Exact sequences, lower central series and representations of surface braid groups

Abstract

We consider exact sequences and lower central series of (surface) braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recently by An and Ko and independently by Blanchet.

Lower central series and split extensions

“…As noted in [4,Proposition 6.3], the property of a split extension B " A ¸ϕ C being an almost direct product does not depend on the choice of splitting. That is, if σ 1 : C Ñ B is another splitting of the projection β : B Ñ C, and if ϕ 1 : C Ñ AutpAq is the corresponding monodromy action, then the split extension B " A¸ϕ1 C is again an almost direct product.…”

“…Arguing as in the proofs of [4,Proposition 6.3] and [3, Proposition 3.2], it is readily seen that the property of a split extension being a rational almost direct product does not depend on the choice of splitting.…”

Alexander invariants and cohomology jump loci in group extensions

“…Among the successive quotients of the derives series of a group G, the second one plays a special role. The Alexander invariant of G is the abelian group (6) BpGq ≔ G 1 {G 2 , viewed as a module over the group-ring ZrG ab s; alternatively, BpGq " G 1 ab " H 1 pG 1 ab ; Zq. Addition in BpGq is induced from multiplication in G, to wit, pxG 2 q `pyG 2 q " xyG 2 for all x, y P G 1 , while scalar multiplication is induced from conjugation in the maximal metabelian quotient, G{G 2 , via the exact sequence (7) 1…”