After a couple years of living out of suitcases, we recently sold our house in Pasadena, and bought a new one in Hyde Park. All our junk was shipped to us, and the boxes we didn’t feel like unpacking are all sitting around in the attic, where the kids have been spending a lot of time this summer. Every so often they root through some box and uncover some archaeological treasure; so it was that I found Lisa and Anna the other day, mucking around with a Rubik’s cube. They had persisted with it, and even managed to get the first layer done.

I remember seeing my first cube some time in early 1980; my Dad brought one home from work. He said I could have a play with it if I was careful not to scramble it (of course, I scrambled it). After a couple of hours of frustration trying to restore the initial state, I gave up and went to bed. In the morning the cube had been solved – I remember being pretty impressed with Dad for this (later he admitted that he had just taken the pieces out of their sockets). Within a year, Rubik’s cube fever had taken over – my Mum bought me a little book explaining how to solve the cube, and I memorized a small list of moves. I remember taking part in an “under 10” cube-solving competition; in the heat of the moment, I panicked and got stuck with only two layers done (since there were only two competitors, I came second anyway, and won a prize: a vinyl single of the Barron Knights performing “Mr. Rubik”). The solution in the book was a procedure for completing the cube layer by layer, by judiciously applying in order some sequence of operations, each of which had a precise effect on only a small number of cubelets, leaving the others untouched. In retrospect I find it a bit surprising – in view of how much effort I put into memorizing sequences, reproducing patterns (from the book), and trying to improve my speed – that I never had the curiosity to wonder how someone had come up with this list of “magic” operations in the first place. At the time it seemed a baffling mystery, and I wouldn’t have known where to get started to come up with such moves on my own. So the appearance of my kids playing with a cube 33 years later is the perfect opportunity for me to go back and work out a solution from first principles.

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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.

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A few weeks ago, Ian Agol, Vlad Markovic, Ursula Hamenstadt and I organized a “hot topics” workshop at MSRI with the title Surface subgroups and cube complexes. The conference was pretty well attended, and (I believe) was a big success; the organizers clearly deserve a great deal of credit. The talks were excellent, and touched on a wide range of subjects, and to those of us who are mid-career or older it was a bit shocking to see how quickly the landscape of low-dimensional geometry/topology and geometric group theory has been transformed by the recent breakthrough work of (Kahn-Markovic-Haglund-Wise-Groves-Manning-etc.-) Agol. Incidentally, when I first started as a graduate student, I had a vague sense that I had somehow “missed the boat” — all the exciting developments in geometry due to Thurston, Sullivan, Gromov, Freedman, Donaldson, Eliashberg etc. had taken place 10-20 years earlier, and the subject now seemed to be a matter of fleshing out the consequences of these big breakthroughs. 20 years and several revolutions later, I no longer feel this way. (Another slightly shocking aspect of the workshop was for me to realize that I am older or about as old as 75% of the speakers . . .)

The rationale for the workshop (which I had some hand in drafting, and therefore feel comfortable quoting here) was the following:

Recently there has been substantial progress in our understanding of the related questions of which hyperbolic groups are cubulated on the one hand, and which contain a surface subgroup on the other. The most spectacular combination of these two ideas has been in 3-manifold topology, which has seen the resolution of many long-standing conjectures. In turn, the resolution of these conjectures has led to a new point of view in geometric group theory, and the introduction of powerful new tools and structures. The goal of this conference will be to explore the further potential of these new tools and perspectives, and to encourage communication between researchers working in various related fields.

I have blogged a bit about cubulated groups and surface subgroups previously, and I even began this blog (almost 4 years ago now) initially with the idea of chronicling my efforts to attack Gromov’s surface subgroup question. This question asks the following:

Gromov’s Surface Subgroup Question: Does every one-ended hyperbolic group contain a subgroup which is isomorphic to the fundamental group of a closed surface of genus at least 2?

The restriction to one-ended groups is just meant to rule out silly examples, like finite or virtually cyclic groups (i.e. “elementary” hyperbolic groups), or free products of simpler hyperbolic groups. Asking for the genus of the closed surface to be at least 2 rules out the sphere (whose fundamental group is trivial) and the torus (whose fundamental group \mathbb{Z}^2 cannot be a subgroup of a hyperbolic group). It is the purpose of this blog post to say that Alden Walker and I have managed to show that Gromov’s question has a positive answer for “most” hyperbolic groups; more precisely, we show that a random group (in the sense of Gromov) contains a surface subgroup (in fact, many surface subgroups) with probability going to 1 as a certain natural parameter (the “length” n of the random relators) goes to infinity. (update April 8: the preprint is available from the arXiv here.)

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The purpose of this brief blog post is to advertise that I wrote a little piece of software called wireframe which can be used to quickly and easily produce .eps figures of surface for inclusion in papers. The main use is that one can specify a graph in an ASCII file, and the program will then render a nice 3d picture of a surface obtained as the boundary of a tubular neighborhood of the graph. The software can be downloaded from my github repository at

https://github.com/dannycalegari/wireframe 

and then compiled on any unix machine running X-windows (e.g. linux, mac OSX) with “make”.

The program is quite rudimentary, but I believe it should be useful even in its current state. Users are strenuously encouraged to tinker with it, modify it, improve it, etc. If you use the program and find it useful (or not), please let me know.

A couple of examples of output (which can be created in about 5 minutes) are:

braid_iso

and

punct

(added Feb. 20, 2013): I couldn’t resist; here’s another example:

hand

(update April 12, 2013:) Scott Taylor used wireframe to produce a nice figure of a handlebody (in 3-space) having the Kinoshita graph as a spine. He kindly let me post his figure here, as an example. Thanks Scott!

KinoshitaHandlebody

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:

lines_1

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.

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Last week while in Tel Aviv I had an interesting conversation over lunch with Leonid Polterovich and Yaron Ostrover. I happened to mention the following gem from the remarkable book A=B by Wilf-Zeilberger. The book contains the following Theorem and “proof”:

Theorem 1.4.2. For every triangle ABC, the angle bisectors intersect at one point

Proof. Verify this for the 64 triangles for which the angle at A and B are one of 10, 20, 30, \cdots, 80. Since the theorem is true in these cases it is always true.

We are asked the provocative question: is this proof acceptable? The philosophy of the W-Z method is illustrated by pointing out that this proof is acceptable if one adds for clarity the remark that the coordinates of the intersections of the pairs of angle bisectors are rational functions of degree at most 7 in the tangents of A/2 and B/2; hence if they agree at 64 points they agree everywhere.

Leonid countered with a personal anecdote. Recall that an altitude in a triangle is a line through one vertex which is perpendicular to the opposite edge. Leonid related that one day his geometry class (I forget the precise context) were given the problem of showing that the altitudes in a hyperbolic triangle (i.e. a triangle in the hyperbolic plane) meet at a single point — the orthocenter of the triangle. After the class had struggled with this for some time, the professor laconically informed them that the result obviously followed immediately from the corresponding fact for Euclidean triangles “by analytic continuation”. Philosophically speaking, this is not too far from the W-Z example, although the details are slightly more shaky — in particular, the class of Euclidean triangles are not Zariski dense in the class of triangles in constant curvature spaces, so a little more remains to be done.

Actually, one might even go back and rethink the W-Z example — how exactly are we to verify that the angular bisectors intersect at a point for the triangles in question without doing a calculation no less complicated that the general case? Let’s raise the stakes further. After some thought, we see that not only will the intersections of pairs of angle bisectors be given by rational functions of the tangents of A/2 and B/2, but the (algebraic) heights of the coefficients of these rational functions can be easily estimated, and one can therefore compute an effective lower bound on how far apart the intersections of the angle bisectors would be if they were not equal. We can then literally draw the triangles on a piece of physical paper using a protractor, and verify by eyesight that the angle bisectors appear to coincide to within the necessary accuracy. After rigorously estimating the experimental errors, we can write qed.

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The other day by chance I happened to look at Richard Kenyon’s web page, and was struck by a very beautiful animated image there. The image is of a region tiled by colored squares, which are slowly rotating. As the squares rotate, they change size in such a way that the new (skewed, resized) squares still tile the same region. I thought it might be fun to try to guess how the image was constructed, and to produce my own version of his image.

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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.

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