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Virtually fibred Montesinos Links

William Thurston conjectured over twenty years ago that every compact hyperbolic 3manifold whose boundary is a possibly empty union of tori is finitely covered by a manifold which fibres over the circle. If true, it provides a significant amount of global information about the topology of such manifolds. The first non-trivial examples supporting the conjecture were obtained by Gabai [Ga1] (1986) and Reid [Re] (1995). In the 1999 paper [AR], Aitchison and Rubinstein found combinatorial conditions on certain polyhedral decompositions of 3-manifolds which guarantee the existence of a finite cover which fibres over the circle. Chris Leininger showed that every manifold obtained by Dehn filling one component of the Whitehead link exterior is finitely covered by a surface bundle [Le] (2002), and more recently Genevieve Walsh verified Thurston’s conjecture for the exteriors of spherical Montesinos link exteriors [Wa] (2005), which includes all 2-bridge links. Jack Button [Bu] (2005) has determined many examples of nonfibred virtually fibred 3-manifolds in the CallahanHildebrand-Weeks and Hodgson-Weeks censuses, including over 100 closed examples. Most recently Ian Agol [A] (2008) gave some group theoretic criteria for a 3-manifold group which imply the virtually fibering property of the underlying manifold. 3-manifolds satisfying the criteria include all arithmetic hyperbolic link complements.