The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group
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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.References
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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
- Received by editor(s): May 17, 2007
- Published electronically: November 5, 2008
- © Copyright 2008 American Mathematical Society
- 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
- MathSciNet review: 2465830
Dedicated: Dedicated to Professor Shigeyuki Morita on the occasion of his 60th birthday