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

Abstract

We determine the lower central and derived series of the n-string braid groups B_n(RP^2) of the real 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. For n=1,2, B_n(RP^2) is finite and its lower central and derived series are known. If n>2 (resp. n>4), we show that the lower central (resp. derived) series of B_n(RP^2) is constant from the commutator subgroup onwards, and we exhibit a presentation of the commutator subgroup. In the exceptional cases n=3,4, we determine explicitly the complete derived series of B_3(RP^2), we calculate the derived series of B_4(RP^2) up to and including its fifth term, and we obtain many of the derived series quotients in these two cases.

arXiv comment35 pages

References in corpus (4)

• The braid groups of the projective plane
• The lower central and derived series of the braid groups of the sphere and the punctured sphere
• Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence
• The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

Cited by in corpus (5)

• A survey of surface braid groups and the lower algebraic K-theory of their group rings
• The inclusion of configuration spaces of surfaces in Cartesian products, its induced homomorphism, and the virtual cohomological dimension of the braid groups of S^2 and RP^2
• When the lower central series stops: a comprehensive study for braid groups and their relatives
• The lower central and derived series of the braid groups of compact surfaces
• Lower central series, surface braid groups, surjections and permutations