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

- Date: 03/22/2007

Steve Boyer (Université du Québec à Montréal)

University of British Columbia

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.

Algebraic Topology Seminar 2007