The A-polynomial detects the trivial knot and other applications of character variety methods in 3-dimensional topology

William Thurston conjectured over twenty years ago that every compact hyperbolic 3-manifold whose boundary is a possibly empty union of tori is virtually fibred, that is, has a finite cover which fibres over the circle. If true, it provides a significant amount of global information about the topology of such manifolds. To date, there has been remarkably little evidence to support the conjecture. For instance, there is only one published non-trivial example of a closed virtually fibred hyperbolic rational homology 3-sphere. ( Non-trivial in this context means that the manifold neither fibres nor semi-fibres.) In this talk I will report on joint work with Xingru Zhang which shows that the conjecture holds for the exteriors of many Montesinos links. As a consequence, we construct an infinite family of closed virtually fibred hyperbolic rational homology 3-spheres. Another byproduct of the construction is that we are able to verify that the fundamental groups of the exteriors of many Montesinos links have a finite index bi-orderable subgroup.