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For almost 50 years, Paul Sally was a towering figure in mathematics education at the University of Chicago. Although he was 80 years old, and had two prosthetic legs and an eyepatch (associated with the Type 1 diabetes he had his whole life), it was nevertheless a complete shock to our department when he passed away last December, and we struggled just to cover his undergraduate teaching load this winter and spring. As my contribution, I have been teaching an upper-division undergraduate class on “topics in geometry”, which I have appropriated and repurposed as an introduction to the classical geometry and topology of surfaces.

I have tried to include at least one problem in each homework assignment which builds a connection between classical geometry and some other part of mathematics, frequently elementary number theory. For last week’s assignment I thought I would include a problem on the well-known connection between Pythagorean triples and the modular group, perhaps touching on the Euclidean algorithm, continued fractions, etc. But I have introduced the hyperbolic plane in my class mainly in the hyperboloid model, in order to stress an analogy with spherical geometry, and in order to make it easy to derive the identities for hyperbolic triangles (i.e. hyperbolic laws of sines and cosines) from linear algebra, so it made sense to try to set up the problem in the language of the orthogonal group $O(2,1)$, and the subgroup preserving the integral lattice in $\mathbb{R}^3$.

When I started in graduate school, I was very interested in 3-manifolds, especially Thurston’s geometrization conjecture. Somehow in dimension 3, there is a marvelous marriage of flexibility and rigidity: generic 3-manifolds are flexible enough to admit hyperbolic structures — i.e. Riemannian metrics of constant curvature -1, modeled on hyperbolic space. But these structures are so rigid that they are determined up to isometry (!) entirely by the fundamental group of the manifold, and provide a bridge from topology to the rigid world of number fields and arithmetic. 3-manifolds, especially the hyperbolic ones, display an astonishing range of interesting phenomena, so that even though the individual manifolds are discrete and rigid, they come in infinite families parameterized by Dehn surgery. When Perelman proved Thurston’s conjecture, I gradually moved away from 3-manifold topology into some neighboring fields such as dynamics and geometric group theory; subjectively this move felt to me like a transition from a baroque world of highly intricate, finely tuned and beautiful objects to more rough and disordered domains in which the rule was chaos and disorder, and where one had to restrict attention and focus to find the kinds of structured objects that one can say something about mathematically. In these new domains my familiarity with 3-manifold topology was always extremely useful to me, but almost always as a source of inspiration or analogy or example, rather than that some specific theorem about 3-manifolds could be used to say something about groups in general, or dynamical systems in general, or whatever. Many important recent developments in geometric group theory are generalizations of geometric ideas which were first identified or studied in the world of 3-manifolds; but there was not much connection at the deepest level, at least as far as I could see.

This impression was dramatically shaken by Agol’s proof of the virtual Haken conjecture and virtual fibration conjectures in 3-manifold topology by an argument which depends for one of its key ingredients on the theory of non-positively curved cube complexes — a subject in geometric and combinatorial group theory which, while inspired by key examples in low-dimensions (especially surfaces in the hands of Scott, and graphs in the hands of Stallings), is definitely a high-dimensional theory with no obvious relations to manifolds at all. Even so, the transfer of information in this case is still from the “broad” world of group theory to the “special” world of 3-manifolds. It shows that 3-manifold topology is even richer than hitherto suspected, but it does not contradict the idea that the beautiful edifice of 3-manifold topology is an exceptional corner in the vast unstructured world of geometry.

I have just posted a paper to the arXiv, coauthored with Henry Wilton, and building on prior work I did with Alden Walker, that aims to challenge this idea. Let me quote the first couple of paragraphs of the introduction:

Geometric group theory was born in low-dimensional topology, in the collective visions of Klein, Poincaré and Dehn. Stallings used key ideas from 3-manifold topology (Dehn’s lemma, the sphere theorem) to prove theorems about free groups, and as a model for how to think about groups geometrically in general. The pillars of modern geometric group theory — (relatively) hyperbolic groups and hyperbolic Dehn filling, NPC cube complexes and their relations to LERF, the theory of JSJ splittings of groups and the structure of limit groups — all have their origins in the geometric and topological theory of 2- and 3-manifolds.

Despite these substantial and deep connections, the role of 3-manifolds in the larger world of group theory has been mainly to serve as a source of examples — of specific groups, and of rich and important phenomena and structure. Surfaces (especially Riemann surfaces) arise naturally throughout all of mathematics (and throughout science more generally), and are as ubiquitous as the complex numbers. But the conventional view is surely that 3-manifolds per se do not spontaneously arise in other areas of geometry (or mathematics more broadly) amongst the generic objects of study. We challenge this conventional view: 3-manifolds are everywhere.

It’s been a while since I last blogged; the reason, of course, is that I felt that I couldn’t post anything new before completing my series of posts on Kähler groups; but I wasn’t quite ready to write my last post, because I wanted to get to the bottom of a few analytic details in the notorious Gromov-Schoen paper. I am not quite at the bottom yet, but maybe closer than I was; but I’m still pretty far from having collected my thoughts to the point where I can do them justice in a post. So I’ve finally decided to put Kähler groups on the back burner for now, and resume my usual very sporadic blogging habits.

So the purpose of this blog post is to advertise that I wrote a little piece of software called kleinian which uses the GLUT tools to visualize Kleinian groups (or, more accurately, interesting hyperbolic polyhedra invariant under such groups). The software can be downloaded from my github repository at

https://github.com/dannycalegari/kleinian

and then compiled from the command line with “make”. It should work out of the box on OS X; Alden Walker tells me he has successfully gotten it to compile on (Ubuntu) Linux, which required tinkering with the makefile a bit, and installing freeglut3-dev. There is a manual on the github page with a detailed description of file formats and so on.

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 $CAT(K_0)$.