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

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

2010-05-21

arxiv.org/abs/1005.3906v1

Journal of Algebra 331, 1 (2011) 96-129

Mathematics

Geometric Topology

35 pages

Scientific paper

10.1016/j.jalgebra.2010.12.007

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.

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