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

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 class of m-string braid groups B_m(S^2)
{x_1,…,x_n} of the n-punctured sphere includes the Artin braid groups B_m, those of the annulus, and certain Artin and affine Artin groups. We extend results of Gorin and Lin, and show that the LCS (resp. DS) of B_m is determined for all m (resp. for all m\neq 4). For m=4, we obtain some elements of the DS. When n\geq 2, we prove that the LCS (resp. DS) of B_m(S^2)
{x_1,…,x_n} is constant from the commutator subgroup onwards for all m\geq 3 (resp. m\geq 5). We then show that B_2(S^2{x_1,x_2}) is residually nilpotent, that its LCS coincides with that of Z_2*Z, and that the \Gamma_i/\Gamma_{i+1} are 2-elementary finitely-generated groups. For m\geq 3 and n=2, we obtain a presentation of the derived subgroup and its Abelianisation. For n=3, we see that the quotients \Gamma_i/\Gamma_{i+1} are 2-elementary finitely-generated groups.