## Algebraic & Geometric Topology

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

Takao Satoh

#### 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 $ℤ$–free part in each of the augmentation quotients, which can not be detected by the abelianization of the IA-automorphism group.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 2 (2012), 1239-1263.

Dates
Revised: 22 March 2012
Accepted: 23 March 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715388

Digital Object Identifier
doi:10.2140/agt.2012.12.1239

Mathematical Reviews number (MathSciNet)
MR2928912

Zentralblatt MATH identifier
1261.20036

#### Citation

Satoh, Takao. On the augmentation quotients of the IA-automorphism group of a free group. Algebr. Geom. Topol. 12 (2012), no. 2, 1239--1263. doi:10.2140/agt.2012.12.1239. https://projecteuclid.org/euclid.agt/1513715388

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