ICERM Fall 2013: Topology, geometry, and dynamics

I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics.   You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other.   The deadline for the postdoctoral positions is January 14, 2013; the early deadline for everyone else is December 1, 2012 and the second deadline March 15, 2013.

There will also be three week-long workshops associated with this, so mark your calendars for these exciting events:

1. Exotic Geometric Structures. September 15-20, 2013.
2. Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
3. Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.

Bill Thurston is dead at age 65.

Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years.   I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology.  Almost everything we blog about here has the imprint of his amazing mathematics.    Bill was always very generous with his ideas, and his presence in the community will be horribly missed.    Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:

• Wikipedia.
• 2010 lecture on The mystery of 3-manifolds.
• On proof and progress in mathematics.

The virtual Haken conjecture

Agol’s preprint, which includes a long appendix joint with Groves and Manning, is now on the arXiv.

Agol’s work on the Virtual Haken Conjecture

Over at Geometry and the Imagination, Danny Calegari is reporting live from Paris on talks by Agol and Manning on the announced proof of the VHC:  Part I, Part II, Part III. 

… or Agol’s Theorem?

This just in: Ian Agol (UC Berkeley), speaking at the Institut Henri Poincaré today, has announced a proof of the very same Wise’s Conjecture that I blogged about just last week! In particular, this implies the Virtually Haken Conjecture.  His proof is based on joint work with Daniel Groves (UI Chicago) and Jason Manning (SUNY Buffalo).  It makes heavy use of the work of Dani Wise (McGill) on the Virtually Fibred Conjecture, as well as the proof of the Surface Subgroup Conjecture by Jeremy Kahn (Brown) and Vlad Markovic (Caltech).

Wise’s Conjecture

At the end of his monumental preprint addressing the Virtually Fibred Conjecture for Haken 3-manifolds [7], Wise makes a remarkably bold conjecture.  (Nathan Dunfield blogged about Wise’s work here.) The purpose of this post is to highlight that conjecture and explain what it means. It’s such a remarkable conjecture that it’s difficult to believe it’s true, but it’s also a win-win in the sense that either a positive or a negative answer would be a huge advance in geometric group theory.

Wise’s Conjecture (Conjecture 20.5 of [7]):  Let be a word-hyperbolic group which is also the fundamental group of a compact, non-positively curved cube complex .  Then has a finite-sheeted covering space which is special.

Most of the rest of this post will be an attempt to explain what ‘special’ means, but let me first whet your appetite by giving some consequences.

Videos for NSF-CBMS Cubulationathon

I’ve mentioned here several times the work of Wise on residual properties of certain word-hyperbolic groups, specifically those of Haken hyperbolic 3-manifolds. You can now view all 10 of Dani’s talks at the NSF-CBMS conference (as well as all the other talks) at the conference webpage. The picture and audio quality is quite reasonable considering the setup that was used, and they are certainly watchable.

I really wish more conferences did this. While it’s certainly true that the benefits of attending a conference go far beyond the content of the talks themselves, I think it’s still quite valuable to have this online for those who weren’t able to addend.

Mineyev and the Hanna Neumann Conjecture

You may have heard the big news – Igor Mineyev has announced a proof of the Hanna Neumann Conjecture. I’d like to  quickly remind you what the conjecture says.  It concerns the rank of a free group, by which I mean the minimal number of generators.

Let F be a non-abelian free group, which we may take to be of rank two.  The starting point is  an old theorem of Howson.

Theorem (Howson, 1954). If H,K are finitely generated subgroups of F then the rank of the intersection of H and K is finite.

The obvious question is: ‘What is the best bound on the rank of the intersection, in terms of the ranks of H and K?’  Shortly after Howson’s paper, Hanna Neumann conjectured the answer.

Conjecture (Hanna Neumann, 1956). If H,K are non-trivial subgroups of F, then

.

She also proved that .

I want to quickly convey why the Hanna Neumann Conjecture is reasonable.  Specifically, I want to explain why it holds for finite-index subgroups of F, using a topological argument of Stallings.

As usual, the idea is to think of F as the fundamental group of a graph X (equipped with some basepoint).  We have that , where is Euler characteristic.  The subgroups H and K correspond to (based) covering spaces and respectively.  Because the correspondence between subgroups and covering spaces is functorial, we can study the intersection of H and K by studying the covering space Y that is the pullback of the following diagram.

The space Y can be seen explicitly as the fibre product .  Specifically,

where and are the covering maps.  The intersection of H and K is the fundamental group of the component of Y picked out by the basepoints of and .

Remarks.

• The maps and are covering maps.
• (and, symmetrically, ).

At this point, it’s easy to prove the Hanna Neumann Conjecture for subgroups of finite index.

Proof of the HNC for .

It follows immediately from the fact that Euler characteristic is multiplicative that

.

Let Y’ be the component of Y with .  Because is a covering map of finite degree, each component of Y has non-positive Euler characteristic.  Therefore

,

which is exactly the statement of the conjecture. QED

If you were reading the above proof carefully, you may have been a little dissatisfied when we passed from Y to Y’, as we lost a lot of information at that point.  Indeed, the the fundamental groups of the other components of Y are conjugate to subgroups of F of the form .  The above proof naturally gave the following.

.

These sorts of considerations motivate the Strengthened Hanna Neumann Conjecture, proposed by Walter Neumann.  A little care is needed, because if H and K are both of infinite index in F then will often be trivial, and so of rank 0; but we do not want such intersections to contribute negative terms to our sum, as this would make the ‘strengthened’ conjecture weaker than the original!  Therefore, the strengthening takes the following form.

Strengthened Hanna Neumann Conjecture (Walter Neumann, 1989). If H,K are subgroups of F, then

.

The Hanna Neumann Conjecture has provided endless fun for group theorists over the past 50 years.  I was told that Benson Farb used to set it as a problem to his incoming students.  Much of the appeal lies in its elementary statement, and it always seemed likely that what was needed was one ingenious idea, rather than a big new piece of technology.

Well, Igor Mineyev has announced a proof of the Strengthened Hanna Neumann Conjecture.  I haven’t looked at his papers on that topic, but at a recent conference in Southampton, Yago Antolin presented a simplification of Mineyev’s proof due to Warren Dicks.  Slides of his talk are available here, and I heartily recommend you take a look.  The proof is elementary (though also ingenious!), and can be read quite quickly.

I haven’t yet had time to absorb the proof fully.  If I do, then perhaps I’ll blog about it again.  But let me highlight two features.

• It uses the fact that free groups are left-orderable.  I don’t think anyone saw that coming. (Feel free to boast in the comments if you did!)
• The idea of the proof is to construct a new, in some sense better, family of trees on which H, K and their intersection act.  The proof then uses an estimate like the one derived from the fibre product above.

Wise’s work on groups with quasiconvex hierarchies (again)

This blog has mentioned several times Dani Wise’s work on subgroup separability properties for certain word-hyperbolic groups [1, 2, 3]. In August, there will be a CBMS-NSF conference at CUNY focusing on this work, and the reason for this post is that one major part of Dani’s work is now available on the conference website.

(Thanks to Jason Behrstock and Ian Agol for telling me that the preprint had been posted there.)

Virtually geometric words

A couple of years ago, I wrote about virtually geometric words, in connection with finding surface subgroups of certain hyperbolic groups. A conjugacy class, represented by an element w, in a non-abelian free group F is called geometric if we can write F as the fundamental group of a handlebody H in such a way that w is represented by an embedded curve on the boundary. (More generally, everything here will apply just as well to finite sets of conjugacy classes.) The word w is virtually geometric if the handlebody H has a finite-sheeted cover in which the total preimage of a submanifold representing w is homotopic to an embedded multicurve on the boundary. Cameron Gordon and I were surprised to discover that there are words that are virtually geometric but not geometric [1].

Example: The Baumslag–Solitar relators (for coprime) are virtually geometric but not geometric.

Prompted by this discovery, we asked (perhaps rather hopefully) whether every word is virtually geometric.

Even before our paper was published, Jason Manning had answered our question by finding examples of words that are not virtually geometric [2]. More recently, Chris Cashen has explained exactly when a word is virtually geometric [3], building on work of Jean-Pierre Otal [4] , as well as on joint work that Chris did with Natasa Macura [5]. I’d like to take some time to explain the solution, which turns out to be really beautiful.