Daniel Moskovich suggested that, in light of the recent advances, I summarize the state of the Virtual Haken Conjecture.

So let *M * be a closed hyperbolic 3-manifold. Then one has the following sequence of increasingly strong conjectures.

**Conjecture 1**: contains a surface subgroup.

**Conjecture 2:** *M* has a finite cover *N* which is Haken, i.e. contains a closed incompressible surface.

**Conjecture 3:** *M* has a finite cover *N* with .

**Conjecture 4:** For each , the manifold *M* has a finite cover *N* with .

**Conjecture 5**: *M* has a finite cover *N* which is large, i.e. surjects onto a free group of rank 2.

Here Conjecture 2 is the classical form of the VHC. Conjecture 3 could also be stated as *N* contains a finite cover which contains a non-separating incompressible surface. In addition one has the question about virtual fibering

**Conjecture 6:** *M* has a finite cover *N* which fibers over the circle.

Notice that Conjecture 6 immediately implies Conjectures 1-3, though it is not currently know to imply Conjectures 4-5. A surface subgroup of is either geometrically finite or geometrically infinite; in the former case, it’s a quasi-fuchsian group, and in the latter, Thurston and Bonahon showed it comes from a virtual fiber. Thus, in the language of Conjecture 1, Conjecture 6 says that contains a *geometrically infinite* surface group.

Tying these these questions together is

**Conjecture 7:** The fundamental group of *M* is subgroup seperable (a.k.a LERF), i.e. finitely generated subgroups are closed in the profinite topology.

In particular, Conjectures 1 and 7 imply all the rest; for Conjectures 2-5 a nice argument for this was given in [Lubotzky 1996], for Conjecture 6 this is the recent work of Agol and Bergeron-Wise.

The recent announcements are: Kahn-Markovic claim a proof of Conjecture 1, and Wise claims a proof of Conjecture 7 in the special case that M is a Haken manifold containing an embedded quasi-fuchsian surface. Assuming both proofs are correct, this would reduce all of the above questions to:

**Conjecture 8:** At least one of the quasi-fuchsian surface groups constructed by Kahn-Markvoic has a finite index subgroup which corresponds to an embedded surface in some finite cover of *M*.

Wise’s claimed result almost gives that the classical VHC (Conjecture 2) implies all the rest, but not quite; it mean that Conjecture 2 implies Conjectures 3 and 6, but not Conjectures 4, 5, and 7 in certain special cases. In particular the cover manifold *N* given by Conjecture 2 could fiber over the circle with and then Wise’s result does not apply. Conjecture 4, even for , does suffice to prove them all, since not every class in can come from a fibration when . This suggests it would be particularly interesting to show that Conjecture 6 implies Conjecture 4, but currently this is only known when the fiber has genus 2, by a nice result of Masters.

**Edit:** Last paragraph clarified.

You state that Wise’s claimed result gives that the VHC (conjecture 2) implies conjecture 6. Doesn’t your statement then imply that Wise’s claimed result currently gives conjecture 6 for all Haken manifolds?

Comment by Mayer A. Landau — September 1, 2009 @ 11:45 pm |

Yes, that’s correct.

Comment by Nathan Dunfield — September 2, 2009 @ 7:08 am |

We live in exciting times…

Markovic will be talking here in Kyoto next week, and I’ll certainly try to understand as much as I can of his argument (already well-outlined in Danny Calegari’s blog)… Wise’s claim seems much more mysterious at the moment.

Comment by dmoskovich — September 6, 2009 @ 2:56 am |