On the stable cohomology of the IA-automorphism groups of free groups

TL;DR

This paper investigates the stable rational cohomology of IA-automorphism groups of free groups, combining classical theorems and spectral sequences to propose conjectural algebraic structures and relate to surface Torelli groups.

Contribution

It computes the twisted first cohomology of automorphism groups of free groups and proposes a new conjectural algebraic structure for their stable rational cohomology.

Findings

Computed the twisted first cohomology of Aut(F_n)

Studied the stable rational cohomology of IA_n groups

Proposed conjectural algebraic structures for cohomology

Abstract

Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. By combining the Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$. We also study the stable rational cohomology of the IA-automorphism group $\mathrm{IA}_n$ of $F_n$. We propose a conjectural algebraic structure of the stable rational cohomology of $\mathrm{IA}_n$, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces.