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

Borel’s stability and vanishing theorem gives the stable cohomology of $\mathrm{G}\mathrm{L}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{G}\mathrm{L}(n,\mathbb{Z})$-representations. By combining the Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ of the free group $F_n$ of rank $n$. We also study the stable rational cohomology of the IA-automorphism group ${\mathrm{I}\mathrm{A}}_n$ of $F_n$. We propose a conjectural algebraic structure of the stable rational cohomology of ${\mathrm{I}\mathrm{A}}_n$, 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.