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

Abstract

Borel's stability and vanishing theorem gives the stable cohomology of GL(𝑛, ℤ) with coefficients in algebraic GL(𝑛, ℤ)-representations. By combining the Borel theorem with the Hochschild–Serre spectral sequence, we compute the twisted first cohomology of the automorphism group Aut(𝐹_𝑛) of the free group 𝐹_𝑛 of rank 𝑛. We also study the stable rational cohomology of the IA-automorphism group IA_𝑛 of 𝐹_𝑛. We propose a conjectural algebraic structure of the stable rational cohomology of 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.