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

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.