New Obstructions for the Surjectivity of the Johnson Homomorphism of the Automorphism Group of a Free Group

Satoh, Takao

doi:10.1112/s0024610706022976

Cited by 32 publications

(52 citation statements)

References 13 publications

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“…In the free group case G 1 /G 2 , G 2 /G 3 and G 3 /G 4 are known [28,29] but as yet there is no general formula. In this paper we restrict ourselves to studying the abelianisation of T (A Γ ), using the generating set M Γ defined in Section 2.5.…”

Section: 1mentioning

confidence: 99%

Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups

Wade

2013

Journal of the London Mathematical Society

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Abstract. Let G be a real semisimple Lie group with no compact factors and finite centre, and let Λ be an irreducible lattice in G. Suppose that there exists a homomorphism from Λ to the outer automorphism group of a right-angled Artin group A Γ with infinite image. We give a strict upper bound to the real rank of G that is determined by the structure of cliques in Γ. An essential tool is the Andreadakis-Johnson filtration of the Torelli subgroup T (A Γ ) of Aut(A Γ ). We answer a question of Day relating to the abelianisation of T (A Γ ), and show that T (A Γ ) and its image in Out(A Γ ) are residually torsion-free nilpotent.

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Section: 1mentioning

confidence: 99%

Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups

Wade

2013

Journal of the London Mathematical Society

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“…Satoh [26] In general, however, to determine the structure of the image and the cokernel of τ k is quite difficult.…”

Section: Johnson Homomorphismsmentioning

confidence: 99%

“…The study of the Johnson homomorphisms was originally begun in 1980 by D. Johnson [10] who determined the abelianization of the Torelli subgroup of a mapping class group of a surface in [11]. Recently, the study of the Johnson filtration and the Johnson homomorphisms of Aut F n achieved good progress through the work of many authors, for example, [7], [12], [18], [19], [20], [24] and [26]. Through the images of the Johnson homomorphisms, we can study IA n using infinitely many pieces of a free abelian group of finite rank.…”

Section: Introductionmentioning

confidence: 99%

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

Satoh

2010

Trans. Amer. Math. Soc.

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Abstract. Let F n be a free group of rank n and F N n the quotient group of, where Γ 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 → F N n , we see that there appear only two irreducible components, the Morita obstruction S k H Q and the Schur-Weyl module of type H, in the cokernel of the rational Johnson homomorphism τ k,Q = τ k ⊗ id Q for k ≥ 5 and n ≥ k + 2.

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“…Moreover, by independent works of Cohen-Pakianathan [4,5], Farb [6] and Kawazumi [10], it is known that gr 1 (A n ) is isomorphic to the abelianization of IA n . For k = 2 and 3, the rank of gr k (A n ) is determined by Pettet [19] and Satoh [21] respectively. For k ≥ 4, however, it seems that there are few results for the structure of gr k (A n ).…”

Section: Introductionmentioning

confidence: 99%

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

Satoh

2011

Proc. Amer. Math. Soc.

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Abstract. Let F n be a free group of rank n with basis x 1 , x 2 , . . . , x n . We denote by S n the subgroup of the automorphism group of F n consisting of automorphisms which fix each of x 2 , . . . , x n and call it the McCool stabilizer subgroup. Let IS n be a subgroup of 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 IS n and its graded quotients. Then we show that the Johnson filtration of S n coincides with the lower central series of IS n .

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New Obstructions for the Surjectivity of the Johnson Homomorphism of the Automorphism Group of a Free Group

Abstract: In this paper we construct new obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group. We also determine the structure of the cokernel of the Johnson homomorphism for degrees 2 and 3.

Cited by 32 publications

References 13 publications

Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups

Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups

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

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

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