Elevator pitch for the Virtual Fibering Theorem

Question

There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm guessing they will soon be proven in full generality). The proof is lucidly discussed in Danny Calegari's blog. The theorems state that every compact orientable irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken or a surface bundle over a circle, correspondingly. This implies various good things for a 3-manifold with fundamental group π, including:

1. π is large, meaning that π has a finite index subgroup which maps onto a free group with at least 2 generators. In particular the Betti numbers of finite covers can become arbitrarily large.
2. π is linear over $\mathbb{Z}$, i.e. π admits a faithful representation $\pi \to \mathrm{GL}(n,\mathbb{Z})$ for some $n$. (Thurston conjectured that $n\leq 4$ is sufficient).
3. π is virtually biorderable.

Stefan Friedl, from whose comment the above list is an excerpt, summarizes the situation as follows:

It seems like every nice property of fundamental groups which one can possibly ask for either holds for π or a finite index subgroup of π.

All well and good. But how could you `sell' that to somebody who isn't a classically-oriented 3-dimensional topologist? An elevator pitch is defined by Wikipedia as follows:

An elevator pitch is a short summary used to quickly and simply define a product, service, or organization and its value proposition. The name "elevator pitch" reflects the idea that it should be possible to deliver the summary in the time span of an elevator ride, or approximately thirty seconds to two minutes. In The Perfect Elevator Speech, Aileen Pincus states that an elevator speech should "sum up unique aspects of your service or product in a way that excites others."

The Virtual Fibering Conjecture (or the Virtual Haken Conjecture) was the grand conjecture in 3-manifold topology following Geometrization, and thus must have/ should have/ ought to have (I believe) a compelling elevator pitch. For contrast, Geometrization is easy to `sell' because it directly applies to the Homeomorphism Problem in 3-manifold topology: Given two 3-manifolds, determine whether or not they are homeomorphic. Geometrization allows you to canonically decompose both manifolds into submanifolds with geometric structure, and then to compare geometric invariants. In terms of "The Goals of Mathematical Research" as given in the introduction to The Princeton Companion to Mathematics, this corresponds to the goal of Classifying.

Question: What is a good elevator pitch for Virtual Fibering (or for Virtual Haken), explaining the utility of these results in terms of "the fundamental goals of mathematical research" (Solving Equations, Classifying, Generalizing, Discovering Patterns, Explaining Patterns and Coincidences, Counting and Measuring, and Finding Explicit Algorithms). The target would be mathematicians who are not 3-dimensional topologists.

Everyone in the approximate vicinity of the field instinctively feels that these are historic results, but I'd like to be able to justify that feeling (in the abovementioned sense) to myself and to others.

Answer 1

OK, I will also give it a shot.

First of all, I don't like to sell Geometrization because it helps with the homeomorphism problem. The Geometrization Theorem is an object of stunning beauty ("most 3-manifolds are hyperbolic" should be an exciting statement for anybody in an elevator who has seen the art of M.C.Escher), and beauty in mathematics is usually a sign that we are on the right track. And indeed, the beauty of Geometrization begets all kinds of results.

Regarding the new results of Agol, Wise et al. one should perhaps not jump right to "virtual Haken" or "virtually fibered" but one should look at the "real theorem", the Virtually Compact Special Theorem which goes as follows:

If $N$ is a finite volume hyperbolic 3-manifold, then $\pi_1(N)$ is virtually compact special, i.e. $\pi_1(N)$ is virtually a quasi-convex subgroup of a Right Angled Artin Group (RAAG).

One can explain a RAAG to anybody who has seen group theory in an elevator between about 3 floors. The fact that "simple" objects like RAAGs contain all hyperbolic 3-manifold groups (up to going to a finite index subgroup) is stunning and beautiful. All the goodies, e.g. largeness, linear over $\mathbb{Z}$, virtual fibering, LERF, virtually biorderable etc come from that statement (well, together with Agol's fibering theorem, tameness etc.). This can be seen clearly by looking at Diagram 4 in a recent survey paper on 3-manifold groups by authors whose names escape me at the moment. It is really stunning how the Virtually Compact Special Theorem answers all open questions at once. It is one of the great achievements of Dani Wise to have found the "right statement".

(Note that largeness, linear over $\mathbb{Z}$, biorderable do NOT follow from virtual fibering or virtual Haken alone.)

Back to the elevator:

The results make me think that hyperbolic 3-manifolds are like Jack in the Box. If you take a hyperbolic integral homology sphere you look at a tiny manifold, but when you press a button (i.e. go to an appropriate finite cover), the 3-manifold suddenly becomes a grand object of beauty (e.g. has as many fibered faces in the Thurston norm ball as you could wish).

(This analogy also works with tiny seed, a bit of water, blooming flower etc. for the botanically minded elevator companion)

So to conclude, I think the Geometrization Theorem and the Virtually Compact Special Theorem of Agol-Wise are stunningly beautiful results. The fact that the statements are so beautiful made it highly plausible that they were right, even before they were proved (I can't imagine that any serious person doubted the Poincare conjecture after Thurston stated the Geometrization conjecture). And ideally it's this beauty which I would like to communicate.

Answer 2

I suppose I'm obligated to make a stab at answering this :)

First, the virtual fibering question was asked by Thurston: "Does every hyperbolic 3-manifold have a finite-sheeted cover which fibers over the circle? This dubious-sounding question seems to have a definite chance for a positive answer."

When I originally read this, I found the question intriguing, partly because it is surprising as made evident in the phrasing. Also, certainly questions and conjectures of Fields medalists receive sometimes an inordinate amount of attention (Thurston has been correct for most of his conjectures and questions). However, from the perspective of Kleinian groups, this question is quite interesting, since doubly degenerate Kleinian groups have a nice structure (e.g. they give rise to group-invariant Peano curves in $S^2$).

There are some group-theoretic consequences of virtual fibering:

• If a hyperbolic 3-manifold is virtually fibered, its fundamental group does not have the finitely generated intersection property, which seems to be of interest to some group theorists.

• Virtual fibered 3-manifold fundamental groups have property FD introduced by Lubotzky and Shalom (finite reps. are dense in the unitary dual with respect to the Fell topology).

• Virtual fibering or virtual Haken implies the group is "good", which roughly means its cohomology is reflected in that of its profinite completion.

• The Lubotzky-Sarnak conjecture holds, namely hyperbolic 3-manifold groups do not have property $(\tau)$. In fact, the Heegaard gradient is zero.

I think there may be some other applications, and certainly will be more in the future.

Finally, the proof of virtual fibering in full generality relies on the group-theoretic property I call the "RFRS" condition (for residually finite rational solvable). This condition has other consequences, such as the twisted Alexander polynomials detect the Thurston norm, which has been applied by Friedl and Vidussi to study minimal genus surfaces in 4-manifolds which fiber over 3-manifolds.

I should also mention that recently Przytycki and Wise have proved that a 3-manifold with torus boundary components is virtually RFRS if and only if it admits a non-positively curved metric, which greatly generalizes Thurston's question.

OK, I think the elevator has gone all the way up to the top of the Sears tower and come back down.

Answer 3

The Virtual Fibring theorem provides the topological classification of closed 3-manifolds.

Very roughly, after passing to finite covers, we can build 3-manifolds by gluing together simpler pieces that we understand, in close analogy with the topological classification of surfaces. So the main extra subtlety as we pass to the 3-dimensional case is having to pass to finite covers.

Likewise, of course, the elevator pitch for the geometrisation theorem is that it's the 3-dimensional analogue of the uniformisation theorem for surfaces.

There are a few caveats:

• The statement is subtle, because not all 3-manifolds virtually fibre and so it's crucial that we also have an exact gluing description of the ones that don't.

• Arguably the Virtual Haken theorem provides another version of the same thing: after passing to finite covers, every irreducible closed 3-manifold has a Haken hierarchy. I think the point is that, from the "long range" point of view needed for an elevator pitch, the Virtual Haken and Virtually Fibred theorems are difficult to distinguish.

• A sceptic might argue that the mere existence of a Heegaard decomposition is a sort of topological classification. To this I'd respond that gluing along inessential things gives a much less useful classification. Heegaard decompositions are analogous to the statement that every surface is triangulable, which is a bit weaker than the topological classification of surfaces.