Abelian and Metabelian Quotient Groups of Surface Braid Groups

Abstract

In this paper we study Abelian and metabelian quotients of braid groups fn oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell-Neuwirth fibrations. 2000

Lower central series, surface braid groups, surjections and permutations

“…For part (b), we just give the proof in the case b = 1. The general case may be obtained in a similar manner using the presentation of B 2 (Σ g,b ) given in [8]. As we mentioned above, a presentation of B 2 (Σ g,1 ) may be obtained by deleting relation (8) from the presentation of Theorem 2.2.…”

“…The general case may be obtained in a similar manner using the presentation of B 2 (Σ g,b ) given in [8]. As we mentioned above, a presentation of B 2 (Σ g,1 ) may be obtained by deleting relation (8) from the presentation of Theorem 2.2. Thus the proof given in Proposition 2.7 for Σ g is also valid in the case of Σ g,b , except that we can no longer conclude that σ 2 1 is of finite order, so the second factor in the direct product decomposition of B 2 (Σ g,1 )/Γ 3 (B 2 (Σ g,1 )) is Z.…”

“…Let Σ g,b be a compact, connected orientable surface of genus g with b ≥ 0 boundary components. A presentation for B n (Σ g,b ) may be found in [8,Proposition 3.1], and in the case b = 1, a presentation for B n (Σ g,1 ) may be obtained from that of B n (Σ g ) given in Theorem 2.2 by deleting relation (8). The case b = 0 was dealt with in Section 2, so we shall assume henceforth that b ≥ 1.…”

“…(iii) Suppose that n > m ≥ 3 and n = 4. Using the presentation of B m (Σ g,b ) given in [8,Proposition 3.1], the proof of Theorem 1.4 goes through in this case, the only difference being that B n (Σ g,b )/ σ 1 is isomorphic to Z 2g+b−1 . The result then follows by an argument similar to that given in case (3) of the proof of Theorem 1.1(a)(ii) in Section 2.2.…”

“…We define a map θ : B n (T 2 )/Γ 3 (B n (T 2 )) −→ S 8 on the generators of B n (T 2 )/Γ 3 (B n (T 2 )) by: θ(a 1 ) = (1, 3)(2, 4), θ(b 1 ) = (1, 5)(2, 6)(3, 7)(4, 8) and θ(σ) = (1, 2, 3, 4) (5,6,7,8).…”