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A reduction of the target of the Johnson homomorphisms of the automorphism group of a free group


Author: Takao Satoh
Journal: Trans. Amer. Math. Soc. 363 (2011), 1631-1664
MSC (2000): Primary 20F28; Secondary 20F12
DOI: https://doi.org/10.1090/S0002-9947-2010-05227-4
Published electronically: October 13, 2010
MathSciNet review: 2737281
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Abstract: Let $ F_n$ be a free group of rank $ n$ and $ F_n^N$ the quotient group of $ F_n$ by a subgroup $ [\Gamma_n(3), \Gamma_n(3)][[\Gamma_n(2),\Gamma_n(2)],\Gamma_n(2)]$, where $ \Gamma_n(k)$ denotes the $ k$-th subgroup of the lower central series of the free group $ F_n$. In this paper, we determine the group structure of the graded quotients of the lower central series of the group $ F_n^N$ by using a generalized Chen's integration in free groups. Then we apply it to the study of the Johnson homomorphisms of the automorphism group of $ F_n$. In particular, under taking a reduction of the target of the Johnson homomorphism induced from a quotient map $ F_n \rightarrow F_n^N$, we see that there appear only two irreducible components, the Morita obstruction $ S^k H_{\mathbf{Q}}$ and the Schur-Weyl module of type $ H_{\mathbf{Q}}^{[k-2, 1^2]}$, in the cokernel of the rational Johnson homomorphism $ \tau_{k, \mathbf{Q}}'=\tau_k' \otimes \mathrm{id}_{\mathbf{Q}}$ for $ k \geq 5$ and $ n \geq k+2$.


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Additional Information

Takao Satoh
Affiliation: Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawaoiwake cho, Sakyo-ku, Kyoto city 606-8502, Japan
Email: takao@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2010-05227-4
Keywords: Automorphism group of a free group, IA-automorphism group, Johnson homomorphisms, Chen’s integration on free groups
Received by editor(s): June 10, 2009
Received by editor(s) in revised form: October 15, 2009
Published electronically: October 13, 2010
Article copyright: © Copyright 2010 American Mathematical Society

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