Cohomology of automorphism groups of free groups with twisted coefficients

TL;DR

This paper computes the stable cohomology groups of automorphism and outer automorphism groups of free groups with twisted coefficients derived from their rational homology and cohomology, revealing complex behaviors related to symmetric power plethysm.

Contribution

It provides the first detailed calculations of these cohomology groups with twisted coefficients, highlighting differences between coefficients in homology and cohomology modules.

Findings

Stable cohomology groups are described via multiplicities in plethysm of symmetric powers.

Coefficients in $H_ ext{Q}$ behave less trivially than those in $H^*_ ext{Q}$.

Stable integral cohomology with coefficients in $H$ or $H^*$ is computed.

Abstract

We compute the groups $H^*(\mathrm{Aut}(F_n); M)$ and $H^*(\mathrm{Out}(F_n); M)$ in a stable range, where $M$ is obtained by applying a Schur functor to $H_\mathbb{Q}$ or $H^*_\mathbb{Q}$, respectively the first rational homology and cohomology of $F_n$. For reasons which are not conceptually clear, taking coefficients in $H_\mathbb{Q}$ and its related modules behaves in a far less trivial way than taking coefficients in $H^*_\mathbb{Q}$ and its related modules. The answer may be described in terms of stable multiplicities of irreducibles in the plethysm $\mathrm{Sym}^k \circ \mathrm{Sym}^l$ of symmetric powers. We also compute the stable integral cohomology groups of $\mathrm{Aut}(F_n)$ with coefficients in $H$ or $H^*$, respectively the first integral homology and cohomology of $F_n$, and compute the stable cohomology with coefficients in Schur functors of $H$ or $H^*$ modulo small primes.