Representations of the Braid Group and Punctured Torus Bundles

Abstract

Abstract. In this short note, we consider a family of linear representations of the braidgroup and the fundamental group of a punctured torus bundle over the circle. We con-struct an irreducible (special) unitary representation of the fundamental group of a closed3-manifold obtained by the Dehn lling. 1. IntroductionRepresentations of the fundamental group have played an important role inthe study of 3-dimensional topology. For instance, studying the structure of theSL(2;C)-representation space gives us information about embedded surfaces in a3-manifold. Representing a knot group into well-known groups, including SU(2),to obtain geometric information of a knot has met with success in various context.In [4], motivated by the volume conjecture due to Kashaev-Murakami-Murakami[7], the rst author et al. introduced an in nite sequence of L 2 -torsion invariants,which approximates the simplicial volume, for a surface bundle over the circle. It isde ned by using the regular representations associated with the lower central seriesof the surface group. One of the results in [4] states that the geometric structureof a punctured torus bundle is detected by our rst invariant corresponding to thehomology representation. More precisely, the invariant is non-trivial if and only ifa punctured torus bundle admits the hyperbolic structure.In view of such a background, it seems natural to ask whether representationsinto other groups with the similar geometric information exist. In particular, wewould like to construct an invariant of 3-manifolds derived from the representation