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

Satoh, Takao

doi:10.2140/agt.2012.12.1239

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

Takao Satoh

Algebraic & Geometric Topology 12 (2012) 1239–1263

DOI: 10.2140/agt.2012.12.1239

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.

Keywords

automorphism group of free groups, augmentation quotient, Johnson homomorphism

Mathematical Subject Classification 2010

Primary: 20F28

Secondary: 16S34

References

Publication

Received: 4 April 2011

Revised: 22 March 2012

Accepted: 23 March 2012

Published: 5 June 2012

Authors

Takao Satoh
Department of Mathematics, Faculty of Science Tokyo University of Science 1-3 Kagurazaka Shinjuku-ku Tokyo 162-8601 Japan

Volume 12, issue 2 (2012)