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The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group


Author: Takao Satoh
Journal: Proc. Amer. Math. Soc. 139 (2011), 1237-1245
MSC (2010): Primary 20F28; Secondary 20F14
DOI: https://doi.org/10.1090/S0002-9939-2010-10550-6
Published electronically: August 27, 2010
MathSciNet review: 2748417
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Abstract: Let $ F_n$ be a free group of rank $ n$ with basis $ x_1, x_2, \ldots , x_n$. We denote by $ \mathrm{S}_n$ the subgroup of the automorphism group of $ F_n$ consisting of automorphisms which fix each of $ x_2, \ldots, x_n$ and call it the McCool stabilizer subgroup. Let $ \mathrm{IS}_n$ be a subgroup of $ \mathrm{S}_n$ consisting of automorphisms which induce the identity on the abelianization of $ F_n$. In this paper, we determine the group structure of the lower central series of $ \mathrm{IS}_n$ and its graded quotients. Then we show that the Johnson filtration of $ \mathrm{S}_n$ coincides with the lower central series of $ \mathrm{IS}_n$.


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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/S0002-9939-2010-10550-6
Keywords: Automorphism group of a free group, IA-automorphism group, Johnson filtration
Received by editor(s): September 6, 2009
Received by editor(s) in revised form: February 19, 2010, and April 20, 2010
Published electronically: August 27, 2010
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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