Stable rational homology of the IA-automorphism groups of free groups

The rational homology of the IA-automorphism group ${\mathrm{IA}}_n$ of the free group $F_n$ is still mysterious. We study the quotient of the rational homology of ${\mathrm{IA}}_n$ that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of ${\mathrm{IA}}_n$. We obtain a representation-stable $\mathrm{GL}(n,\mathbb{Q})$-subquotient of the Albanese homology of ${\mathrm{IA}}_n$, which conjecturally coincides with the entire Albanese homology of ${\mathrm{IA}}_n$. In particular, we obtain a lower bound of the dimension of the Albanese homology of ${\mathrm{IA}}_n$ for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of ${\mathrm{IA}}_n$ for $n\ge 9$. We also study the Albanese homology of an analogue of ${\mathrm{IA}}_n$ to the outer automorphism group of $F_n$ and the Albanese homology of the Torelli groups of surfaces. Moreover, we study the relation between the Albanese homology of ${\mathrm{IA}}_n$ and the cohomology of $\mathrm{Aut}(F_n)$ with twisted coefficients.