For an arbitrary semi-direct product, we give a general description of its lower central series and an estimation of its derived series. In the second part of the paper, we study these series for the full braid group B n (M) and pure braid group P n (M) of a compact surface M , orientable or non-orientable, the aim being to determine the values of n for which B n (M) and P n (M) are residually nilpotent or residually soluble. We first solve this problem in the case where M is the 2-torus. We then use the results of the first part of the paper to calculate explicitly the lower central series of P n (K), where K is the Klein bottle. 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.
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 nonorientable 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.
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