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

2009
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Transactions of the American Mathematical Society
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Our aim is to determine the lower central series (LCS) and derived series (DS) for the braid groups of the sphere and of the finitely-punctured sphere. We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string braid group B\_n(S^2) is constant from the commutator subgroup onwards, and that \Gamma\_2(B\_4(S^2)) is a semi-direct product of the quaternion group by a free group of rank 2. For n=4, we determine the DS of B\_4(S^2), as well as its quotients. For n \geq 1, the

doi:10.1090/s0002-9947-09-04766-7
fatcat:74yx4lasorftzokfkbm47y5wcu