The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group
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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 $\mathrm {IS}_n$.References
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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
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2748417