Brunnian braids and Lie algebras

Li, J.Y.; Vershinin, V. V.; Wu, Jie

doi:10.1016/j.jalgebra.2015.05.013

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…Using Chen groups, Cohen and Suciu gave explicit formulae for rank(L k (P n )) for k ∈ {1, 2, 3} and all n ≥ 2 [CS, Theorem 1.1 and page 46]. More generally, by [LVW,Theorem 4.6], the rank of L k (P n ) is given by equation ( 1). In practice, we may compute these ranks as follows.…”

Section: The Rank Of the Free Abelian Groupmentioning

confidence: 99%

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Almost-crystallographic groups as quotients of Artin braid groups

Gonçalves¹,

Guaschi²,

Ocampo³

2018

Preprint

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Let n, k ≥ 3. In this paper, we analyse the quotient group B n /Γ k (P n ) of the Artin braid group B n by the subgroup Γ k (P n ) belonging to the lower central series of the Artin pure braid group P n . We prove that it is an almost-crystallographic group. We then focus more specifically on the case k = 3. If n ≥ 5, and i f τ ∈ N is such that gcd(τ, 6) = 1, we show that B n /Γ 3 (P n ) possesses torsion τ if and only if S n does, and we prove that there is a oneto-one correspondence between the conjugacy classes of elements of order τ in B n /Γ 3 (P n ) with those of elements of order τ in the symmetric group S n . We also exhibit a presentation for the almost-crystallographic group B n /Γ 3 (P n ). Finally, we obtain some 4-dimensional almost-Bieberbach subgroups of B 3 /Γ 3 (P 3 ), we explain how to obtain almost-Bieberbach subgroups of B 4 /Γ 3 (P 4 ) and B 3 /Γ 4 (P 3 ), and we exhibit explicit elements of order 5 in B 5 /Γ 3 (P 5 ).

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Section: The Rank Of the Free Abelian Groupmentioning

confidence: 99%

“…These quotients have been widely studied, see [Hal, MKS] for example. In the case of P n , it is known that L q (P n ) is a free Abelian group of finite rank, and by [LVW,Theorem 4.6], its rank is given by: rank (L q…”

Section: Introductionmentioning

confidence: 99%

Almost-crystallographic groups as quotients of Artin braid groups

Gonçalves¹,

Guaschi²,

Ocampo³

2018

Preprint

View full text Add to dashboard Cite

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Almost-crystallographic groups as quotients of Artin braid groups

Gonçalves¹,

Guaschi

Ocampo

2019

Journal of Algebra

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On groups Gnk, braids and Brunnian braids

Kim

Manturov

2016

J. Knot Theory Ramifications

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In [7] the second author defined the k-free braid group with n strands G k n . These groups appear naturally as groups describing dynamical systems of n particles in some "general position". Moreover, in [10] the second author and I.M.Nikonov showed that G k n is closely related classical braids. The authors showed that there are homomorphisms from the pure braids group on n strands to G 3 n and G 4 n and they defined homomorphisms from G k n to the free product of Z 2 . That is, there are invariants for pure free braids by G 3 n and G 4 n . On the other hand in [6] D.A.Fedoseev and the second author studied classical braids with addition structures: parity and points on each strands. The authors showed that the parity, which is an abstract structure, has geometric meaning -points on strands. In [4], the first author studied G 2 n with parity and points. the author construct a homomorphism from G 2 n+1 to the group G 2 n with parity.In the present paper, we investigate the groups G 3 n and extract new powerful invariants of classical braids from G 3 n . In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids.

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Brunnian braids and Lie algebras

Abstract: Abstract. Brunnian braids have interesting relations with homotopy groups of spheres. In this work, we study the graded Lie algebra of the descending central series related to Brunnian subgroup of the pure braid group. A presentation of this Lie algebra is obtained.

Cited by 3 publications

References 21 publications

Almost-crystallographic groups as quotients of Artin braid groups

Almost-crystallographic groups as quotients of Artin braid groups

Almost-crystallographic groups as quotients of Artin braid groups

On groups Gnk, braids and Brunnian braids

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