On the mod - p cohomology of Out(F_{2(p-1)}

TL;DR

This paper investigates the mod-p cohomology of Out(F_{2(p-1)}) for prime p, providing explicit calculations for p=3 and recursive descriptions for larger primes, advancing understanding of automorphism groups of free groups.

Contribution

It offers the first complete computation of the mod-3 cohomology of Out(F_4) and introduces a recursive method for larger primes based on elementary abelian p-subgroups.

Findings

Complete cohomology calculation for p=3

Recursive description for p>3

Cohomology of elementary abelian p-subgroups

Abstract

We study the mod-p cohomology of the group Out(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 Out(F_4) (which is 5). More precisley, 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 Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n).