I just got back from an extremely enjoyable meeting in Montreal, where I learned some nice new results about homogeneity, a natural logical property of groups. Now, I realise that logic may seem like a distant and irrelevant subject to many topologists, but I hope you’ll bear with me, as in this case I think there’s a very interesting relationship between logic and geometric group theory. If you need to be further convinced, perhaps it would help if I told you that this circle of ideas is intimately connected to Sela’s solution to the homeomorphism problem for hyperbolic manifolds. (Perhaps I’ll write some more about that on another occasion.) (more…)
October 18, 2010
April 30, 2010
Two weeks ago, there was a very interesting conference in Montréal organized around Dani Wise’s work on subgroup separability properties for certain word-hyperbolic groups, namely those with “quasi-convex hierachies”. I’ve mentioned this work twice before. This time Dani gave about 12-15 hours of lectures. Stefan Friedl took notes and typed then up into a 21-page summary which you can read here.
January 26, 2010
During the nice talk that Ian Biringer gave on the structure of hyperbolic 3-manifolds at Caltech on Friday, a 4-manifold theorist in the back was heard to ask ’3-manifold groups are known, right?’
I know what he meant. Any finitely presented group can arise as the fundamental group of a 4-manifold (this is a nice exercise that you can do for yourself, involving surgery on a connect sum of copies of S3 x S1). With a little care, you can deduce that classifying topological 4-manifolds is at least as impossible as classifying finitely presented groups (which is impossible).
In contrast, there are constraints on the fundamental groups of closed 3-manifolds. One of the first is an easy consequence of the existence of Heegaard splittings: any closed 3-manifold group admits a balanced presentation, meaning that there are no more relations than generators.
What does it mean to ‘know’ a class of finitely presentable groups? Do we really ‘know’ the class of (orientable) 3-manifold groups? The point of view that I will adopt in addressing these questions is algorithmic, and comes from combinatorial group theory.
August 26, 2009
Theorem 6.5. If quasi-fuchsian surface subgroups of the fundamental group of a hyperbolic 3-manifold M are separable, then M is virtually fibered.
This has also been proved by Ian using the same approach, see his Georgia slides. The idea is to apply a construction of Sageev to build a CAT(0)-cube complex from a large collection of quasi-fuchsian surface subgroups, and then apply a theorem of Haglund and Wise to get enough residual control over the fundamental group to apply Ian’s earlier work on virtual fibering.
Another aspect of [BW] was the implicit announcement by Wise of the following result
Theorem. Suppose M is a hyperbolic 3-manifold containing an embedded incompressible quasi-fuchsian surface. Then the fundamental group of M is subgroup separable.
In particular, [BW] refers to a 175 page(?!) preprint by Wise showing this, though it doesn’t seem to be on his webpage yet. If correct, this would be another major breakthrough in the study of the Virtual Haken Conjecture.
August 5, 2009
I just heard that Jeremy Kahn has announced joint work with Vladimir Markovic showing that every hyperbolic 3-manifold contains an immersed, -injective surface. This is equivalent to showing that the fundamental group of every hyperbolic 3-manifold contains a subgroup isomorphic to the fundamental group of a surface. The abstract for Jeremy’s talk is on the conference web page. This is a long standing open problem that Henry Wilton wrote about on this blog back in February. I’m not at the conference, so I didn’t see the talk, but if any of you readers were ther and have anything to add, feel free to write in in the comments.
February 26, 2009
Not long ago, Jesse was kind enough to blog about my recent work with Cameron Gordon. Now that the preprint has finally appeared on the arXiv, I thought I’d say a few more words about what we do, and why.
The context is Gromov’s famous question about surface subgroups.
“Does every one-ended word-hyperbolic group contain a subgroup isomorphic to the fundamental group of a closed surface?”
January 7, 2009
Cameron Gordon asked an interesting question in his talk yesterday morning (at 8 AM!) here at the JMM. Given a non-primitive element g of a free group F, which we identify with the fundamental group of a handlebody H, you can ask whether there is a simple loop in the boundary of H representing the element g in the fundamental group. There is always an infinite family of curves representing g, but in general, these curves may all have self intersections. If you do happen to find a simple representative then you can glue a 2-handle to H along this curve. The fundamental group of the resulting 3-manifold with boundary M is the 1-relator group that you get by quotienting F by the relation g. If you compress the boundary of M into the manifold as much as possible, the result is an incompressible surface, implying that the 1-relator group contains a surface subgroup (i.e. a subgroup isomorphic to the fundamental group of a surface.) This property is important to geometric group theorists for various reasons. (Perhaps I can convince Henry, who has signed up to write occasional posts, to explain why…)
Of course, most elements of F don’t have simple representatives so here’s something else to try: If we could find a subgroup of the 1-relator group that contains a surface subgroup then we’d be done. A finite index subgroup of the 1-relator group is determined by a finite cover of H in which there is a closed curve whose projection into H represents g. If we can find such a finite cover of H in which there is a simple closed curve that projects to g then the fundamental group of the manifold it defines (which is a subgroup of the original 1-relator group) contains a surface subgroup. Cameron asks whether this always happens. In other words, is every non-primitive element of a free group F represented by a simple closed curve in a finite cover of a handlebody with fundamental group F?
Incidentally, the motivation for this question comes from studying certain delta hyperbolic groups in which one takes the free product of two copies of F amalgamated along the subgroup generated by g. He and Henry have shown that the amalgamated free product will contain a surface subgroup if and only if the corresponding 1-relator group does. (At least I think that’s what he said…) He also described another method for finding surface subgroups that I didn’t understand as well, though it sounded like this other method has been more effective.
December 1, 2008
In light of the sad news about John Stallings reported by Jesse, I thought I’d remember him by recounting my favourite proof. I don’t know if I love it because it inspired my thesis or vice versa, but it’s been an important part of my mathematical life so far.
In ‘Topology of Finite Graphs’ , Stallings exploited the observation that free groups are precisely the fundamental groups of graphs to give elementary topological proofs of lots of classical theorems about subgroups of free groups. This simple idea enables ignoramuses like me, whose group theory is pretty ropey, to prove nice theorems about free groups using nothing more than undergraduate topology!
The paper is a goldmine: among various highlights, Stallings gives a proof of Howson’s Theorem that the intersection of two finitely generated subgroups of a free group is finitely generated, and I think it’s the first place that his famous folds appear. The theorem I want to focus on was first proved by Marshall Hall Jr in 1949 .
Theorem(M. Hall Jr): If F is a free group and H is a finitely generated subgroup then H is a free factor in a finite-index subgroup of F.
I’d like to outline Stallings’ proof. As we go, we’ll see that this theorem is as much a statement about the topology of graphs as it is about group theory. The only prerequisite should be covering space theory. Here goes.
The ambient free group F may as well be finitely generated – any finitely generated subgroup is contained in a finitely generated free factor. So, as in all these proofs, we identify F with the fundamental group of some finite graph. For definiteness, if F is of rank n then let G be the ‘rose’ with n petals – that is, the graph with one vertex and n edges. To keep track of things, let’s orient the edges of G – in other words, put an arrow on each edge, so one end is leaving the vertex and the other is entering it.
The idea of the proof is to analyse immersions into G. Recall that a map of graphs f from D to G (sending vertices to vertices and edges to edges) is an immersion if it is locally injective – so whenever v is a vertex of D, f embeds a neighbourhood of v into a neighbourhood of f(v). If f is a local homeomorphism then f is a covering map. The next theorem is the heart of Stallings’ proof.
Theorem(Stallings): For any immersion f from a finite graph D to G there is a finite-sheeted covering space D’ of G that extends f. More precisely, there is an embedding of D into D’ and the restriction of the covering map to D coincides with f.
The proof of this theorem is a fun exercise. The idea is to look at the vertices of D in turn and to count the number of ‘missing’ edges that need to be included to turn f into a covering. Here’s an example. The colours and arrows are meant to indicate where each map sends each edge. So red edges get sent to red edges, in such a way that the arrows match up. The dotted edges are the ones that need to be added to obtain a covering space.
Armed with Stallings’ theorem, Marshall Hall’s Theorem is quite easy.
Proof of Marshall Hall’s Theorem: As above, identify F with the fundamental group of G. Let G’ be the covering space corresponding to the subgroup H. Now G’ is an (in general infinite) graph, but because H is finitely generated, its fundamental group is carried by a finite, connected subgraph D. Pictorially, G’ looks like D with some infinite trees attached.
The restriction of the covering map to D is an immersion. By Stallings’ theorem, D can be extended to a finite-sheeted covering space D’. This corresponds to a finite-index subgroup K of F, and as H is carried by the embedded subgraph D, it corresponds to a free factor in K. QED
So go forth, topologists, and prove theorems about free groups!
Remarks: 1. These ideas were floating around for a while before the paper appeared, and I believe a similar proof is given in Hempel’s 3-manifolds book. I did some investigation while writing my thesis, and there seems to be general agreement that these ideas are essentially due to Stallings.
2. Stallings’ ideas are the beginning of a long story. The first and perhaps most notable proof in a similar vein is Scott’s theorem that finitely generated subgroups of surface groups are virtually geometric . His argument makes really elegant use of hyperbolic geometry. Haglund and Wise’s ‘Special cube complexes’  is a very recent highlight. It seems to simultaneously unite Scott’s and Stallings’ approaches and to open the door to proving similar theorems for much more general classes of groups. I may write something about this in the future.
3. For any topologists out there wondering why they should care about free groups, the corresponding theorem for 3-manifolds would imply that every immersed surface can be lifted to an embedding in a finite-sheeted covering space. There are examples of graph manifolds where this doesn’t hold, but the question is still open for hyperbolic 3-manifolds. The link to the Virtual Haken Conjecture is obvious.
I gave a talk about the analogous theorem for residually free groups  at MSRI in the autumn, and was really gratified when Stallings came to talk to me afterwards about it. He was an idol of mine, and I’m sad not to have had the opportunity to talk maths with him more often.