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

Daciberg Lima Gonçalves (1) , John Guaschi (2)

1 IME - Instituto de Matemática e Estatística
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: surface braid group; projective plane braid group; lower central series; derived series; configuration space; exact sequence

Domaines: Topologie géométrique [math.GT]; Théorie des groupes [math.GR]

Daciberg Lima Gonçalves, John Guaschi. The lower central and derived series of the braid groups of the projective plane. Journal of Algebra, 2011, 331 (1), pp.96-129. ⟨10.1016/j.jalgebra.2010.12.007⟩. ⟨hal-00485339⟩