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

Abstract: 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 of the projective plane were originally studied by Van Buskirk during the 1960s, and are of particular interest due to the fact that they have torsion. The group (resp. ) is isomorphic to the cyclic group of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If , we first prove that the lower central series of is constant from the commutator subgroup onwards. We observe that is isomorphic to , where denotes the free group of rank k, and denotes the quaternion group of order 8, and that is an extension of an index 2 subgroup K of by . As for the derived series of , we show that for all , it is constant from the derived subgroup onwards. The group being finite and soluble for , the critical cases are . We are able to determine completely the derived series of . The subgroups , and are isomorphic respectively to , and , and we compute the derived series quotients of these groups. From onwards, the derived series of , as well as its successive derived series quotients, coincide with those of . We analyse the derived series of and its quotients up to , and we show that is a semi-direct product of by . Finally, we give a presentation of . [Copyright &y& Elsevier]