The walled Brauer category and stable cohomology of $\mathrm{IA}_n$

Abstract: The IA-automorphism group is the group of automorphisms of the free group $F_n$ that act trivially on the abelianization $F_n^{\mathrm{ab}}$. This group is in many ways analoguous to Torelli groups of surfaces and their higher dimensional analogues. In recent work, the stable rational cohomology of such groups was studied by Kupers and Randal-Williams, using the machinery of so-called Brauer categories. In this paper, we adapt their methods to study the stable rational cohomology of the IA-automorphism group. We obtain a conjectural description of the algebraic part of the stable rational cohomology and prove that it holds up to degree $Q+1$, given the assumption that the stable cohomology groups are stably finite dimensional in degrees up to $Q$. In particular, this allows us to compute the algebraic part of the stable cohomology in degree 2, which we show agrees with the part generated by the first cohomology group via the cup product map and which has previously been computed by Pettet. In the appendix, written by Mai Katada, it is shown how the results of the paper can be applied to compute the stable Albanese (co)homology of the IA-automorphism group.