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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). 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