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

Satoh, Takao

doi:10.1090/S0002-9939-2010-10550-6

The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group
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Proc. Amer. Math. Soc. 139 (2011), 1237-1245 Request permission

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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