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.

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Can you give a “big picture” summary of where things stand now? How far are we from proving virtual fibering (or virtual Haken)? It’s a stupid question, but do either of these imply the Borel conjecture?

Comment by Daniel Moskovich — August 31, 2009 @ 8:49 am |

That’s an excellent idea, I’ll do that in an upcoming post.

As for the Borel conjecture in dimension 3, it follows from Perelman’s Geometrization Theorem, independent of the VHC.

Comment by Nathan Dunfield — August 31, 2009 @ 8:56 am |

If you want the latest on the Borel conjecture, see the paper

The Borel Conjecture for hyperbolic and CAT(0)-groups

Authors: Arthur Bartels, Wolfgang Lueck

at http://lanl.arxiv.org/PS_cache/arxiv/pdf/0901/0901.0442v1.pdf

and the paper

Survey on aspherical manifolds

Authors: Wolfgang Lueck

at http://lanl.arxiv.org/PS_cache/arxiv/pdf/0902/0902.2480v3.pdf

Comment by Mayer A. Landau — September 2, 2009 @ 2:48 pm |

I just noticed (two weeks late) that the Bergeron & Wise withdrew their paper…

Comment by Dave Futer — September 18, 2009 @ 3:49 pm |

Yes, I noticed that too. As I understand it, they are still claiming the particular result Theorem 6.5 above; rather its a different part of the paper that had an error in it. Regardless, Ian also has a proof of Theorem 6.5, so…

Comment by Nathan Dunfield — September 18, 2009 @ 3:57 pm |