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

TL;DR

This paper investigates the structure of the rational homology of the IA-automorphism groups of free groups, introducing the concept of Albanese homology and providing new bounds and explicit calculations in stable ranges.

Contribution

It defines and analyzes the Albanese homology of IA-automorphism groups, establishing stability properties, lower bounds, and explicit computations for the third homology group.

Findings

Established a stable representation of the Albanese homology

Provided a lower bound for the dimension of Albanese homology in stable range

Determined the third Albanese homology for groups with n ≥ 9

Abstract

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