Linear representations of Aut(F_r) on the homology of representation varieties

TL;DR

This paper investigates how the automorphism group of a free group acts on the homology of representation varieties of a compact semisimple Lie group, revealing that the action's image depends only on the Lie algebra's rank.

Contribution

It computes the image of the Aut(F_r) action on homology and shows the kernel is the Torelli subgroup IA_r, depending only on the Lie algebra's rank.

Findings

The image of the representation depends solely on the rank of the Lie algebra.

The kernel of the representation is the Torelli subgroup IA_r.

The action's structure is explicitly characterized.

Abstract

Let G be a compact semisimple linear Lie group. We study the action of Aut(F_r) on the space H_*(G^r;\QQ). We compute the image of this representation and prove that it only depends on the rank of the Lie algebra of G. We show that the kernel of this representation is always the Torrelli subgroup IA_r of Aut(F_r).