More on the Virtual Haken Conjecture and friends.

Nicolas Bergeron and Dani Wise posted an interesting preprint “A boundary criterion for cubulation” on the arXiv this morning. Using Kahn and Markovic, and work of Ian Agol, they show

Theorem 6.5. If quasi-fuchsian surface subgroups of the fundamental group of a hyperbolic 3-manifold M are separable, then M is virtually fibered.

This has also been proved by Ian using the same approach, see his Georgia slides. The idea is to apply a construction of Sageev to build a CAT(0)-cube complex from a large collection of quasi-fuchsian surface subgroups, and then apply a theorem of Haglund and Wise to get enough residual control over the fundamental group to apply Ian’s earlier work on virtual fibering.

Another aspect of [BW] was the implicit announcement by Wise of the following result

Theorem. Suppose M is a hyperbolic 3-manifold containing an embedded incompressible quasi-fuchsian surface. Then the fundamental group of M is subgroup separable.

In particular, [BW] refers to a 175 page(?!) preprint by Wise showing this, though it doesn’t seem to be on his webpage yet. If correct, this would be another major breakthrough in the study of the Virtual Haken Conjecture.