 22. For p=3p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F4)Out(F4) (which is 5). More precisely, we calculate the equivariant cohomology of the pp-singular part of outer space for p=3p=3. For a general prime p>3p>3 we give a recursive description in terms of the mod-pp cohomology of Aut(Fk)Aut(Fk) for k≤p−1k≤p−1. In this case we use the Out(F2(p−1))Out(F2(p−1))-equivariant cohomology of the poset of elementary abelian pp-subgroups of Out(Fn)Out(Fn).