The lower central and derived series of the braid groups of the projective plane

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell–Neuwirth short exact sequences, but the problem is interesting in its own right.The n-string braid groups Bn(RP2) of the projective plane RP2 were originally studied by Van Buskirk during the 1960s, and are of particular interest due to the fact that they have torsion. The group B1(RP2) (resp. B2(RP2)) is isomorphic to the cyclic group Z2 of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n>2, we first prove that the lower central series of Bn(RP2) is constant from the commutator subgroup onwards. We observe that Γ2(B3(RP2)) is isomorphic to (F3⋊Q8)⋊Z3, where Fk denotes the free group of rank k, and Q8 denotes the quaternion group of order 8, and that Γ2(B4(RP2)) is an extension of an index 2 subgroup K of P4(RP2) by Z2⊕Z2. As for the derived series of Bn(RP2), we show that for all n⩾5, it is constant from the derived subgroup onwards. The group Bn(RP2) being finite and soluble for n⩽2, the critical cases are n=3,4. We are able to determine completely the derived series of B3(RP2). The subgroups (B3(RP2))(1), (B3(RP2))(2) and (B3(RP2))(3) are isomorphic respectively to (F3⋊Q8)⋊Z3, F3⋊Q8 and F9×Z2, and we compute the derived series quotients of these groups. From (B3(RP2))(4) onwards, the derived series of B3(RP2), as well as its successive derived series quotients, coincide with those of F9. We analyse the derived series of B4(RP2) and its quotients up to (B4(RP2))(4), and we show that (B4(RP2))(4) is a semi-direct product of F129 by F17. Finally, we give a presentation of Γ2(Bn(RP2)).