The lower central and derived series of the braid groups of compact surfaces

Abstract

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.

Lower central series, surface braid groups, surjections and permutations

“…The proof of the 'only if' part is similar to that of Proposition 2.8, and is left to the reader. (d) This follows from [30,Theorem 1.4].…”

“…The following theorem summarises some of the known results about the lower central series of braid groups of non-orientable surfaces without boundary [26,30,37], and is the analogue of Theorems 2.3 and 3.1. One may consult [6] for the case of pure braid groups.…”

“…(a) The statement follows in a straightforward manner using the presentation of Theorem 4.1. [30,Theorem 1.4]. The proof of the 'only if' part is similar to that of Proposition 2.8, and is left to the reader.…”

“…If g ≥ 2, the residual nilpotence of B 2 (U g ) follows by arguing as in the proof of Proposition 3.2(a), using the fact that P 2 (U g ) is residually 2-finite [6, Theorem 1.1]. Note that B 2 (U 2 ) is not nilpotent, since otherwise P 2 (U 2 ) would be nilpotent, but we know from [30,Theorem 5.4] that this is not the case. If g ≥ 3, the centre of the group B 2 (U g ) is trivial [39, Proposition 1.6], and as in Remark 2.6, we can prove that the group (17), these commutators are of order at most 2.…”

“…Define the homomorphism θ 3 : B n (Σ 3 )/Γ 3 (B n (Σ 3 )) −→ S 32 by θ 3 (a 1 ) = (2, 0, 2, 0, 2, 0, 2, 0), θ 3 (a 2 ) = (2, 2, 0, 0, 2, 2, 0, 0), θ 3 (a 3 ) = (2, 2, 2, 2, 0, 0, 0, 0), θ 3 (σ) = (1, 1, 1, 1, 1, 1, 1, 1), regarded as elements of S 32 , where each factor 1 denotes the cyclic permutation of length 4 associated to the four integers corresponding to these four positions, 2 denotes the square of this cyclic permutation, and 0 denotes the identity permutation associated to these four integers. Finally, let θ (13,14,15,16) (17,18,19,20) (21,22,23,24)• (25,26,27,28) (29,30,31,32).…”