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

TL;DR

This paper analyzes the structure of the lower central and derived series of braid groups on the real projective plane, revealing their behavior for different numbers of strands and providing explicit presentations and series calculations.

Contribution

It provides the first detailed determination of the lower central and derived series of B_n(RP^2) for all n, including explicit presentations and series for special cases.

Findings

For n=1,2, the series are known due to finiteness.

For n>2, the series stabilize from the commutator subgroup onward.

Explicit derived series are computed for n=3,4, with series quotients identified.

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.