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Last week I was at Oberwolfach for a meeting on geometric group theory. My friend and collaborator Koji Fujiwara gave a very nice talk about constructing actions of groups on quasi-trees (i.e. spaces quasi-isometric to trees). The construction is inspired by the famous subsurface projection construction, due to Masur-Minsky, which was a key step in their proof that the complex of curves (a natural simplicial complex on which the mapping class group acts cocompactly) is hyperbolic. Koji’s talk was very stimulating, and shook up my thinking about a few related matters; the purpose of this blog post is therefore for me to put some of my thoughts in order: to describe the Masur-Minsky construction, to point out a connection to certain geometric phenomena like winding numbers of curves on surfaces, and to note that a variation on their construction gives rise directly to certain natural chiral invariants of surface automorphisms (and their generalizations) which should be relevant to 4-manifold topologists.

There is an old puzzle which starts by asking: what is the next number in the sequence 1,2,4,? We are supposed to recognize the start of the sequence and answer that the next number is surely 8, because the first three numbers are consecutive powers of 2, and so the next number should be the cube of 2 which is 8. The puzzler then explains (contrary to expectations) that the successive terms in the sequence are actually the number of regions into which the plane is divided by a collection of lines in general position (so that any two lines intersect, and no three lines intersect in a single point). Thus:

So the “correct” answer to the puzzle is 7 (and the sequence continues 11, 26, $\cdots (n^2+n+2)/2$). This is somehow meant to illustrate some profound point; I don’t quite see it myself. Anyway, I would like to suggest that there is a natural sense in which the “real” answer should actually be 8 after all, and it’s the point of this short blog post to describe some connections between this puzzle, the theory of cube complexes (which is at the heart of Agol’s recent proof of the Virtual Haken Conjecture), and the location of the missing 8th region.

In winter and spring of 2001, Nathan Dunfield and I ran a seminar at Harvard whose purpose was to go through Thurston’s proof of the geometrization theorem for Haken manifolds. This was a very useful and productive exercise, and there was wide participation from faculty and students. As well as talks by Nathan and myself, there were talks by David Dumas, Laura de Marco, Maryam Mirzakhani, Curt McMullen, Dylan Thurston, and John Holt. At the conclusion of the semester, Bill Thurston agreed to come out and lead a discussion on geometrization, in which he ended up talking a bit about what had led him to formulate the conjecture in the first place, what ideas had played into it, how and when he had gone about proving it, his ideas about exposition, and so on.

I had recently bought a video camera, and decided to tape Bill’s talk. I never did anything with it until now (in fact, I don’t think I ever re-watched anything that I taped), but it turned out to be not too difficult to transfer the file from tape to computer. Since this seems like an interesting fragment of intellectual history, I thought it might be worthwhile to post the result to YouTube — the video link is here.

My eldest daughter Lisa recently brought home a note from her school from her computer class teacher. Apparently, the 5th grade kids have been learning to program in Logo, in the MicroWorlds programming environment. I have very pleasant memories of learning to program in Logo back when I was in middle school. If you’re not familiar with Logo, it’s a simple variant of Lisp designed by Seymour Papert, whereby the programmer directs a turtle cursor to move about the screen, moving forward some distance, turning left or right, etc. The turtle can also be directed to raise or lower a pen, and one can draw very pretty pictures in Logo as the track of the turtle’s motion.

Let’s restrict our turtle’s movements to alternating between taking a step of a fixed size S, and turning either left or right through some fixed angle A. Then a (compiled) “program” is just a finite string in the two letter alphabet L and R, indicating the direction of turning at each step. A “random turtle” is one for which the choice of L or R at each step is made randomly, say with equal probability, and choices made independently at each step. The motion of a Euclidean random turtle on a small scale is determined by its turning angle A, but on a large scale “looks like” Brownian motion. Here are two examples of Euclidean random turtles for A=45 degrees and A=60 degrees respectively.

The purpose of this blog post is to describe the behavior of a random turtle in the hyperbolic plane, and the appearance of an interesting phase transition at $\sin(A/2) = \tanh^{-1}(S)$. This example illustrates nicely some themes in probability and group dynamics, and lends itself easily to visualization.

I am currently teaching a class at the University of Chicago on hyperbolic groups, and I have just introduced the concept of $\delta$-hyperbolic (geodesic) metric spaces. A geodesic metrix space $(X,d_X)$ is $\delta$-hyperbolic if for any geodesic triangle $abc$, and any $p \in ab$ there is some $q \in ac \cup bc$ with $d_X(p,q)\le \delta$. The quintessential $\delta$-hyperbolic space is the hyperbolic plane, the unique (up to isometry) simply-connected complete Riemannian 2-manifold of constant curvature $-1$. It follows that any simply-connected complete Riemannian manifold of constant curvature $K<0$ is $\delta$-hyperbolic for some $\delta$ depending on $K$; roughly one can take $\delta \sim (-K)^{-1/2}$.

What gives this condition some power is the rich class of examples of spaces which are $\delta$-hyperbolic for some $\delta$. One very important class of examples are simply-connected complete Riemannian manifolds with upper curvature bounds. Such spaces enjoy a very strong comparison property with simply-connected spaces of constant curvature, and are therefore the prime examples of what are known as CAT(K) spaces.

Definition: A geodesic metric space $(X,d_X)$ is said to be $CAT(K)$, if the following holds. If $abc$ is a geodesic triangle in $X$, let $\bar{a}\bar{b}\bar{c}$ be a comparison triangle in a simply connected complete Riemannian manifold $Y$ of constant curvature $K$. Being a comparison triangle means just that the length of $\bar{a}\bar{b}$ is equal to the length of $ab$ and so on. For any $p \in bc$ there is a corresponding point $\bar{p}$ in the comparison edge $\bar{b}\bar{c}$ which is the same distance from $\bar{b}$ and $\bar{c}$ as $p$ is from $b$ and $c$ respectively. The $CAT(K)$ condition says, for all $abc$ as above, and all $p \in bc$, there is an inequality $d_X(a,p) \le d_Y(\bar{a},\bar{p})$.

The term CAT here (coined by Gromov) is an acronym for Cartan-Alexandrov-Toponogov, who all proved significant theorems in Riemannian comparison geometry. From the definition it follows immediately that any $CAT(K)$ space with $K<0$ is $\delta$-hyperbolic for some $\delta$ depending only on $K$. The point of this post is to give a short proof of the following fundamental fact:

CAT(K) Theorem: Let $M$ be a complete simply-connected Riemannian manifold with sectional curvature $\le K_0$ everywhere. Then $M$ with its induced Riemannian (path) metric is $CAT(K_0)$.

Ian gave his second and third talks this afternoon, completing his (quite detailed) sketch of the proof of the Virtual Haken Theorem. Recall that after work of Kahn-Markovic, Wise, Haglund-Wise and Bergeron-Wise, the proof reduces to showing the following:

Theorem (Agol): Let G be a hyperbolic group acting properly discontinuously and cocompactly on a CAT(0) cube complex X. Then there is a finite index subgroup G’ so that X/G’ is special; in other words, G is virtually special.

I am in Paris attending a workshop at the IHP where Ian Agol has just given the first of three talks outlining his proof of the Virtual Haken Conjecture and Virtual Fibration Conjecture in 3-manifold topology (hat tip to Henry Wilton at the Low Dimensional Topology blog from whom I first learned about Ian’s announcement last week). I think it is no under overstatement to say that this marks the end of an era in 3-manifold topology, since the proof ties up just about every loose end left over on the list of problems in 3-manifold topology from Thurston’s famous Bulletin article (with the exception of problem 23 — to show that volumes of closed hyperbolic 3-manifolds are not rationally related — which is very close to some famous open problems in number theory). The purpose of this blog post is to say what the Virtual Haken Conjecture is, and some of the background that goes into Ian’s argument. I hope to follow this up with more details in another post (after Agol gives talks 2 and 3 this coming Wednesday). Needless to say this post has been written in a bit of a hurry, and I have probably messed up some crucial details; but if that caveat is not enough to dissuade you, then read on.

Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3-manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flow-invariant way into stable, unstable and flow directions) and contact (i.e. they preserve a contact form — that is, a 1-form $\alpha$ for which $\alpha \wedge d\alpha$ is a volume form). Their preprint gives some very interesting new constructions of such flows, obtained by surgery along a Legendrian knot (one tangent to the kernel of the contact form) which is transverse to the stable/unstable foliations of the Anosov flow.