The stable Albanese homology of the IA-automorphism groups of free groups

Abstract

The IA-automorphism group ${\mathrm{IA}}_n$ of the free group $F_n$ of rank $n$ is a normal subgroup of the automorphism group $\mathrm{Aut}(F_n)$ of $F_n$. We study the Albanese homology of ${\mathrm{IA}}_n$, which is the quotient of the rational homology of ${\mathrm{IA}}_n$ defined as the image of the map induced by the abelianization map of ${\mathrm{IA}}_n$ on homology. The Albanese homology of ${\mathrm{IA}}_n$ is an algebraic $\mathrm{GL}(n,\mathbb{Q})$-representation. We determine the representation structure of the Albanese homology of ${\mathrm{IA}}_n$ for $n$ greater than or equal to three times the homological degree. We also determine the structure of the stable Albanese homology of the analogue of ${\mathrm{IA}}_n$ to the outer automorphism group of $F_n$. Moreover, we identify the relation between the stable Albanese (co)homology of ${\mathrm{IA}}_n$ and the stable cohomology of $\mathrm{Aut}(F_n)$ with certain twisted coefficients.

12 pages