Cubulation and the proof of the VHC/VFC have been featured prominently on this blog, and quite rightly so. It constitutes a major advance in low dimensional topology, and the proof itself is such a happy story. Thurston’s prophetic unexpected 1982 conjectures of how nice the fundamental group of a hyperbolic 3-manifold must be all proved to be true (such a group turns out to be LERF, virtually fibred, virtually biorderable, linear over the integers, etc.). And it happened in the nicest way possible, without huge mathematical machines (as was the case for *e.g.* Hironaka’s resolution of singularities), without huge computer assistance (as was the case for *e.g.* the Four Colour Theorem), without controversity over correctness of the proof (compare *e.g.* Wiles’s first proof of Fermat’s Last Theorem), with a great deal of conceptual understanding gained (contrast again with the Four Colour Theorem), without hard-to-read outlined arguments which require non-trivial unpacking (contrast with Perelman’s proof of Geometrization), without being unreadably long (contrast with classification of finite simple groups), with essential new ideas and new tools developed, together with a theorem which might be more valuable than the conjecture itself (the Virtually Compact Special Theorem- compare with Geometrization being more valuable than the Poincaré Conjecture).

Indeed, for me this might be the happiest end ever for a quest to prove a mathematical conjecture. What more could one wish for?

I just got back from a trip to North America, where I visited Toronto, Kansas, and Montreal. I have a lot to say about my visits to all 3 places really, but today I want to talk about my visit to The Cubulator Dani Wise, and some snippets of what he taught me, and some of my vague mathematically unmeaningful thoughts about some key ideas of the cubulation story (for a more professional account of related things, read Danny Calegari’s coverage).

### The ideas in cubulation are simple

There really aren’t any big machines which you need to understand in order to grasp cubulation. I posit that a graduate student, without any special combinatorial/geometric group theoretical training, could probably completely understand the proof of the Virtually Compact Special Theorem in a one-year seminar (although not having myself read all of it, this may be total rubbish). Indeed, what we have here is an example of a big theorem being proven by finding good **definitions**, as opposed to by working through a technically formidable proof. With hindsight, Virtual Fibering turned out not to be such a difficult conjecture after all- to me, this is the highest possible praise to those who played a role in its proof.

### The definition of a special cube complex is not obvious

All pictures taken from Dani Wise’s summer school notes.

Coming from the outside and reading superficially, I had thought that a special cube complex was “a nonpositively curved cube complex without the obvious pathologies”:

But it isn’t. The definition is really delicate. The following configuration, for example, is not prohibited:

Meanwhile, the “no interosculation” condition has more to it than meets the eye.

This delicate balance between generality (including enough cube complexes to be useful) and exclusion of pathologies (not including configurations which turn out to ruin the theory) is the hallmark of a truly masterful mathematical definition.

Coincidentally, Haglund and Wise called such cube complexes “special” because they thought that they were unusual, whereas in fact they are in some sense virtually generic. Wise told me that, with hindsight, they should have been named “organized cube complexes”. They can’t do it anymore because “special” has become established usage, but in a blog post I can do whatever I want, so henceforth “organized cube complex” means “special cube complex”.

The main property of organized cube complexes, which could almost be their definition, is:

If a cube complex is organized, then is a subgroup of a RAAG.

### RAAGs are good and so are their quasiconvex subgroups

To remind you, a Right-Angled Artin Group (RAAG, which Dani Wise pronounces Rag to rhyme with Bag, but which I would pronounce Rog to rhyme with Bog), is a group associated with a graph, with a generator for each vertex set, two generators commuting if a vertex is connected by an edge, and no other relations. So a RAAG corresponding to a graph with no edges is a free group, and a RAAG corresponding to a complete graph is a free abelian group.

Another major insight is that RAAGs are super-well-behaved (linear over integers, biorderable, bla bla bla), and that these properties are shared by their quasiconvex subgroups (quasiconvex is a geometric group theory term, but for us low-dimensional topologists interested in the VFC for hyperbolic manifolds of finite volume, it basically boils down to saying that a subgroup H of a hyperbolic group G is quasiconvex if it acts properly and cocompactly on a CAT(0) cube complex of the universal cover, obtained from lifting surfaces which Kahn-Markovic gave us).

### Proof by universal reciever

So RAAGs emerge as some sort of universal organizing object for a certain sort of group theory. I think this is rather unexpected. A huge part of the proof of the VFC ends up being a proof that something is a quasiconvex subgroup of a RAAG, which I’m sure isn’t what anyone expected (say, 10 years ago) a proof of the VFC to look like.

This whole concept of “proof by universal receiver” intrigues me. It’s quite amazing that one can prove a major conjecture about objects by showing that each object is a `nice’ subobject of some unexpected universal object, from which all good properties must follow. Maybe one can compare with proofs which use Cayley’s Theorem that any group embeds in a permutation group, except that this embedding is a great deal more routine and canonical. I asked an MO question about this phenomenon, and Terry Tao gave a better example, Lie’s Third Theorem, which is proven using Ado’s Theorem that every finite-dimensional Lie algebra embeds into the Lie algebra of a general linear group. This still doesn’t have quite the umph of the Virtually Compact Special Theorem.

### The Malnormal Special Quotient Theorem is important

Agol-Groves-Manning’s proof of the Weak Separation Theorem, a key step in Agol’s proof of the VFC, uses the Malnormal Special Quotient Theorem in an essential way, which I hadn’t really appreciated. This is discussed by Danny Calegari.

### RAAGs are better than riches

3-manifolds are beautiful wonderful objects, but for many purposes it might be worth trading them for cube complexes. A cube complex can be high-dimensional, can have mixed dimension, and can be ugly and horrible even for a simple-looking 3-manifold; but they behave much better with respect to the kinds of properties which we are now concerned with. This is another major conceptual breakthrough- rather than staying in the 3-dimensional world, forget dimensional considerations and go to cube complexes. Now that this idea has proven its value, I imagine it will spread throughout quite a bit of low-dimensional topology. Dani thinks that cube complexes might be underlying many of the structures that we study.

### 3-manifold groups are either good or good

A week ago, I thought that 3-manifold groups were either good (*e.g* the fundamental group of a lens space) or bad (*e.g.* fundamental group of some hyperbolic 3-manifold), but cubulation shows that they’re good or good. Either they have “Property T” which turns out to imply that they are finite groups (observation of Koji Fujiwara- see Henry Wilton’s comment), or they have satisfy a strong negation of it, meaning thay they’re somehow very free like RAAGS. Highly structure or highly free- nothing in between. How unexpected and wonderful!

### It will get even better!

As simple and transparent as the proof now looks (although I have gone through only a small part of it), there are still simplifications which can and will be made. Rough edges in the proof will be smoothed, everything will be coordinated in a unified language, hard steps will be reformulated, simple shortcuts will be found… when it’s over, this has real potential to be a “Proof from the Book”.

### My middle school classmates are strong mathematicians

This is entirely unrelated to the previous discussion, but Lior Silberman, Mike Hochman, and I attended the same class in middle school (and the small part of high school I attended). We talked about math quite a bit then… anyway, both of them went on to Princeton, great mathematical discoveries, and fame and fortune. I hadn’t seen either of them in 20 years or so, but in Montreal I had the good fortune to attend a talk by Lior about crazy pathological groups with expander graphs embedded in their presentations which don’t act nicely on anything. Lior is primarily a number theorist, or so I had believed, but his talk was really good and easy to understand. This is the mark of a great mathematician.

Nice post, Daniel. It’s a beautiful theory, and I agree that it’s very accessible (in fact, I said something similar here).

I have two small comments:

In fact, it follows from Geometrization that if they have Property T then they are in fact finite! This was observed by Koji Fujiwara.

It’s true that the distinctions between A-special, C-special and special are quite delicate, but the key property is being ‘virtually special’, and this is much more robust. For instance, the ‘indirect self-osculation’ that you illustrate can be removed by passing to a finite-index subgroup.

Comment by Henry Wilton — July 12, 2012 @ 12:58 pm |

It’s seems to me a little optimistic to get within one year to the proof that hyperbolic 3-manifold groups are virtually compact special and therefore virtually fibered. Not only would this seminar have to cover all the work of Dani Wise and Ian Agol, which might not be so easy.

The seminar would also have to cover the theorem of Kahn-Markovic which is an essential ingredient in Ian’s proof of the Virtually Compact Special Theorem.

For the record, when you write “the fundamental group of a hyperbolic 3-manifold […] turn out to be residual finiteness, virtually fibred, biorderable, linear, etc.”, the residual finiteness and linearity follows immediately from being hyperbolic. What follows from the Virtually Compact Special Theorem is that the fundamental groups are linear over the integers (which seemingly was not even conjectured before 2009) and that the fundamental groups are LERF (a very strong form of being residually finite, but this relies on the Tameness Theorem). Also, hyperbolic 3-manifold groups are now known to be *virtually* biorderable, but they are usually certainly not biorderable on the nose.

Comment by Stefan Friedl — July 12, 2012 @ 5:06 pm |

Thanks for this! Post edited.

I should point out to the reader (here in comments), that you can see what the consequences of the Virtually Compact Special Theorem are in Section 6 of the survery paper by Aschenbrenner-Friedl-Wilton. Incidentally, the first paragraph wasn’t to imply that all of these are Thurston’s conjectures- the comment in the brackets is independent. I lumped major easy to understand consequences together.

I had taken Kahn-Markovic into account… Covering the full proof in a year is certainly easier said than done, but it would be a great hypothesis to test on motivated graduate students! Even if it didn’t go the whole way, everyone would learn a lot.

Comment by dmoskovich — July 12, 2012 @ 7:25 pm

Hey Daniel,

One small observation – the universal receiver in this case (for hyperbolic 3-manifold groups) is virtual RAAGs. In fact, when one proves that a hyperbolic cubulated group is virtually special, one can find a finite special regular cover of the cube complex, together with a local isometry to a RAAG. Since the cover is regular, the covering automorphisms give automorphisms of the special cube complex, and therefore of the RAAG which it maps into functorially. So the equivariant map descends to a map of the cube complex associated to the cube complex to an (orbicomplex) quotient of the Salvetti complex of the RAAG. Thus, one embeds the cubulated group in a virtual RAAG.

Also, the notion of embedding hyperbolic 3-manifold groups in right-angled reflection groups might have originated in a paper of myself, Long and Reid, where we prove the Bianchi groups (cusped arithmetic 3-orbifold groups) virtually embed in a 6 dimensional hyperbolic reflection group. The problem is that there is a paucity of hyperbolic reflection group lattices in high dimensions (Potyagailo-Vinberg proved they can’t exist in dims. > 14).

To extend Peter Scott’s arguments, one must then search for embeddings into abstract reflection groups. Thus it is natural to analyze locally isometrically immersed subcomplexes of Salvetti complexes associated to RAAGs and RARGs. Haglund-Wise realized that these subcomplexes satisfy certain combinatorial conditions which are also sufficient for embedding. Thus, the notion of a special complex is forced upon you when you study quasiconvex subgroups of RAAGs. This was the great idea of Haglund-Wise.

Returning to the hyperbolic case, it would be interesting to me if one could prove that hyperbolic 3-manifolds virtually embed in right-angled hyperbolic reflection groups (not finite co-volume of course). Unfortunately, the special complexes produced by Haglund-Wise probably won’t give embeddings into hyperbolic right-angled reflection groups, so one might need a new idea to analyze this problem.

best,

Ian

Comment by Ian Agol — July 15, 2012 @ 12:30 pm |

This is an extremely nice comment- thanks! In particular, the first paragraph cleared up something for me which I hadn’t understood (the big-picture role of the orbicomplex quotient of the Salvetti complex).

Thank you for the explanation of the heuristic for RAAGs to be relevant. Wise’s approach explains why being virtually special is a good property to look for in general when a group acts on a CAT(0) cube complex, but not its relationship to hyperbolic 3-manifolds specifically. Indeed, I don’t understand this at all (it looks “just lucky”). Perhaps it might make more sense in the version which you are proposing, when we are embedding into hyperbolic right-angled reflection groups? Do you have a heuristic argument for why such a conjecture might hold?

Comment by dmoskovich — July 15, 2012 @ 8:13 pm |

This question is to Stefan Friedl’s comment “linearity follows immediately from being hyperbolic”. I assume here that you mean linearity over the complex numbers (which follows from your comment D.8 on page 39 of your “3 manifold groups paper” Version 2). Is that a correct statement, i.e. that you mean linear over C?

Comment by Mayer A. Landau — August 14, 2012 @ 11:49 pm |

yes, I meant linearity over the complex numbers. Linearity of hyperbolic 3-manifold groups over the integers follows from Agol and Wise, and was a huge surprise to me (and I think it’s fair to say, to most other people).

Comment by stefanfriedl — August 22, 2012 @ 10:21 am |

[…] These were apparently the outstanding open conjectures after the proof of geometrization. This post in particular describes the techniques involved in the proof. To see how fast things changed over […]

Pingback by Virtually Haken Conjecture | Ars Mathematica — December 30, 2012 @ 8:05 am |

[…] In 2012, Ian Agol proved, building on work of Wise and Groves-Manning, that every cubulated hyperbolic 3-manifold group is special, or in other words embeds in a RAAG. In combination with Kahn-Markovic’s resolution of the surface subgroup conjecture and the Haglund-Wise-Sageev construction for cubulating hyperbolic 3-manifold groups, this proved the Virtually Haken Conjecture, and with further work of Agol and Wise involving special cube complex fundamental groups, the Virtually Fibered Conjecture. That proof, in its entirety, involved an extraordinary sweep of ideas, and in part demonstrates the mathematical value of geometric group theory ideas in general and cubulation in particular. […]

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