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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Daciberg Lima Gonçalves and John Guaschi
Journal: Trans. Amer. Math. Soc. 361 (2009), 3375-3399
MSC (2000): Primary 20F36, 20F14; Secondary 20F05, 55R80, 20E26
Published electronically: March 3, 2009
MathSciNet review: 2491885
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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 $ \mathbb{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\in\mathbb{N}$, the lower central series of the $ n$-string braid group $ B_n(\mathbb{S}^2)$ is constant from the commutator subgroup onwards. We obtain a presentation of $ \Gamma_2(B_n(\mathbb{S}^2))$, from which we observe that $ \Gamma_2(B_4(\mathbb{S}^2))$ is a semi-direct product of the quaternion group $ \mathcal{Q}_8$ of order $ 8$ by a free group $ \mathbb{F}_2$ of rank $ 2$. As for the derived series of $ B_n(\mathbb{S}^2)$, we show that for all $ n\geq 5$, it is constant from the derived subgroup onwards. The group $ B_n(\mathbb{S}^2)$ being finite and soluble for $ n\leq 3$, the critical case is $ n=4$ for which the derived subgroup is the above semi-direct product $ \mathcal{Q}_8\rtimes \mathbb{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(\mathbb{S}^2)$ which from $ (B_4(\mathbb{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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Additional Information

Daciberg Lima Gonçalves
Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de Saõ Paulo, Caixa Postal 66281, Ag. Cidade de São Paulo, CEP: 05314-970, São Paulo, SP, Brazil
Email: dlgoncal@ime.usp.br

John Guaschi
Affiliation: Laboratoire de Mathématiques Emile Picard, UMR CNRS 5580, UFR-MIG, Université Toulouse III, 31062 Toulouse Cedex 9, France
Address at time of publication: Laboratoire de Mathématiques Nicolas Oresme, UMR CNRS 6139, Université de Caen BP 5186, 14032 Caen Cedex, France
Email: guaschi@math.unicaen.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04766-7
PII: S 0002-9947(09)04766-7
Keywords: Surface braid group, sphere braid group, lower central series, derived series, configuration space, exact sequence
Received by editor(s): April 15, 2006
Published electronically: March 3, 2009
Article copyright: © Copyright 2009 American Mathematical Society