The Lower Central and Derived Series of the Braid Groups of the Torus and of the Klein bottle

We are interested in studying the lower central and derived series of the braid group (resp. pure braid group) of the torus, Bn (T) (resp. Pn (T)), or of the Klein bottle, Bn (K) (resp. Pn (K)). For the braid groups of surfaces, these series have been studied in the case of the disc, sphere and the projective plane. Further, the lower central series of Bn (T) was studied by P. Bellingeri, S. Gervais et J. Guaschi where the authors show that Bn (T) is residually nilpotent if and only if n ≤ 2, and Pn (T) is residually nilpotent for all n. For K we have the same result, that Bn (K) is residually nilpotent if and only if n ≤ 2. As in the case of the torus, we conjecture that Pn (K) is residually nilpotent for all n, unfortunately, we have not been able to prove this conjecture, but we have been able to show a slightly weaker property, that Pn (K) is residually soluble for all n. We also show that Bn (T) and Bn (K) are residually soluble if and only if n ≤ 4.