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The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group


Author: Takao Satoh
Journal: Trans. Amer. Math. Soc. 361 (2009), 2085-2107
MSC (2000): Primary 20F28; Secondary 20J06
DOI: https://doi.org/10.1090/S0002-9947-08-04767-3
Published electronically: November 5, 2008
MathSciNet review: 2465830
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Abstract: In this paper, we determine the cokernel of the $ k$-th Johnson homomorphisms of the automorphism group of a free metabelian group for $ k \geq 2$ and $ n \geq 4$. As a corollary, we obtain a lower bound on the rank of the graded quotient of the Johnson filtration of the automorphism group of a free group. Furthermore, by using the second Johnson homomorphism, we determine the image of the cup product map in the rational second cohomology group of the IA-automorphism group of a free metabelian group, and show that it is isomorphic to that of the IA-automorphism group of a free group which is already determined by Pettet. Finally, by considering the kernel of the Magnus representations of the automorphism group of a free group and a free metabelian group, we show that there are non-trivial rational second cohomology classes of the IA-automorphism group of a free metabelian group which are not in the image of the cup product map.


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Additional Information

Takao Satoh
Affiliation: Department of Mathematics, Graduate School of Sciences, Osaka University, 1-16 Machikaneyama, Toyonaka-city, Osaka 560-0043, Japan
Email: takao@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-08-04767-3
Keywords: Automorphism group of a free metabelian group, Johnson homomorphism, second cohomology group, Magnus representation
Received by editor(s): May 17, 2007
Published electronically: November 5, 2008
Dedicated: Dedicated to Professor Shigeyuki Morita on the occasion of his 60th birthday
Article copyright: © Copyright 2008 American Mathematical Society

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