Lower central and derived series of semi-direct products, and applications to surface braid groups

John Guaschi (1) , Carolina de Miranda E Pereiro (2)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 UFES - Universidade Federal do Espirito Santo

Let M be a compact surface, either orientable or non-orientable. We study the lower central and derived series of the braid and pure braid groups of M in order to determine the values of n for which B_n(M) and P_n(M) are residually nilpotent or residually soluble. First, we solve this problem for the case where M is the 2-torus. We then give a general description of these series for an arbitrary semi-direct product that allows us to calculate explicitly the lower central series of P_2(K), where K is the Klein bottle, and to give an estimate for the derived series of P_n(K). Finally, if M is a non-orientable compact surface without boundary, we determine the values of n for which B_n(M) is residually nilpotent or residually soluble in the cases that were not already known in the literature.

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

John Guaschi, Carolina de Miranda E Pereiro. Lower central and derived series of semi-direct products, and applications to surface braid groups. Journal of Pure and Applied Algebra, 2020, 224 (7), pp.106309. ⟨10.1016/j.jpaa.2020.106309⟩. ⟨hal-01714012v2⟩