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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

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

automorphism group of free groups, augmentation quotient, Johnson homomorphism
Mathematical Subject Classification 2010
Primary: 20F28
Secondary: 16S34
Received: 4 April 2011
Revised: 22 March 2012
Accepted: 23 March 2012
Published: 5 June 2012
Takao Satoh
Department of Mathematics, Faculty of Science
Tokyo University of Science
1-3 Kagurazaka
Tokyo 162-8601