On the braid groups of 𝐸² and 𝑆²

Fadell, Edward; Buskirk, James Van

doi:10.1090/s0002-9904-1961-10570-3

Cited by 50 publications

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“…(2) Let n = 3. Then B 3 (S 2 ) is a ZS-metacyclic group (a group whose Sylow subgroups, commutator subgroup and commutator quotient group are all cyclic) of order 12 [14]. It follows from Proposition 2.1(1) that B 3 (S 2 ) Ab ∼ = Z 4 , and hence Γ 2 (B 3 (S 2 )) ∼ = Z 3 .…”

Section: The Lower Central Series Of B N (Smentioning

confidence: 98%

“…Along with the braid groups of the real projective plane, they are of particular interest, notably because they have non-trivial centre (which is also the case for the Artin braid groups) and torsion elements (which were characterised by Murasugi [38]). We briefly recall some of their properties [11,14,19,44]. If D 2 ⊆ S 2 is a topological disc, there is a group homomorphism ι : B n (D 2 ) −→ B n (S 2 ) induced by the inclusion.…”

Section: Generalities and Definitionsmentioning

confidence: 99%

“…where n ≥ 3 if M is the 2-sphere S 2 [11,14], n ≥ 2 if M is the real projective plane RP 2 [44], and n ≥ 1 otherwise [13], and where p * is the group homomorphism which geometrically corresponds to forgetting the last m−n strings, and i * is inclusion (we consider P m−n (M \{x 1 , . .…”

Section: Generalities and Definitionsmentioning

confidence: 99%

“…Fadell and Neuwirth gave various sufficient conditions for the existence of a geometric section for p in the general case [13]. For the sphere, it was known that there exists a section on the geometric level [14]. If M is the 2-torus, then Birman exhibited an explicit algebraic section for (1.3) for m = n + 1 and n ≥ 2 [5].…”

Section: Generalities and Definitionsmentioning

confidence: 99%

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The lower central and derived series of the braid groups of the sphere

Gonçalves¹,

Guaschi

2009

Trans. Amer. Math. Soc.

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Abstract. In this paper, we determine the lower central and derived series for the braid groups of the sphere. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is important in its own right.The braid groups of the 2-sphere S 2 were studied by Fadell, Van Buskirk and Gillette during the 1960s, and are of particular interest due to the fact that they have torsion elements (which were characterised by Murasugi). We first prove that for all n ∈ N, the lower central series of the n-string braid group B n (S 2 ) is constant from the commutator subgroup onwards. We obtain a presentation of Γ 2 (B n (S 2 )), from which we observe that Γ 2 (B 4 (S 2 )) is a semi-direct product of the quaternion group Q 8 of order 8 by a free group F 2 of rank 2. As for the derived series of B n (S 2 ), we show that for all n ≥ 5, it is constant from the derived subgroup onwards. The group B n (S 2 ) being finite and soluble for n ≤ 3, the critical case is n = 4 for which the derived subgroup is the above semi-direct product Q 8 F 2 . By proving a general result concerning the structure of the derived subgroup of a semi-direct product, we are able to determine completely the derived series of B 4 (S 2 ) which from (B 4 (S 2 )) (4) onwards coincides with that of the free group of rank 2, as well as its successive derived series quotients.

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Section: The Lower Central Series Of B N (Smentioning

confidence: 98%

Section: Generalities and Definitionsmentioning

confidence: 99%

Section: Generalities and Definitionsmentioning

confidence: 99%

Section: Generalities and Definitionsmentioning

confidence: 99%

See 2 more Smart Citations

The lower central and derived series of the braid groups of the sphere

Gonçalves¹,

Guaschi

2009

Trans. Amer. Math. Soc.

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“…Van Buskirk [10] in 1961. From this presentation one can easily obtain the presentation with two generators δ 1 , δ and the following relations:…”

Section: Figurementioning

confidence: 99%

On presentations of generalizations of braids with few generators

Vershinin

2007

J Math Sci

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Abstract. In his initial paper on braids E. Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin's presentation for various generalizations of braids.The diverse aspects of presentations of braid groups and their generalizations continue to attract attention [12], [5], [15].The canonical presentation of the braid group Br n was given by E. Artin [1] and is well known. It has the generators σ 1 , σ 2 , . . . , σ n−1 and relationsThere exist other presentations of the braid group. J. S. Birman, K. H. Ko and S. J. Lee [5] introduced the presentation with generators a ts with 1 ≤ s < t ≤ n and relations a ts a rq = a rq a ts for (t − r)(t − q)(s − r)(s − q) > 0, a ts a sr = a tr a ts = a sr a tr for 1 ≤ r < s < t ≤ n.The generators a ts are expressed in the canonical generators σ i as follows: a ts = (σ t−1 σ t−2 · · · σ s+1 )σ s (σ −1 s+1 · · · σ −1 t−2 σ −1 t−1 ) for 1 ≤ s < t ≤ n.

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