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).

]]>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?

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

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.

]]>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.

]]>I have two small comments:

Either they have “Property T” meaning that they’re somehow very structured like finite groups, or they have satisfy a strong negation of it

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

The definition [of a special cube complex] is really delicate.

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.

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