A family of representations of braid groups on surfaces

Abstract

We propose a family of homological representations of the braid groups on surfaces. This family extends linear representations of the braid groups on a disc, such as the Burau representation and the Lawrence-KrammerBigelow representation.

Abelian and Metabelian Quotient Groups of Surface Braid Groups

“…This latter remark was made in [1] under certain conditions of a homological nature. Within a purely algebraic framework, we prove this non-existence result with fewer conditions than those of [1]. Let us start recalling that B n may be seen as the mapping class group of D n , thus giving rise to an action of B n on π 1 (D n ), the latter being isomorphic to the free group F n on n generators [9].…”

“…Finally in Section 5, we describe an algebraic approach to the Burau and Bigelow-Krammer-Lawrence representations that is based on the lower central series, and we explain why it is not possible to extend them to representations of surface braid groups. This latter remark was made in [1] under certain conditions of a homological nature. Within a purely algebraic framework, we prove this non-existence result with fewer conditions than those given in [1].…”

“…In [1,Section 3], the authors consider also a group H Σ that is defined by its group presentation. One may check using this presentation that H Σ is isomorphic to the quotient B k,n ( Σ g )/Γ 3 (B k,n ( Σ g )) / σ=1 .…”

“…In [1], An and Ko described an extension of the Bigelow-Krammer-Lawrence representations of B n to surface braid groups B n ( Σ g ) based on the regular covering associated to a projection map Φ Σ : B k ( Σ g,n ) −→ G Σ of the braid group B k ( Σ g,n ) onto a specific group G Σ , which can be seen as a kind of generalised Heisenberg group and that is constructed as a subgroup of a group H Σ . The group H Σ is defined abstractly in terms of its group presentation, and is chosen to satisfy certain technical homological constraints (Section 3.1 of [1]). An and Ko prove a rigidity result for H Σ , which states intuitively that it is the 'best possible' group that satisfies the constraints.…”

“…We will in fact obtain a stronger result, by proving that some of the above results remain true when the assumptions on Φ G , ρ K and ρ H are relaxed. Remarking that the group H Σ of [1] is a quotient of B k,n ( Σ g )/Γ 3 (B k,n ( Σ g )), in Proposition 4.5, we exhibit an alternative proof of [1,Theorem 4.3]. This paper is organised as follows.…”