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.

Let $F=\langle a,b\rangle$ be the free group on two generators, and let $\phi:F \to F$ be the endomorphism defined on generators by $\phi(a)=ab$ and $\phi(b)=ba$. We define Sapir’s group $C$ to be the ascending HNN extension

$F*_\phi:=\langle a,b,t\; | \; a^t=ab,b^t=ba\rangle$

This group was studied by Crisp-Sageev-Sapir in the context of their work on right-angled Artin groups, and independently by Feighn (according to Mark Sapir); both sought (unsuccessfully) to determine whether $C$ contains a subgroup isomorphic to the fundamental group of a closed, oriented surface of genus at least 2. Sapir has conjectured in personal communication that $C$ does not contain a surface subgroup, and explicitly posed this question as Problem 8.1 in his problem list.

After three years of thinking about this question on and off, Alden Walker and I have recently succeeded in finding a surface subgroup of $C$, and it is the purpose of this blog post to describe this surface, how it was found, and some related observations. By pushing the technique further, Alden and I managed to prove that for a fixed free group $F$ of finite rank, and for a random endomorphism $\phi$ of length $n$ (i.e. one taking the generators to random words of length $n$), the associated HNN extension contains a closed surface subgroup with probability going to 1 as $n \to \infty$. This result is part of a larger project which we expect to post to the arXiv soon.

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 $CAT(K_0)$.

This morning I heard the awful news that Bill Thurston died last night. Many of us knew that Bill was very ill, but we all hoped (or imagined?) that he would still be with us for a while yet, and the suddenness of this is very harsh. As Sarah Koch put it in an email to me, “Although this was not unexpected, it is still shocking.” On the other hand, I am glad to hear that he was surrounded by family, and died peacefully.

I counted Bill as my friend, as well as my mentor, and I have many vivid and happy memories of time I spent with him. I hope that writing down a few of these reminiscences will be cathartic for me, and for others who are coping with this loss.

I am spending a few months in Göttingen as a Courant Distinguished Visiting Professor, and talking a bit to Laurent Bartholdi about rational functions — i.e. holomorphic maps from the Riemann sphere $\widehat{\mathbb C}$ to itself. A rational function is determined (up to multiplication by a constant) by its zeroes and poles, and can therefore generically be put in the form $f:z \to P(z)/Q(z)$ where P and Q are polynomials of degree $d$. If $d=1$ then $f$ is invertible, and is called a fractional linear transformation (or, sometimes, a Mobius transformation). The critical points are the zeroes of $P'Q-Q'P$; note that this is a polynomial of degree $\le 2d-2$ (not $2d-1$) and the images of these points under $f$ are the critical values. Again, generically, there will be $2d-2$ critical values; let’s call them $V$. Precomposing $f$ with a fractional linear transformation will not change the set of critical values.

The map $f$ cannot usually be recovered from $V$ (even up to precomposition with a fractional linear transformation); one needs to specify some extra global topological information. If we let $\overline{C}$ denote the preimage of $V$ under $f$, and let $C$ denote the subset consisting of critical points, then the restriction $f:\widehat{\mathbb C} - \overline{C} \to \widehat{\mathbb C} - V$ is a covering map of degree $d$, and to specify the rational map we must specify both $V$ and the topological data of this covering. Let’s assume for convenience that 0 is not a critical value. To specify the rational map is to give both $V$ and a representation $\rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d$ (here $S_d$ denotes the group of permutations of the set $\lbrace 1,2,\cdots,d\rbrace$) which describes how the branches of $f^{-1}$ are permuted by monodromy about $V$. Such a representation is not arbitrary, of course; first of all it must be irreducible (i.e. not conjugate into $S_e \times S_{d-e}$ for any $1\le e \le d-1$) so that the cover is connected. Second of all, the cover must be topologically a sphere. Let’s call the (branched) cover $\Sigma$ for the moment, before we know what it is. The Riemann-Hurwitz formula lets one compute the Euler characteristic of $\Sigma$ from the representation $\rho$. A nice presentation for $\pi_1(\widehat{\mathbb C}-V,0)$ has generators $e_i$ represented by small loops around the points $v_i \in V$, and the relation $\prod_{i=1}^{|V|} e_i = 1$. For each $e_i$ define $o_i$ to be the number of orbits of $\rho(e_i)$ on the set $\lbrace 1,2,\cdots,d\rbrace$. Then

$\chi(\Sigma) = d\chi(S^2) - \sum_i (d-o_i)$

If each $\rho(e_i)$ is a transposition (i.e. in the generic case), then $o_i=d-1$ and we recover the fact that $|V|=2d-2$.

This raises the following natural question:

Basic Question: Given a set of points $V$ in the Riemann sphere, and an irreducible representation $\rho:\pi_1(\widehat{\mathbb C} - V,0) \to S_d$ satisfying $\sum_i (d-o_i) = 2d-2$, what are the coefficients of the rational function $z \to P(z)/Q(z)$ that they determine (up to precomposition by a fractional linear transformation)?

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.

Today Jason Manning gave a talk on a vital ingredient in the proof of Agol’s theorem, which is a result in geometric group theory. The theorem is a joint project of Agol-Groves-Manning, and generalizes some earlier work they did a few years ago. Jason referred to the main theorem during his talk as the “Goal Theorem” (I guess it was the goal of his lecture), but I’m going to call it the Weak Separation Theorem, since that is a somewhat more descriptive name. The statement of the theorem is as follows.

Weak Separation Theorem (Agol-Groves-Manning): Let G be a hyperbolic group, let H be a subgroup of G which is quasiconvex, and isomorphic to the fundamental group of a virtually special NPC cube complex, and let g be an element of G which is not contained in H. Then there is a surjection $\phi:G \to \bar{G}$ so that

1. $\bar{G}$ is hyperbolic;
2. $\phi(H)$ is finite; and
3. $\phi(g)$ is not contained in $\phi(H)$.

In the remainder of this post I will try to explain the proof of this theorem, to the extent that I understand it. Basically, this amounts to my summarizing Manning’s talk (or the part of it that I managed to get down in my notes); again, any errors, foolishness, silly blog post titles etc. are due to me.