The lower central and derived series of the braid groups of the sphere
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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
- Received by editor(s): April 15, 2006
- Published electronically: March 3, 2009
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 3375-3399
- MSC (2000): Primary 20F36, 20F14; Secondary 20F05, 55R80, 20E26
- DOI: https://doi.org/10.1090/S0002-9947-09-04766-7
- MathSciNet review: 2491885