THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE
Abstract
Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class of m-string braid groups Bm(𝕊2\{x1,…,xn}) of the n-punctured sphere includes the usual Artin braid groups Bm (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type 
(for n = 3).
We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of Bm is completely determined for all m ∈ ℕ (respectively, for all m ≠ 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients.
When n ≥ 2, we prove that the lower central series (respectively, derived series) of Bm(𝕊2\{x1,…,xn}) is constant from the commutator subgroup onwards for all m ≥ 3 (respectively, m ≥ 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B2(𝕊2\{x1,x2}) admits various interpretations, as the Baumslag–Solitar group BS(2,2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B2(𝕊2\{x1,x2}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product ℤ2 * ℤ. Further, its lower central series quotients Γi/Γi + 1 are direct sums of copies of ℤ2, the number of summands being determined explicitly. In the case m ≥ 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Γi/Γi + 1 are 2-elementary finitely-generated groups.
