On the stable cohomology of $\mathrm{GL}(n, \mathbb{Z})$, $\mathrm{Aut}(F_n)$ and $\mathrm{IA}_n$

Kazuo Habiro, Mai Katada

Published 2022-11-24Version 1

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. We improve the stable range in two ways by using ideas of Borel and Kupers-Miller-Patzt. By combining the improved 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.