Low Dimensional Topology

August 31, 2009

Forms of the Virtual Haken Conjecture

Filed under: 3-manifolds,Hyperbolic geometry,Virtual Haken Conjecture — Nathan Dunfield @ 2:04 pm

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: $\pi_1(M)$ 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 $b_1(M) = \mathrm{rank}(H^1(M; Z)) > 0$.
Conjecture 4: For each $n > 0$, the manifold M has a finite cover N with $b_1(N) > n$.
Conjecture 5: M has a finite cover N which is large, i.e. $\pi_1(N)$ 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 $\pi_1(M)$ 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 $\pi_1(M)$ 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 $b_1 = 1$ and then Wise’s result does not apply. Conjecture 4, even for $n = 2$, does suffice to prove them all, since not every class in $H^1(N)$ can come from a fibration when $b_1 > 1$. 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.