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

TL;DR

This paper investigates the algebraic structure of the Albanese homology of the IA-automorphism groups of free groups, revealing their representation properties and connections to stable cohomology of automorphism groups.

Contribution

It determines the representation structure of the Albanese homology for IA-automorphism groups and their stable variants, linking to the cohomology of automorphism groups.

Findings

Representation structure of Albanese homology for n ≥ 3×degree

Stable Albanese homology of outer automorphism groups identified

Relation established between Albanese (co)homology and automorphism group cohomology

Abstract

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