On the Automorphisms of Free Groups and Free Nilpotent Groups

Andreadakis, S.

doi:10.1112/plms/s3-15.1.239

Cited by 162 publications

(281 citation statements)

References 3 publications

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“…(3) Are the groups Cb k n , k ≥ 1 is the subgroup of Aut(F n ), which consists of automorphisms acting trivially modulo the k-th term of the lower central series of F n . This chain was introduced in [1]. For which k ≥ 3, n ≥ 3 the groups IA k n are non-linear?…”

Section: Questionsmentioning

confidence: 99%

On Certain Questions of the Free Group Automorphisms Theory

Bardakov

Mikhailov

2008

Communications in Algebra

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Abstract. Certain subgroups of the groups Aut(F n ) of automorphisms of a free group F n are considered. Comparing Alexander polynomials of two poly-free groups Cb + 4 and P 4 we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohen-Pakianathan-Vershinin-Wu from [8]. The questions of linearity of subgroups of Aut(F n ) are considered. As an application of the properties of poison groups in the sense of Formanek and Procesi, we show that the groups of the type Aut(G * Z) for certain groups G and the subgroup of IA-automorphisms IA(F n ) ⊂ Aut(F n ) are not linear for n ≥ 3. This generalizes the recent result of Pettet that IA(F n ) are not linear for n ≥ 5.

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Section: Questionsmentioning

confidence: 99%

On Certain Questions of the Free Group Automorphisms Theory

Bardakov

Mikhailov

2008

Communications in Algebra

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“…Also, the homomorphism Aut(F n ) → Aut(F n /γ 3 (F n )), determined by the natural homomorphism F n → F n /γ 3 (F n ), is surjective [1] (equivalently, one says that the automorphism groups of finitely generated free nilpotent groups of class two are tame). It follows that in the automorphism group of any infinitely generated free nilpotent group of class two all finitary IA-automorphisms are contained in the normal closure of any analogue of K 12 .…”

Section: Stabilizing Everythingmentioning

confidence: 99%

Infinitely generated free nilpotent groups: completeness of the automorphism groups

Tolstykh¹

2009

Math. Proc. Camb. Phil. Soc.

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Abstract. We transfer the results of Dyer, Formanek and Kassabov on the automorphism towers of finitely generated free nilpotent groups to infinitely generated free nilpotent groups. We prove that the automorphism groups of infinitely generated free nilpotent groups are complete. By combining the results of Dyer, Formanek, Kassabov with the results in the present paper, one gets that the automorphism tower of any free nilpotent group terminates after finitely many steps. IntroductionBaumslag conjectured in the 1970s that the automorphism tower of a finitely generated free group (free nilpotent group) must be very short. Dyer and Formanek [8] justified the conjecture concerning finitely generated free groups in the "sharpest sense" by proving that the automorphism group Aut(F n ) of a non-abelian free group F n of finite rank n is complete. Recall that a group G is said to be complete if G is centreless and all automorphisms of G are inner; it then follows that Aut(G) ∼ = G. Thus Aut(Aut(F n )) ∼ = Aut(F n ), or, in other words, the height of the automorphism tower over F n is two. The proof of completeness of Aut(F n ) given by Dyer and Formanek in [8] has later been followed by the proofs given by Formanek [11], by Khramtsov [14], by Bridson and Vogtmann [3], and by the author [17]. The proof given in [17] works for arbitrary non-abelian free groups; thus the automorphism groups of infinitely generated free groups are also complete.Let F n,c denote a free nilpotent group of finite rank n 2 and of nilpotency class c 2. In [9] Dyer and Formanek studied the automorphism towers of free nilpotent groups F n,2 of class two. They showed that the group Aut(F n,2 ) is complete provided that n = 3. In the case when n = 3 the height of the automorphism tower of F n,2 is three. The main result of [18] states that the automorphism group of any infinitely generated free nilpotent group of class two is complete.In [10] Dyer and Formanek proved completeness of the automorphism groups of groups of the form F n /R ′ where R is a characteristic subgroup of F n which is contained in the commutator subgroup F ′ n of F n and F n /R is residually torsion-free nilpotent.In his Ph. D. thesis [13] Kassabov found an upper bound u(n, c) ∈ N for the height of the automorphism tower of F n,c in terms of n and c, thereby finally proving Baumslag's conjecture on finitely generated free nilpotent groups. By analyzing the function u(n, c) one can conclude that if c is small compared to n, then the height of the automorphism tower of F n,c is at most three.2000 Mathematics Subject Classification. 20F28 (20F18, 03C60).

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“…The group π has lower central series defined recursively by Γ 0 = π and Γ k+1 = [π, Γ k ] for k ≥ 0, where the bracket of two groups denotes their commutator group, and the k-th nilpotent quotient is defined as N k = π/Γ k . Thus, we have the exact sequence (1) 0−→Γ k /Γ k+1 −→N k+1 −→N k −→1…”

Section: Introductionmentioning

confidence: 99%

“…If ϕ ∈ M g, * acts trivially on N k and γ ∈ N k+1 , then ϕ(γ)γ −1 ∈ Ker(N k+1 →N k ), so by exactness of (1) together with a few arguments, there are mappings τ k : I g, * (k)−→Hom(N k+1 , Γ k /Γ k+1 ), and these are called the Johnson homomorphisms introduced in [13] [14] (see also prior works of Andreadakis [1] and Sullivan [39]). See [14] for a survey of the Torelli groups and [28][31] [32] for further results.…”

Section: Introductionmentioning

confidence: 99%

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Morita¹,

Penner²

2008

Math. Proc. Camb. Phil. Soc.

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Abstract. Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N . Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmüller space. The main tool for these results is the known mapping class group invariant ideal cell decomposition of the Teichmüller space.This new 1-cocycle is mapping class group equivariant, so various contractions of its powers yield various combinatorial (co)cycles of the moduli space of curves, which are also new. Our combinatorial construction can be related to former works of Kawazumi and the first-named author with the consequence that the algebra generated by the cohomology classes represented by the new cocycles is precisely the tautological algebra of the moduli space.There is finally a discussion of prospects for similarly finding cocycle lifts of the higher Johnson homomorphisms.

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On the Automorphisms of Free Groups and Free Nilpotent Groups

Cited by 162 publications

References 3 publications

On Certain Questions of the Free Group Automorphisms Theory

On Certain Questions of the Free Group Automorphisms Theory

Infinitely generated free nilpotent groups: completeness of the automorphism groups

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

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