On the augmentation quotients of the IA-automorphism group of a free group

Abstract

We study the augmentation quotients of the IA-automorphism group of a free group and a free metabelian group. First, for any group $G$, we construct a lift of the $k$–th Johnson homomorphism of the automorphism group of $G$ to the $k$–th augmentation quotient of the IA-automorphism group of $G$. Then we study the images of these homomorphisms for the case where $G$ is a free group and a free metabelian group. As a corollary, we detect a $\mathbb{Z}$–free part in each of the augmentation quotients, which can not be detected by the abelianization of the IA-automorphism group.