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 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” of the random relators) goes to infinity. (update April 8: the preprint is available from the arXiv here.)

First let’s start with the precise definition of a random group. There are actually two parameters in the definition — the density and the length . A random group at density and length is obtained by fixing a finite generating set with at least 2 elements, and adding “random” reduced words of length as relators, where the number of relators to add is governed by the density . Precisely, is the multiplicative density of the relators. There are (approximately) cyclically reduced words of length , so we choose subwords, independently and with the uniform measure, as our relators , and then define  to be our “random group”.

Gromov introduced random groups and established some of their basic properties. One talks about a random group at density , and says that it has a certain property with overwhelming probability. What this means is that with fixed , the probability that the property holds goes to 1 as . Gromov showed that there is a remarkable phase transition in this definition. Explicitly, he showed:

Theorem (Gromov): A random group has the following properties with overwhelming probability:

1. At the group is either trivial or isomorphic to ;

2. At the group is infinite, hyperbolic, and 2-dimensional; and

3. At the group satisfies the small cancellation condition .

The story at density is more subtle, and it is not so clear what happens, as far as I know. (update April 6: Piotr Przytycki points out that the one-endedness of random groups is actually due to Dahmani-Guirardel-Przytycki. Thanks Piotr!)

With this definition, the main theorem Alden and I prove is the following:

Theorem (Calegari-Walker): A random group at density contains many quasiconvex surface subgroups, with probability .

In particular, they contain surface subgroups with overwhelming probability. In fact, at the MSRI conference I gave a partial announcement of this theorem, saying only that we could prove the existence of surface subgroups at “some positive density”; I was worried about the fact that at density the group is no longer and therefore not a small cancellation group in the classical sense. However, it turns out that Yann Ollivier developed enough elements of a kind of small cancellation theory for random groups at any that the argument can be pushed all the way.

The proof contains some technical details, but I believe that some of the main ideas of the proof can be given in a blog post. But before I do so, I think it is worth discussing (very) briefly why one might be interested in finding surface subgroups.

For certain classes of hyperbolic groups — for example, fundamental groups of hyperbolic 3-manifolds — finding a surface subgroup was always known to be an important question to give insight into the virtual Haken conjecture. In fact, the Kahn-Markovic construction of such subgroups turned out to be one of the key steps in the eventual proof of that conjecture by Agol. But even beyond 3-manifolds per se, surface subgroups play an important role. At the MSRI conference Vlad Markovic talked about an approach he has to Cannon’s Conjecture — which says if is a hyperbolic group whose boundary is homeomorphic to a 2-sphere, then is virtually isomorphic to a (hyperbolic) 3-manifold group — and his approach depends on being able to find “enough” (quasiconvex) surface subgroups of . I asked Gromov (by email) what had motivated him to pose this question; I don’t think he would mind if I shared his reply, which was:

I do not remember exactly my motivations and heuristic evidence in favor of the existence of “many surface groups in many hyperbolic groups” except for connectedness arguments at the boundaries, but I had  (and am having) a feeling that these are essential structural components of hyperbolic groups.

My own view, and my main interest in this question, is stimulated by a belief that surface groups (not necessarily closed, and possibly with boundary) can act as a sort of “bridge” between hyperbolic geometry and symplectic geometry (through their connection to causal structures, quasimorphisms, stable commutator length, etc). Surface groups are the “simplest” kind of hyperbolic groups after free groups, and surfaces themselves are the “simplest” class of symplectic manifold; any route between the two kinds of geometry must surely say a lot about surfaces. In this vein, I should remark that in the world of 3-manifold topology (where these issues are infinitely better understood), surfaces again play the premier role in both worlds: minimal/pleated/shrinkwrapped surfaces in the hyperbolic world, norm minimizing/pseudoholomorphic/convex in the contact/symplectic world. It is worth remarking that for the longest time embedded surfaces played a preeminent role in both theories, but that recent breakthroughs (on the hyperbolic side) have depended on developing a deep understanding of immersed surfaces. I wonder whether there is an important role for immersed surfaces on the symplectic side (in -manifold topology)? Maybe a reader who is an expert on Heegaard Floer homology can offer an opinion.

OK, let’s move on to the proof of the Random Group Surface Subgroup Theorem. The first step of the proof builds on a construction in our paper Surface subgroups from Linear Programming, where we show that a sufficiently random homologically trivial collection of cyclic words in a free groups can be taken to bound a certain kind of combinatorial object called a Folded Fatgraph (this result also underpins the main theorem in my recent related paper Random graphs of free groups contain surface subgroups, joint with Henry Wilton). A fatgraph is just an ordinary graph together with a choice of cyclic ordering on the edges incident to each vertex. Such a graph can be canonically fattened to a compact surface (with boundary) in which it lies as a spine. Stallings famously observed that an immersion (i.e. a locally injective simplicial map) between graphs is injective on fundamental groups; such a map of graphs is said to be folded. Thus a folded fatgraph gives an injective surface (with boundary!) subgroup of a free group with prescribed boundary.

The first step in our paper is to make this result more quantitative. A trivalent fatgraph with reduced boundary words is necessarily folded. Our first main result is the following

 Thin Fatgraph Theorem: If is a sufficiently random homologically trivial collection of cyclically reduced words in a free group , then for any there is some depending only on so that copies of bounds a trivalent fatgraph in which every edge has length at least .

These fatgraphs have very long edges and are trivalent; hence are “thin”. Let me not say anything about the proof except that the first part of it closely models the proof of Thm 8.9 from our SSLP paper linked above, but the last step (which was done by computer in the SSLP paper) depends on an elementary but complicated combinatorial argument (which takes up almost half the paper!). (It is worth remarking that this last combinatorial step has something morally in common with the Kahn-Markovic proof of the Ehrenpreis conjecture via the theory of “good pants homology”, in that we want to cancel some collection of “superfluous” short loops which can be thought of as random excitations on the surface of a (Dirac) sea of perfectly equidistributed loops. I should also remark that some version of this theory — “pants homology” if you will — was earlier developed by me in my paper Faces of the scl norm ball, in which I showed that every homologically trivial immersed collection of geodesics on a hyperbolic surface virtually cobounds an immersed subsurface with a sufficiently large multiple of the boundary.)

By the way, it is natural to wonder just how “random” the collection needs to be for the conclusion of the theorem to hold (technically, we work with a deterministic property called “pseudorandomness” which is a kind of controlled equidistribution at certain scales). One can ask how long a random cyclically reduced (homologically trivial) word needs to be before it bounds a trivalent fatgraph (with, for the sake of concreteness, no constraint on the length of edges). This is a question that can be addressed experimentally by computer. The results are very interesting. For rank 3, we looked at between 100000 and 400000 such words of each even length from 10 to 120. The proportion of such words that bound trivalent fatgraphs is plotted below:

The first time we did this experiment, we only looked at words up to length 50 or so; needless to say, this gives a somewhat misleading idea of the asymptotic picture!

How can one use thin fatgraphs to build surface subgroups? Before tackling a random group, let’s consider a one-relator group with a single (long, random) relator . We can imagine building a (polygonal) surface out of disks, each of which has either or on its boundary, where the disks are glued to each other along mutually inverse subwords of the boundary words. Since a random word will probably not be homologically trivial, we build a surface out of disks labeled and disks labeled , where is as in the Thin Fatgraph Theorem. The 1-skeleton of is a graph, and the way in which it sits in gives it a fatgraph structure.

The first thing one might think therefore is that one should just apply the Thin Fatgraph Theorem to build a fatgraph bounding . One can do this, but why should one expect the resulting surface to be injective? In order for the surface to fail to be injective there must be some essential loop in the 1-skeleton which bounds a van Kampen disk in the group. Without loss of generality, we can assume that this disk has a minimal number of faces; note that each face has either or on its boundary. A (random) 1-relator group is hyperbolic; in fact, it is for any with overwhelming probability, when the length of the relator gets long. So in such a van Kampen diagram there must be very long subwords (of length ) in or which are subwords of . Of course, does contain long subwords of and ; the boundary of the fattening of consists entirely of such words! But in a minimal van Kampen diagram such “boundary” subwords must not occur, and the question is whether contains long subwords in common with or that are not boundary-parallel.

A counting estimate gives the following heuristic answer. By the defining property of a Thin Fatgraph, for any there are paths in of length starting at any point, and only starting points. On the other hand, there are random reduced words of length , and the relator contains at most of them. The difficulty in making this argument rigorous is that the fatgraph is not independent of ; in fact it is constructed “from” in a direct sense! So the trick is to break up into small subwords, and build thin fatgraphs bounding each subword, and then each small thin fatgraph will be independent of the other subwords.

Explicitly, we find what we call a Bead Decomposition of ; this is a decomposition of into subwords of length which start and end with mutually inverse subwords of length . The inverse subwords at the start of each are paired, to produce a collection of beads of size , separated by intervals of length called necks. Each bead on its own will probably be homologically essential, but we can perform a bead decomposition at “the same” locations in the word to get a collection of pairs of inverse beads of length . Taking copies of each pair of beads, we can build a thin fatgraph that bounds it, and then these thin fatgraphs are joined one to the next along necks. By construction, the subwords contained in the spine of the fatgraph bounding a bead are independent of the subwords in for , so with overwhelming probability, they have no long subwords in common. The necks are sufficiently long that whenever a subword passes over a neck, another copy of that subword cannot appear within distance for some (with high probability). But the existence of a van Kampen diagram would give rise to a long string of such coincidences, and therefore we deduce that no van Kampen diagram exists, and the surface is injective.

We now throw in an additional random relators of length independently, and with the uniform measure. Now the naive counting argument above is rigorous, and each additional relator is unlikely to have a long segment in common with a subpath in the spine . In fact, what can be shown is that for each there are of order relators that have of their boundary in common with a subpath of (this common part does not need to be consecutive, but we do need to bound the number of connected components by some constant independent of ; this is where Ollivier’s small cancellation work comes in to bootstrap such “local” small cancellation estimates to “global” ones). From this argument, and some elementary reasoning with van Kampen diagrams, the result follows.

One subtlety is that it is necessary to control the size of the van Kampen diagrams we consider independently of . A path in a hyperbolic group which is not quasigeodesic can be shortened on a segment of size , where is the constant of hyperbolicity. Ollivier shows that is linear in , for fixed , and therefore we can obtain estimates on the probability that fails to be injective by considering van Kampen diagrams containing a bounded number of disks.

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:

and

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

(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!

So the “correct” answer to the puzzle is 7 (and the sequence continues 11, 26, ). 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.

Actually, there is no great mystery about where the missing 8th region went. To ensure general position, I first needed to choose lines which were not parallel to each other, so let’s suppose that I have chosen a direction for the lines in advance. As I lay them in the plane one by one, I must also make sure that they don’t intersect an existing crossing. Since I have already chosen a direction for the line, I will either lie to the left or to the right of each existing crossing. After laying two lines, there is one crossing, so there are two choices for how I should lay the third line (given its direction); one choice gives the arrangement above; the other choice gives the following arrangement, which contains the missing 8th region:

Here I am thinking of the two different ways of arranging 3 lines in the plane as being related by a certain kind of “move” which translates the lines but does not turn them. The complementary regions are then specified by knowing on which side of each of the 3 lines they lie. We can think of this as a kind of (binary) code: if we orient each of the three lines, we denote a region to the left of it by L and a region to the right of it by R. Thus each region is coded by a three letter word in the alphabet L,R, so there are 8 possible regions which we can put in bijection with the numbers from 1 to 8 however we like.

The move on configurations of 3 lines is very closely related to a certain kind of move on knot diagrams called the “Reidemeister 3 move”. Think of the lines as shadows cast by strands of string, and think of moving one strand over a crossing of two other strands. The result (on shadows) is the Reidemeister 3 move:

What about 4 lines? We suppose that the 4 lines are ordered by increasing angle from horizontal, and give each complementary region a binary code depending on which side of the lines it’s on, so that there are 16 regions, which we give labels from 0 to 15. There are 8 configurations of 4 lines with given directions, indicated in the figure.

The unbounded regions — 0, 1, 3, 7, 15, 14, 12, 8 — are present in each configuration. As we “cycle” through these 8 configurations, in the order indicated in the figure, one bounded region appears and one disappears, in the cyclic order 10, 2, 6, 4, 5, 13, 9, 11.

For 5 lines there are many more combinatorial possibilities (even up to topological symmetries of the plane), but we can still get between any two configurations by a sequence of Reidemeister 3 moves, and all 32 regions appear in some configuration (actually, in many configurations). For more than 5 lines the story is similar.

There is a nice duality between arrangements of lines (or more generally, arrangements of hyperplanes) and zonohedra. If you recall from a previous post, a zonohedra is a polyhedron obtained as the Minkowski sum of a collection of intervals; that is, if are intervals in some vector space, the zonohedron is the set of points of the form where each . Zonohedra are simply the (linear) projections to lower dimensional spaces of higher dimensional cubes. Given a zonohedron Z in a Euclidean space, there is a corresponding hyperplane arrangement in a projective space of one dimension lower, defined as follows: for each face F of the zonohedron, one considers the collection of supporting hyperplanes for Z that contain F. This is a polyhedron in projective space of dimension dim(Z) – dim(F) -1. So top dimensional faces give rise to points, codimension two faces give rise to segments, and so on. Each zone of the zonohedron (that is, each equivalence class of parallel edges, corresponding to one of the ) gives rise to the hyperplane in projective space corresponding to hyperplanes in the Euclidean space containing . What is nice about this correspondence is that complementary regions to the hyperplane arrangement correspond to pairs of opposite vertices of the zonohedron. If we consider oriented hyperplanes then we get an arrangement of great spheres on a sphere; in the 2-dimensional case, an arrangement of great circles on the 2-sphere. So the possible regions that can occur (for all configurations) correspond to the vertices of the high dimensional cube, some subset of which project down to become the vertices of the zonohedron. (Note that I have swept under the rug the fact that we are now interested in configurations of great circles on the 2-sphere rather than straight lines in the plane. Three generic great circles on the 2-sphere decompose it into 8 regions, so this is another way of saying where the missing 8th region went: it was hiding round the back of the sphere).

Zones in the zonohedron correspond to midcubes in the high dimensional cube that projects to it — that is, the codimension one cubes that slice symmetrically through the center, parallel to a pair of opposite top-dimensional faces. So we can see a direct correspondence between midcubes in a high dimensional cube, and lines in the plane, or great circles in the sphere for that matter. Since we are already considering straight lines in non-Euclidean geometries, let’s ask what happens if we consider generic configurations of (straight) lines in the hyperbolic plane. Now things get much more interesting. Two generic lines in the hyperbolic plane might intersect, or they might be disjoint. There is an abstract graph whose vertices correspond to the straight lines in the configuration, and whose edges correspond to the pairs of straight lines which intersect. A complete in this graph — i.e. a configuration of n lines each of which intersects the other — gives rise in a canonical way to an n-dimensional cube, whose shadow is the 3-dimensional zonohedron that parameterizes the combinatorics of the configuration. If some is contained as a subgraph of two distinct , we obtain a complex by gluing together the j-cube and k-cube along their corresponding sub i-cube. The resulting space is a cube complex — a combinatorial complex built from cubes by gluings which respect the cubical structure on faces. Unions of midplanes glue together to make combinatorial hyperplanes which correspond precisely to the lines in the configuration. If the arrangement of lines was invariant under some group of hyperbolic isometries, then this group acts naturally and combinatorially on the associated cube complex. For example, if we start with a closed hyperbolic surface and a finite configuration of immersed closed geodesics on , the universal cover is the hyperbolic plane with an interesting arrangement of lines which is invariant under the action of the fundamental group .

In fact, it turns out that the key point is not that the arrangement is of lines, but that it is of codimension one objects. If G is a (finitely generated) group, and H is a subgroup, we say that H is codimension 1 if the quotient of the Cayley graph of G by H has at least two ends. If it does, we can divide the Cayley graph into two H-invariant subsets, so that the frontier has finitely many orbits under the H action; this frontier is a kind of combinatorial hyperplane in G. The G translates of this hyperplane might intersect each other in a complicated way in the Cayley graph. As before, we can build a cube complex, where (roughly speaking) the n-cubes are the collections of n translates of the hyperplane each of which intersects the other in an essential way. The details of this construction can be found in a paper of Sageev (who first thought it up) and the end result is that one obtains a natural action of G on a cube complex. In fact, this cube complex is very nice geometrically — it is simply connected, and non-positively curved, so that if we make it a metric space by declaring that every cube is Euclidean with side lengths of edge 1, the result is CAT(0). The construction works just as well with a finite collection of codimension 1 subgroups instead of just one, and under suitable hypotheses, one shows that the group G is isomorphic to the (orbifold) fundamental group of a compact non-positively curved cube complex. This now becomes extremely relevant to Agol’s proof of the VHC — if G is the fundamental group of a hyperbolic 3-manifold, the surface subgroups constructed by Kahn-Markovic (see these blog posts) provide the raw material from which one builds a cube complex on which G acts, and this is the starting point for Agol’s work; see here for an introduction. I don’t know if Sageev was led from combinatorial hyperplane arrangements to cube complexes via zonohedra, but it’s plausible; and in any case thinking of it in these terms (at least for low dimensional examples) helps me to more easily see where the cubes “come from”.

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, , 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.

While I am off on a tangent, this reminds me of a discussion I once had with Michael Aschbacher about the status of arguments (in topology, say) using diagrams. This could be a computation of the fundamental group of a knot complement by Wirtinger’s algorithm, for example, or a proof that some topological 4-sphere is smoothly standard via Kirby moves. He took what I think is an extreme view, that such arguments are never mathematically valid. This is a bit of a fuzzy argument to have if one is not careful to define precisely what one means by a “diagram” — suppose (as is in fact the case) I draw a diagram by writing a (finite) .eps file in ASCII. Then a “diagram” can be taken to be a certain kind of string in a finite alphabet, and the kinds of reasoning about diagrams one is prepared to accept could also be precisely specified and formalized, and could presumably be shown to be consistent with ZFC. This shows (in some very weak sense) that it is possible to conceive of a theory of “reasoning by diagrams” which must be respectable even to Michael Aschbacher. However, in practice one “reasons using diagrams” (just as one reasons in every other context) by a combination of explicit formal rules and pre logical “leaps”: if I extend this line indefinitely, it will intersect that line here; or, if I pick up this strand and pull it behind the other strand, it will eliminate these three crossings and introduce a new crossing here. And so on. If one pursues this line of reasoning too far it starts to degenerate into questions about the reliability of short term memory, or the psychophysics of perception, which throw any kind of mathematical reasoning in question. But before reaching that point, one can argue (and Aschbacher did argue) that arguments involving diagrams are “special” because of the sheer quantity and sophistication of the pre logical leaps involved. Anyone who has seen how much effort is involved in translating e.g. the Jordan curve theorem into a formal proof system like HOL light might be prepared to concede that Aschbacher has a point.

Anyway, back to hyperbolic orthocenters. If one substitutes spherical for hyperbolic geometry, there is quite a cute proof of the existence of an orthocenter as follows. Let’s fix the unit sphere in 3-space, and let ABC be a Euclidean triangle in a plane tangent to the unit sphere and touching it exactly at the orthocenter O of ABC. Radial projection of the vertices determines a spherical triangle A’B'C’. I claim that the radial projection of the altitudes of ABC become altitudes of A’B'C’, and therefore these altitudes intersect in O, which turns out also to be the (spherical) orthocenter of the (spherical) triangle A’B'C’. To see the validity of the claim, observe first that the radial projection of a straight line in to the sphere is a great circle on the sphere; so if L is any straight line in through O, the radial projection L’ is a great circle through O. Second, note that if M is a straight line in perpendicular to L (as above), the radial projection M’ is a great circle perpendicular to L’; this follows by symmetry: reflection in the plane through the origin containing L takes M to itself and therefore M’ to itself, while fixing L’ pointwise. This proves the claim, and therefore that the (spherical) altitudes of A’B'C’ intersect at O’. By a dimension count, all spherical triangles arise in this way; qed. At this point the appeal to analytic continuation (from spherical to hyperbolic geometry) is more persuasive.

I already know a little bit about square tilings. This is a subject with a history, going back at least to the work of Tutte and his colleagues. The basic problem is just to tile a rectangular region by squares. Easy enough, you say.

Well, yes; if the rectangle has sides which are rationally related, in can be filled up by squares with commensurable side lengths pretty easily. Here a 4 by 8 rectangle is filled with 7 squares of edge length 1, 1 square of edge length 3, and 1 square of edge length 4. It’s more amusing to look for a tiling in which all the squares have different lengths. One well-known tiling, found by Tutte’s colleague Stone, is as follows:

Obviously the problem becomes more interesting and challenging if one starts in advance with the combinatorics of a square tiling, and then tries to assign edge lengths to squares in such a way that they fit together nicely. One elegant method, developed by Brooks, Smith, Stone and Tutte, assigns a directed graph to the tiling, with one vertex for each vertical edge (say) and one directed edge for each square. Here’s the graph associated to Stone’s tiling:

The condition that the sum of square lengths on either side of a vertical edge sum to the length of that edge implies that the incoming edge weights and the outgoing edge weights at each vertex sum to the same value (except for at the leftmost and rightmost vertices). On the other hand, the fact that the squares are all square implies that each edge weight (as above) is equal to the length of its projection to a horizontal line; this means that the sum of edge weights around each loop in the graph (with sign changed when the orientation disagrees with the orientation on the loop) is equal to zero. These two conditions are precisely Kirchoff’s two laws for the current flowing through an electrical network where every edge has resistance 1, and the voltage difference between left and right vertices is the width of the rectangle. There is a unique solution; it might have some weights negative, in which case we can reverse the orientation of the edge so that the weights are all positive, and determine a square tiling with slightly different combinatorics. By the way, the uniqueness of the solution has an interesting (and well-known) consequence: since Kirchoff’s laws both impose linear conditions on the edge weights, the space of solutions is a rational affine space (in units for which the width is equal to 1). Since this space of solutions consists of a single point, this point has rational coordinates; this implies in particular that the height of the rectangle is a rational multiple of the width, and so are the widths of the squares.

In more homological language, the assignment of weights to edges is a (simplicial) 1-chain. The condition that the incoming and outgoing edge weights at each vertex have equal sum says that this 1-chain is actually a (relative) 1-cycle; i.e. that it is closed. The condition that the sum around every loop is zero says that if we think of this 1-chain as a 1-cochain it is actually a 1-cocycle; i.e. it is co-closed. A (co)-chain which is both closed and co-closed is said to be harmonic, and the uniqueness of a solution corresponds to the uniqueness of a harmonic representative of a (relative co-) homology class.

Incidentally, if we form the graph with one vertex for each vertical horizontal line and one edge for each square, this will be the (planar) dual to the graph above. Edges in one graph correspond to edges in the other, and the closed condition for one set of edge weights becomes the co-closed condition for the other, and vice versa.

Now instead of considering a square tiling of a rectangle, let’s consider a square tilings of a Euclidean torus. A combinatorial tiling gives us a graph, and a harmonic 1-cycle gives us a square tiling with the desired combinatorics. Changing the 1-cycle by rescaling it just rescales the torus and all the squares by the same factor, which is not very interesting. However, there is something interesting we can do. The homology of a torus is 2-dimensional, so we can consider a 1-parameter family of homology classes whose projective classes are changing, and a 1-parameter family of harmonic 1-cycles and of square tilings.

Let’s start with the simplest possible example. We fix a graph G embedded in the torus. Since we want G to be able to carry every homology class, we need at least two edges. So let’s take as G the graph with one vertex and two edges, one of which wraps horizontally once around the torus, and one of which wraps vertically around. Any assignment of weights to the edges will be both closed and co-closed, so a 1-parameter family is given by taking weights cos(t), sin(t) for t in the unit circle. The resulting square tilings of the torus have two squares, one of side length cos(t) and one of side length sin(t). The total area of the torus is thus normalized to be 1. The pattern of tilings “rotates” with t as follows:

(click on the image to see it rotate)

OK, how about a more complicated example? Let’s let G be some complicated embedded graph on the torus (so that it can carry any homology class). For the sake of concreteness, let’s let G be the following graph:

G has 10 edges (corresponding to 10 squares in the tiling), 5 vertices and 5 complementary faces. There are 5 vertex conditions and 5 face conditions; however, this system of 10 equations is redundant, and has a 2 dimensional space of solutions.

Weights on the edges of G form a vector space, and there is an inner product on this space which is just the ordinary Euclidean inner product with co-ordinates the weights on each each edge. We want to normalize our weights to have length (i.e. square root of their inner product with themselves) equal to 1, so that the resulting torus will have area 1. All we need to do is find two orthogonal weights M and L which are closed and co-closed, orthogonal to each other (i.e. the inner product of M and L is zero) and of length 1, and then we can form the family cos(t)M + sin(t)L of weights, and the associated square tilings.

The resulting rotating family of tilings is as follows:

(click on image to see it rotate)

Something else is needed to get the “spiraling” evident in Kenyon’s picture. For our square tilings of a torus above, the result of laying down a sequence of squares that winds once around a loop in the torus is to displace the tiling by a translation of the plane; this translation is called the holonomy around the loop, and only depends on its homotopy class (actually: on its homology class). Essentially, this is the result of integrating the (dual) 1-form associated to the weight. An educated guess is that in Kenyon’s picture, the holonomy is not a translation, but rather a dilation of the plane, centered at some point. At the level of homology, one can think of the dilation factor around a loop as a representation of the fundamental group, and we need to consider (harmonic) 1-cycles with coefficients twisted by this representation.

How to translate this into the language of square tilings and weights? Instead of thinking of a weight on the graph G, let’s let G~ denote the lift of G to the universal cover of the torus; i.e. G~ is a periodic graph in the plane. A twisted weight on G with coefficients in a representation is the same thing as a weight on G~ that transforms according to the given representation. For the sake of simplicity, let’s work with the graph G with one vertex and two edges as in the first example above, so that G~ has one vertex, one horizontal edge, and one vertical edge for each pair of integers. Pick a pair of edges H,V of G~, going to the right and up incoming to the vertex (0,0) respectively and let h,v be the weights on these edges.

If we let A denote the multiplication factor for horizontal translation, and B the multiplication factor for vertical translation, the vertex equation at (0,0) is

The vertex equations at every other vertex are obtained from this one by scaling by power of A and B, so they are satisfied if this one is. The face equation for the face with vertices (-1,-1), (0,-1), (0,0), (-1,0) is

Eliminating h from this pair of equations and dividing out by v gives

In order to enforce spiraling, we would like moving “horizontally” some fixed number of steps to be the same as moving “vertically” some (other) fixed number of steps; this can be imposed by setting for some coprime integers p,q. With these constraints, there is a unique solution h,v in complex numbers, up to scale. The real  part of any such solution gives a “spiral” tiling, and the 1-parameter family obtained by multiplying by before taking the real part gives a rotating spiral.

Let’s try an example. Taking p=2,q=1 gives and . There is a totally real solution, giving rise to the following “degenerate” spiral:

Since this solution is totally real, it can’t be “rotated”. Hmm, I wasn’t expecting that. OK, taking p=3,q=1 gives and .

(click on image to see it rotate)

Success!

Getting more squares in the picture is a matter of spiraling slower, which can be achieved by taking p and q bigger. Let’s try p=7,q=1.

(click on image to see it rotate)

If you want to have a play with this yourself, the source of the .eps file that generated these figures is below. To change the amount of spiraling, change the values of A and B, subject to the constraint that . The resulting .eps file can be transformed to a layered .pdf (eg using Preview on a Mac) then to a .gif (eg in gimp). The case q=1 is pretty easy, since then A is the root of with smallest (nonzero) argument, and . Wolframalpha will cough up the values of A and B if you coax it long enough.

(Update January 16): Just for fun, here’s the tiling with p=101, q=1 (warning: the .gif file is quite large!)

(click on image to see it rotate)


%!PS-Adobe-2.0 EPSF-2.0
%%BoundingBox: 0 0 400 400
gsave
400 400 scale
1 20 div setlinewidth
1 setlinejoin
0.5 0.5 translate
/square{4 dict begin
/z exch def
/y exch def
/x exch def
gsave
newpath
rand 10 mod 10 div rand 10 mod 10 div rand 10 mod 10 div setrgbcolor
x y moveto
x z add y lineto
x z add y z add lineto
x y z add lineto
closepath
fill
stroke
grestore
end } def
/simple_edge_squares{4 dict begin
/v exch def
/n v length def
0 1 n 1 sub{
/i exch def
0 0 v i get square
0 v i get translate
} for
end} def
/rcmul{2 dict begin
/t exch def
/z exch def
[ z 0 get t mul z 1 get t mul]
end} def
/ccmul{2 dict begin
/w exch def
/z exch def
[
z 0 get w 0 get mul z 1 get w 1 get mul sub
z 0 get w 1 get mul z 1 get w 0 get mul add
]
end} def
/cconj{1 dict begin
/z exch def
[
z 0 get 0 z 1 get sub
]
end} def
/cnorm{1 dict begin % |z|^2
/z exch def
z 0 get dup mul z 1 get dup mul add
end} def
/ccdiv{2 dict begin % w/z = w*zbar/|z|^2
/z exch def
/w exch def
w z cconj ccmul 1 z cnorm div rcmul
end} def
/ccadd{2 dict begin
/w exch def
/z exch def
[ z 0 get w 0 get add z 1 get w 1 get add ]
end} def
/creal{1 dict begin
/z exch def
z 0 get
end} def
/cimag{1 dict begin
/z exch def
z 1 get
end} def
0 5 355 {
/t exch def
0 srand
gsave
/A [0.71469 -0.870643] def % A is root of x^14+x^8-4x^7+x^6+1=0
/B [0.432505 -0.0429583] def % B = A^-7
% check: A^3B=1
/h [t cos t sin] def
/v h A [-1 0] ccadd ccmul [1 0] B -1 rcmul ccadd ccdiv def %
% h*(A-1)/(1-B)
/Ainv [1 0] A ccdiv def
/Aser [1 0] Ainv ccadd Ainv Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul Ainv ccmul Ainv ccmul ccadd def
/Acom [1 0] Ainv -1 rcmul ccadd def
/htran h Acom ccdiv def
/vtran v Acom ccdiv def
t rotate
htran creal vtran creal -1 mul translate
[h A ccmul creal v B ccmul creal h [-1 0] ccmul creal v [-1 0] ccmul creal] simple_edge_squares
1 1 50{
h -1 rcmul creal v creal translate
/h h A ccdiv def
/v v A ccdiv def
[h A ccmul creal v B ccmul creal h [-1 0] ccmul creal v [-1 0] ccmul creal] simple_edge_squares
} for
showpage
grestore
} for
grestore
%eof

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.

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 . This example illustrates nicely some themes in probability and group dynamics, and lends itself easily to visualization.

Let’s work in the Poincaré unit disk model of hyperbolic geometry. In this model, the hyperbolic plane is thought of as the interior of the unit disk in the Euclidean plane, and the hyperbolic metric is related to the Euclidean metric by multiplying distances infinitesimally by at a point whose (Euclidean) distance from the origin is . In this model, the hyperbolic distance between a point at the origin and a point at Euclidean distance away is . So, at the risk of being slightly confusing, let me say that a hyperbolic random turtle has “step size S” if the first step, starting at the origin, lands on the Euclidean circle of radius S.

I wrote a little program called turtle to illustrate the motion of a random turtle for various values of S and A; it can be downloaded from my github repository if you want to play with it. The figures below are all produced with it. Let’s look at a few examples.

The phase transition alluded to earlier is very evident in these pictures: for large S and small A, the turtle zooms off in an almost straight line to the boundary, with very little wiggling along the way. For small S and large A, the turtle meanders around aimlessly, filling up lots of space, intersecting its path many times, until eventually wandering off to the boundary in a more or less random direction.

For a given length, what is the critical turning angle? The “worst case” scenario is a turtle which always turns left (or always turns right). For such a turtle there is a critical angle (for a given length) such that the trajectory of the turtle just fails to close up. Technically, the hyperbolic isometry describing the turtle’s motion at each step is parabolic, and fixes a unique point at infinity. The segments of the turtle’s trajectory will then osculate an invariant horocycle for the parabolic isometry, when the (discrete) atoms of positive turning curvature at the vertices exactly balance the negative curvature of the hyperbolic plane.

A critical turtle trajectory osculates a horocycle

The critical relationship is precisely that , with our convention about the relationship between S and the hyperbolic length of the segments. For angles smaller than this value, the trajectory is a quasigeodesic — i.e. it stays within a bounded (hyperbolic) distance of an honest geodesic, and does not wind around at all. For angles bigger than this value, there is a definite probability at every stage that the trajectory will undergo some number of complete full turns, and it might return to some region it has visited before. The trajectory still converges to a point at infinity with probability one (this is a very robust feature of random walk in negatively curved spaces) but it makes deviation of order from this geodesic in the first steps.

One interesting statistic for an immersed path in the plane is the winding number. If we trivialize the unit tangent bundle, the derivative can be thought of as a map to the circle, and we can ask how many times it winds around. In the Euclidean plane there is a natural trivialization of the unit tangent bundle via parallel transport, because of the flatness; technically there is a flat orthogonal connection. In the hyperbolic plane any orthogonal connection must have curvature, but there is a flat connection with structure group equal to the group of (hyperbolic) isometries, by identifying the unit circle in each tangent bundle with the circle at infinity. Explicitly: every tangent vector is tangent to a unique oriented geodesic which limits to a unique point in the circle at infinity. This identification is global, and respected by the natural action of the isometry group.

For a random turtle in the Euclidean plane, the trajectory turns left or right through angle A at every step, and the winding number after some number of steps is distributed like simple random walk on the integers. That is, if denotes the winding number after steps, then the random variable converges to a normal distribution with mean zero and standard deviation A. The point is that the increments at every stage are independent and identically distributed. On the other hand, for a random turtle in the hyperbolic plane, each step induces an isometry of the hyperbolic plane, and thereby a projective transformation of the boundary circle. There is no natural invariant metric on this boundary circle, and therefore it is more subtle to compute winding number from this action.

Let’s abstract the discussion somewhat. Suppose we are given a finite collection of (orientation-preserving) homeomorphisms of the circle. The circle is covered by the line, and the group of orientation-preserving homeomorphisms of the circle is covered by the group of orientation-preserving homeomorphisms of the line that commute with integer translation.  Call this covering group , where the tilde denotes central extension. Poincaré’s rotation number is a function from  to the real numbers, whose reduction mod the integers is the usual rotation number for a circle homeomorphism. Thinking of our turtle as turning left or turning right continuously implicitly determines a lift of the motion to the universal covering group, so we can suppose that we are given a finite collection of lifts of . Now we consider some random walk where each is drawn independently and uniformly from , and we ask about the distribution of the random variable , which is defined to be the (real valued) rotation number of the composition .

Now, although there is typically no metric/measure on the circle left invariant by there is a natural measure — the so-called harmonic measure — which is invariant on average. If is a probability measure on the circle, we can define , and then let . The have a subsequence converging to a fixed point for the operator ; such a fixed point is a harmonic measure. Note that such a harmonic measure is quasi-invariant under every . The measure pulls back to a locally finite measure on the real line, and this pullback is harmonic for the action of . We can define a function as follows. For each choose some and define . Then is monotone nondecreasing, and for any and any integer . In particular, the winding number satisfies for any .

Now, by the definition of a harmonic measure, for any and for random , there is an equality (here the notation means the expectation of a random function). In particular, is constant independent of . We call this constant quantity the drift and denote it by . Define a sequence of random variables by . By the calculation above we see that for each , the expectation of conditioned on a particular value of is equal to the given value of . More informally, we could just write and say that at every step, the expected change in the value of is zero. This is a familiar object in probability theory, and is known as a martingale. One can think of the values of the martingale as the wealth of a gambler who makes a succession of fair bets. The wealth of such a gambler over time looks roughly like a simple random walk, after reparameterizing time proportional to the rate at which the gambler takes risks (as measured by the variance of the outcomes of each bet). For our random product of homeomorphisms, there are two possibilities: either the martingale converges, as successive “bets” become smaller and smaller, and the winding number converges to some final value (this happens in the case that the length of the turtle’s steps are big compared to the turning angle), or else the position of the point is equidistributed in the circle with respect to , and there is a central limit theorem: converges to a Gaussian.

Returning to our original setup, the left-right symmetry forces the drift to equal zero, and we can identify with the winding number up to a constant. How does the variance of depend on the variables S and A? The following figure shows a graph of the variance as a function of S and A. The red line marks the phase transition from zero variance (i.e. quasigeodesic turtle trajectories) to strictly positive variance.

As one sees from the figure, the phase transition is not something sharp that can be easily seen experimentally, and in fact, the graph looks completely smooth along the phase locus (although we know it can’t be real analytic there). This experimental observation can be theoretically confirmed, as follows.

Consider the behavior of a random turtle, with fixed stepsize, for some turning angle A’ just marginally bigger than the critical angle A. The critical turtle trajectory bounds an infinite polygon with edges of length and external angles A; this polygon can be decomposed into semi-ideal triangles with internal angles and finite side length . As we deform the angle we get a new triangle with angles where , and the angle is opposite a side of fixed length . The hyperbolic law of cosines says in this context that . Since is fixed, and is small, we can approximate ; in other words, the angle is of polynomial (actually, quadratic) order in the difference . Now, suppose for some very large . A turtle trajectory with the property that there is at least one left and at least one right turn in every steps will be quasigeodesic; the only full twists will occur when there is a sequence of at least left turns or right turns in a row. This is a very rare occurrence — it will typically only happen twice in a sequence of steps. Hence the variance of the winding number is of order . In particular, the graph of the variance is tangent to zero to infinite order along the phase locus, as claimed.

(Update:) At Dylan’s request I’ve added a slice of the variance graph, at with angle varying from 0 to 0.2. The vertical axis has been stretched (relative to the 3d graph above) for legibility. The phase transition is at angle 0.1000417 and I must say the graph looks pretty flat there.

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 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 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 , 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 of finite rank, and for a random endomorphism of length (i.e. one taking the generators to random words of length ), the associated HNN extension contains a closed surface subgroup with probability going to 1 as . This result is part of a larger project which we expect to post to the arXiv soon.

The context of this problem is Gromov’s notorious question:

Question(Gromov): Does every 1-ended hyperbolic group contain a surface subgroup?

Actually, it is not at all clear if Gromov really asked this question, or what sort of answer he expected. There is a discussion of this in the introduction to a recent paper by Henry Wilton. A positive answer to this question is known in only a few special cases, including

• Coxeter groups (Gordon-Long-Reid)
• Graphs of free groups with cyclic edge groups and (Calegari)
• Fundamental groups of hyperbolic 3-manifolds (Kahn-Markovic)
• Certain doubles and graphs of free groups with cyclic edge groups (Kim-Wilton, Kim-Oum, Kim, Wilton)

(this list is not exhaustive). One strategy to find a surface subgroup is to define a class of groups with the property that every one-ended hyperbolic group contains a subgroup in the class , and then to show that every group in this class further contains a surface subgroup. A reasonable candidate for the class is the class of one-ended graphs of free groups. The logic behind this choice is that it is very easy to produce many free subgroups of a one-ended hyperbolic group (in fact, this is more or less the only kind of subgroup one knows how to produce) by Klein’s pingpong argument, and one could perhaps argue that because there are so many such subgroups, that intersect in quite rich and interesting ways, a sufficiently rich collection is one-ended while at the same time has the structure of a graph of groups. On the other hand, the structure of a graph of free groups is similar in some ways to the structure of a Haken 3-manifold, and one knows enough about the components of the graph (i.e. the free factors) that one can try to build a surface subgroup by amalgamating surface-with-boundary subgroups along cyclic subgroups of the edge groups.

Anyway, this is more philosophy than mathematics, but it does partly explain why the class has been widely studied by geometric group theorists interested in Gromov’s question. One important class of graphs of groups are the HNN extensions, whose underlying graphs consist of a single vertex and a single edge joining this vertex to itself. An (injective) endomorphism of a free group thus gives rise to an HNN extension in the class .

Now, suppose is a map from a surface subgroup to . There is a homomorphism sending to 0 and the conjugating element to . The kernel intersected with the image of will determine an infinite cyclic cover of , and one would like to determine whether this map is injective. We can think of as an infinite union of subsurfaces with boundary, where each is attached to and , and contained in a conjugate of the subgroup . If we identify each with for , then we can think of . Let denote the union of the with . Evidently it is sufficient to show that the inclusion of to is injective, since any loop in the kernel of is conjugate into . This is convenient, since we can discuss surface-with-boundary subgroups of a fixed free group, and essentially ignore the endomorphism .

The first thing to check is that each separate inclusion is injective. Each may be represented by a certain kind of diagram, called a fatgraph. Basically, a fatgraph is a graph in the usual sense, together with a choice of cyclic ordering of the edges incident to each vertex. A fatgraph embeds canonically as the spine of some surface which itself deformation retracts back to , in such a way that the cyclic order on edges inherited from the embedding agrees with the fatgraph structure. The oriented edges of are labeled with reduced words in in such a way that the labels on opposite sides of an edge of are inverse in . In this way, a fatgraph “represents” a surface-with-boundary mapping to . Here is an example of a (disconnected) fatgraph, whose underlying surface is homeomorphic to the union of two 4-punctured spheres:

Now, the fundamental group of every (component of every) fatgraph is free, but the map to is not necessarily injective. Stallings gave a celebrated criterion for a simplicial map from a graph to a rose (i.e. a standard graph with fundamental group ) to be injective, namely that the map should be folded, or equivalently, that the map should be an immersion on the link of every vertex. In terms of fatgraphs, this means that there should be at most one incoming edge at each vertex with each label. The graph pictured above is folded in this sense. Notice if every boundary word is reduced, a 2- or 3-valent vertex is necessarily (locally) folded.

OK, this is a criterion that will certify that an individual might be injective, when represented as a fatgraph. What about the dynamics of ? Notice that the endomorphism has a particularly nice property: if we think of it as representing a self-map of the standard rose to itself, then the map is an immersion, in the sense of Stallings. This means that if each of the surfaces is represented by a folded fatgraph , then each will be folded if is. This suggests the following definition:

Definition. A fatgraph with associated surface is -folded if there is a decomposition of its boundary into and in such a way that (with the opposite orientation), and satisfying the following properties:

1. The graph is Stallings folded
2. Every -vertex in (i.e. the images under of the vertices of ) is associated to a 2-valent vertex of
3. No vertex of is associated to more than one -vertex in
4. No vertex of is associated to more than one vertex in

When we talk about a vertex of being “associated” to a vertex of we mean that the vertex of maps to the given vertex of under the deformation retraction of to (this deformation retraction is simplicial when restricted to ).

Now, suppose is -folded. We can glue to by gluing to . Condition 4 implies that the resulting surface is where . In a similar way we can define

and . Now, conditions 2 and 3 imply that every vertex of is obtained by gluing some vertex of to a sequence of 2-valent vertices in various with . In particular, since every vertex of is locally folded, the same is true of every vertex of , and therefore also of . Hence is folded, and thus injective. Since as above, it follows that the suspension of an -folded surface is injective in .

The definition of -folded can be modified for an endomorphism which is not an immersion of . One of the main theorems Alden and I prove is that a “random” endomorphism admits many -folded surfaces in this sense, and therefore the associated HNN extension has (many) surface subgroups. For a random endomorphism of length , the genus of these surfaces will typically be of order at least , but the number will grow at least like for genus .

Now, Sapir’s group is certainly not random in any sense; nevertheless, it is possible to search for an -folded surface. A priori finding an -folded surface with given boundary seems to require trying exponentially many gluings, and is apparently impractical. However, Alden and I are able to show that the search for such a surface can be reduced to a linear programming problem, and thus becomes eminently practical. Sure enough, a computer search rapidly found the following example of an -folded surface in Sapir’s group:

In a bit of detail: the picture above is a fatgraph whose boundary decomposes into three components labeled and one component labeled . There is a 3-fold cover whose boundary decomposes into consisting of three copies of and consisting of three components labeled . The components of are the ones indicated by the blue circles, and one can see that they are embedded, satisfying condition 4. The red dots are the -vertices, and one can check that they are distinct and on 2-valent vertices of the fatgraph. Finally, one can check that the surface is folded in the usual sense of Stallings. It follows that the suspension is an injective surface in Sapir’s group, of genus 31.

(added Thursday, February 21, 2013): Jack Button has just posted a paper to the arXiv making the observation that a random HNN extension of a free group (in the sense of Alden and I, as above) will satisfy the small cancellation condition for any as , with probability , and therefore will be the fundamental group of a special cube complex, by a result of Wise. This is good to know, and underlines the extent to which such HNN extensions resemble 3-manifold groups.

What gives this condition some power is the rich class of examples of spaces which are -hyperbolic for some . 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 is said to be , if the following holds. If is a geodesic triangle in , let be a comparison triangle in a simply connected complete Riemannian manifold of constant curvature . Being a comparison triangle means just that the length of is equal to the length of and so on. For any there is a corresponding point in the comparison edge which is the same distance from and as is from and respectively. The condition says, for all as above, and all , there is an inequality .

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 space with is -hyperbolic for some depending only on . The point of this post is to give a short proof of the following fundamental fact:

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

This theorem is very familiar to people working in coarse geometry, especially geometric group theorists. Because it is really a theorem in Riemannian geometry, rather than coarse geometry per se, its proof is often omitted in expositions of the theory; for example, I don’t believe there is a proof in Gromov-Ballmann-Schroeder or Ballmann (I think it is relegated to the exercises), nor is there a proof in Cheeger-Ebin, although one can piece together an argument from some of the ingredients in this last volume. Therefore I thought it might be a useful exercise to give a more-or-less complete exposition, which is reasonably self-contained and complete (Update: Daniel Groves tells me there is a proof in Bridson-Haefliger, which is good to know).

Part of what makes this a slightly fiddly theorem to prove is that one must somehow connect up the algebraic language of local Riemannian geometry with the metric language of distances, triangles, convexity and so on. The argument breaks up nicely into two parts — an infinitesimal comparison which is proved algebraically, and a global comparison which is derived from the local comparison by a “soft” argument. The first, algebraic part is not very deep, but it does contain an interesting nugget or two, which I will try to explain as I go along.

First, let’s briefly recall some of the ingredients of elementary Riemannian geometry. Given a Riemannian metric, there is a unique connection — the Levi-Civita connection — which is torsion-free, and compatible with the metric. We denote this by , so that denotes the covariant derivative of the vector field along the vector field . For three vector fields one defines the curvature tensor . Geometrically, this measures how rotates as one takes holonomy transport around an infinitesimal negatively oriented loop in the - plane. The sectional curvature in the - plane is the ratio

The denominator of this expression is the area of a parallelogram spanned by and , so if are orthogonal and of length 1, it reduces to 1.

If is a point, and is a tangent vector at that point, there is a unique geodesic with and . If is complete, is defined; thus there is an exponential map from to taking to . If is the subspace of spanned by a vector , and , then we can define a vector field along by setting it equal to at , for some constant and for all . The exponential map pushes this vector field forward to a vector field on along , called a Jacobi field; by its construction, a Jacobi field is tangent to a 1-parameter variation of geodesics. A Jacobi field satisfies the Jacobi equation . By abuse of notation, one identifies the frames along by parallel transport, and writes this as .

The easiest way to connect up the notions of curvature and comparison geometry is in the observation that for a manifold of nonpositive curvature, the norm of a Jacobi field is convex (as a function along a parameterized geodesic). We compute . Using the Jacobi equation, the second term can be rewritten, so this is equal to  . By the hypothesis that curvature is nonpositive, this is . We compute

where the last inequality is just Cauchy-Schwarz.

OK, we are now ready to begin in earnest. Consider a geodesic from to , and a geodesic through making some angle with at . Parameterize by arc length so that , and consider a 1-parameter family of geodesics from to . Note that . If denotes the length of , then the derivative ; in particular, it does not depend on the curvature of the space in question. The curvature manifests itself in second order information. The one-parameter family of geodesics is tangent along to a Jacobi field , where and . Denote the vector field tangent to the s by . The second variation formula (see e.g. Cheeger-Ebin pp. 20-21) says

Now, vanishes at , since vanishes there; moreover at it is tangent to , and therefore vanishes there too. So the first term is zero. Furthermore, the term (since because is tangent to geodesics) and

along , by the Jacobi equation applied to . Hence is constant along , and one sees that it contributes a term which depends only on the angle . Lets abbreviate . Another simple calculation (see Cheeger-Ebin pp.24-25) shows that if for any function with then ; this is one of the fundamental (and standard) index lemmas, which say that in a suitable sense, Jacobi fields minimize the form .

We are now ready to compare second derivatives in and in our comparison space . Let and be geodesics as above in a comparison space of constant curvature with the same lengths as and making the same angle at their intersection. Let be the analogous 1-parameter family of geodesics, and let denote the length of . We know that the first derivatives of and agree, and would like to compare second derivatives. Apart from the term that depends only on the angle, this means comparing and . This is basically a special case of the Rauch comparison theorem, and our argument is a simplification of Rauch. Let’s suppose for simplicity that both and are 2-dimensional. Parallel transport along and identifies the tangent spaces along these geodesics with the tangent spaces at and respectively. Choosing an isometry between these tangent spaces which takes to , we can define the “pushforward” to be a vector field along satisfying and . By construction we can write where is tangent to , and where . Thus . On the other hand, at comparable points by definition, and

pointwise by the hypothesis comparing the curvature of and . Hence

and we conclude that the distance function to geodesics is more convex in than in the comparison space . This is the desired infinitesimal comparison theorem; it remains to bootstrap it to a global comparison theorem.

Right; let’s look at our comparison triangles and . By the hypothesis that is simply-connected, we can actually map a disk into spanning the geodesic triangle; a minimal area such disk will have intrinsic curvature bounded above by that of , and distances in this disk between points on the boundary will be at least as large as they are in . So without loss of generality, we may assume that is 2-dimensional, and that is spanned by an honest triangular disk. Parameterize the side by length, and let be the point on with . Let be the analogous point on . Define

and .

We know and . We would like to show pointwise. Suppose not, and restrict to a maximal connected interval on which this fails. By the infinitesimal comparison theorem  proved above, this interval must have nonempty interior. Let and be the points on and corresponding to the endpoints of the interval. Evidently the triangles and are also comparison triangles; so WLOG we may just take , and so on.

We now employ a trick. Consider a 1-parameter family of comparison triangles in spaces of constant curvature . The CAT(K) Theorem for spaces of constant curvature reduces to an explicit calculation, since the function as above can be computed exactly, and we suppose the theorem proved for such spaces. It follows that as increases, the function also increases monotonically. By assumption, for small there is some with . Eventually therefore we get some and some intermediate where and for all points near . But this contradicts the infinitesimal comparison theorem proved above. qed.

The figure above illustrates the meaning of the last step. The blue curve is the graph of , and the red curves are the graphs of for various . As is increased, the red curves move upward in a family. There is some biggest for which the red curve is not entirely above the blue curve, and for that curve, the red and blue curves have a point of tangency. But at that point of tangency we would have , contrary to the infinitesimal comparison theorem which shows with equality iff the curvatures along the corresponding geodesics are pointwise equal, which they are not for .

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 remember seeing Bill for the first time when I arrived at Berkeley in 1995; at the start of the academic year, all the incoming graduate students were ushered into the colloquium room to meet some of the senior personnel. Bill was there in his capacity as director of MSRI (the Mathematical Sciences Research Institute). He was wearing jeans with big holes at the knees. He made a speech about MSRI, inviting us all to come up the hill and interact with the visitors there. He also encouraged us to pronounce it as “emissary”, rather than “misery”; it didn’t work — we all called it “misery” (and still do).

I remember actually taking the bus up the hill (maybe a few months later?) in the vague hope of running into Bill and asking him to be my advisor (people had warned me against this, saying that Bill “wasn’t taking students”, because he was too busy running MSRI). I don’t think I had a very clear plan about how this was going to work out. I walked in and saw Bill chatting with Richard Kenyon about entropy of dimer tilings, and hyperbolic volume; at this point I basically froze, turned around and walked out again.

I remember Bill giving a few talks at MSRI during the special program on low-dimensional topology and combinatorics which ran during the academic year 1996-7. He gave one talk explaining his idea of a new proof of (some version of) the Robertson-Seymour theorem on the well quasi-ordering of graphs partially ordered by taking graph minors; he explained this as a kind of compactness result (any class of graphs closed under taking minors is characterized by not containing a certain finite list of excluded minors). A simple version of this compactness concerns the relation on strings (in a finite alphabet), where one string U contains another string V if the letters of V appear in U in the same order, but not necessarily consecutively; I remember Bill explaining this with the example that the string “topology” contains the string “poo”.

I remember Bill at a one-day conference at MSRI on mathematics and the media, with both mathematicians and journalists in attendance. Bill explained some of his ideas about communicating mathematics; he started by drawing a picture (on a big sheet of butcher paper hanging on an easel), explaining the “evolution of mathematical thought”. It was basically a horizontal line; at the left hand side he drew some sort of lizard, and on the right hand side, a monkey and then an upright stick figure representing the modern mathematician.

I remember Bill running the “very informal foliations seminar” at MSRI with Dave Gabai, Joe Christy, and a few other people. This seminar was not advertised; I basically wandered in off the street into the middle of a 3-hour lecture by Bill, explaining his new ideas about universal circles, and how they might be used to approach the geometrization conjecture for 3-manifolds with taut foliations. By the time he was done, I had decided I wanted to work on foliations, and I more or less had my thesis problem.

I remember when Bill moved to Davis. This was the only time I ever saw him in his office at Berkeley — when he was cleaning it out. I remember the little photo that used to be on the door, the one that’s on the cover of “More Mathematical People”, of Bill as a child working at a desk. He saw me watching him carrying his boxes out of his office and looking at the photo, and gave a slightly embarrassed smile.

I remember emailing Bill in early 1998, to explain a few of my tentative ideas about foliations, which had been inspired by his slitherings paper. He invited me to come out to visit him at Davis and talk to him in person. Over the next year or so, I drove out there perhaps a couple of times per month, struggling up the freeway in my third-hand lemon, with the wind rushing in through the bad seals in the door frame. We would have conversations that lasted for hours; stopping occasionally for lunch and coffee. Bill basically became my “unofficial advisor” (my real advisor Andrew Casson was meanwhile going through a tough divorce, and moving to Yale), and perhaps because he did not have many “real” students at Davis at the time, I got a lot of his attention. We spent a lot of time working through the theory of universal circles; I learned a huge amount of mathematics, not only stuff obviously connected to foliations (or even low-dimensional topology), but combinatorics, analysis, group theory, and so on. And yet, Bill listened very carefully to my ideas, and always gave them his full attention and consideration. At the time I don’t think I appreciated how rare this attitude is in a senior mathematician towards a graduate student.

I remember running into Bill in Black Oak Books in Berkeley — in the legendary math book section, of course.

I remember arriving one day a bit early to find Bill waggling his tongue through a gap where one of his teeth had fallen out. He kept making slightly funny expressions on his face thereafter, and it was hard to stay focussed on mathematics for the rest of the day.

I remember when we were trying to work out the details of some construction, Bill got very enthusiastic and we went to the campus store to buy some enormous sheets of paper and a few sets of colored pencils, bringing them all back to Bill’s office and laying the paper out on the floor. Bill was really excited by this episode; he remarked that he used to do this sort of thing “all the time” when he was at Princeton. I got the impression he hadn’t done it for a while.

I remember one day Bill was with a crowd of graduate students, and he was talking about how intimidating it is to start out in mathematics. He thought more senior people contributed to the difficulty, by trying to give the impression that they understood everything, and he wished that people would be more forthright in admitting when they didn’t understand something. I admitted that I had never really learned the details of Galois theory, and Bill exclaimed “that’s great; that’s the sort of thing I mean. Everyone should know that Danny Calegari doesn’t understand Galois theory”. He repeated it several times, to quite a few people. I waited for him to add some things that he had found hard to understand, but that seemed to be it. (Years later in an email he confessed that he had “never really come to grips with the Burau representation” . . .)

I remember working to try to get a project finished in the week before Bill’s daughter was born (we didn’t make it in time). My wife and I were thinking about having kids at the time, and I shyly asked him about the experience. He became very emotional and tender, and talked about what it was like to hold a newborn and have them lie in your arms, trusting you completely.

I remember seeing Bill in 2007 at the Cornell topology conference, and noticing that he looked kind of shaggy, with a few days growth of stubble. I remember giving a talk about immersed curves in surfaces, and mentioning that there are examples of such curves which do not bound an immersed surface, but which “virtually” bound such a surface (i.e. they have a finite cover which bounds). I remember being startled when, after a few minutes, Bill exclaimed, “well, don’t leave us in suspense! what are some examples?” I remember giving Bill the advance copy of my “Foliations” book (it had just arrived) as a late 60th birthday present (I hadn’t gone to his birthday conference earlier that year), and he seemed really pleased to get it, and immediately started looking through it, especially at the pictures. Later on someone told me he had been “showing it to everybody”, which cheered me immensely.

I remember visiting Bill in winter of 2008. At the time my family and I were on a vegan kick, and I remember discussing veganism, and Colin Campbell’s book “The China Study” with Bill, while waiting for the cafeteria people to make us our vegan burritos for lunch. Bill’s wife happened to be very sick that week, and in addition they were moving house, so Bill was very distracted. I remember the class Bill taught one morning that week, in which he gave some beautiful constructions of families of projective structures on some surfaces; and the next class, when he explained how to immerse a projective plane in 3-space with a single triple point. When I left at the end of my visit, Bill apologized for being distracted with so many other things, but hoped that I’d visit again soon. Of course I told him not to apologize, that I’d had a great visit (which was true), and that I hoped I would come again soon when we both had more free time. That was the last time I saw him.

The map cannot usually be recovered from (even up to precomposition with a fractional linear transformation); one needs to specify some extra global topological information. If we let denote the preimage of under , and let denote the subset consisting of critical points, then the restriction is a covering map of degree , and to specify the rational map we must specify both 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 and a representation (here denotes the group of permutations of the set ) which describes how the branches of are permuted by monodromy about . Such a representation is not arbitrary, of course; first of all it must be irreducible (i.e. not conjugate into for any ) so that the cover is connected. Second of all, the cover must be topologically a sphere. Let’s call the (branched) cover for the moment, before we know what it is. The Riemann-Hurwitz formula lets one compute the Euler characteristic of from the representation . A nice presentation for  has generators represented by small loops around the points , and the relation . For each define to be the number of orbits of on the set . Then

If each is a transposition (i.e. in the generic case), then and we recover the fact that .

This raises the following natural question:

Basic Question: Given a set of points in the Riemann sphere, and an irreducible representation satisfying , what are the coefficients of the rational function that they determine (up to precomposition by a fractional linear transformation)?

Note that we would like to recover the coefficients numerically (i.e. as numbers with decimal points). And we are really interested in finding a practical algorithm, and then implementing it on computer. One obvious (and bad) idea is to just solve for the coefficients of P and Q subject to the constraint that the set of critical values is V (after normalizing so that three of the critical points are 0, 1 and infinity to remove the ambiguity of the precomposition). The problem is that the number of such solutions is exponential in the degree, and although Newton’s method will quickly find some solution, it is very, very unlikely to be the solution with the correct combinatorial data.

Another idea — and one that leads to the point of this blog post — is to try to build the branched cover directly as a Riemann surface together with a holomorphic map with the correct critical values and combinatorics, and then uniformize it as to determine the numerical location of the zeros and poles. This sounds more promising, since there is an obvious way to build piecewise from copies of regions in glued together by very explicit maps. The problem is that (numerical) uniformization itself is very slow, at least if one wants any kind of accuracy. On the other hand, we do not need to know the values of the uniformization map everywhere, only the locations of the zeros and poles. So we can try to ask for a fast and approximate method of uniformization which gives sufficiently accurate values of these numbers, that they can then be adjusted quickly to very accurate values by Newton’s method.

One potential idea is to use the method of circle packing. A circle packing is a (rigid) configuration of round circles with disjoint interiors and prescribed combinatorial pattern of tangencies. Abstractly, the circle packing determines a triangulation, with one vertex for each circle, one edge for each tangency, and one triangle for each triple of mutually tangent circles. Implicitly, by using the term “round circle”, the domain in which the circles are packed should be a Riemann surface together with a complex projective structure; for example, the Riemann sphere, or the Euclidean or hyperbolic planes. Given a projective surface and a triangulation satisfying mild topological conditions, such a circle packing exists and is unique; this is known as the Circle Packing Theorem (aka the Koebe-Andreev-Thurston theorem). One can also solve for configurations of circles intersecting at prescribed angles; for instance, one can look for a configuration of round circles with prescribed combinatorics, and meeting always at right angles. Such a configuration in the Riemann sphere can be interpreted as the boundaries of a collection of geodesic planes in hyperbolic 3-space meeting in right angles, and cutting out a compact right-angled polyhedron. The existence of such a polyhedron is the base step in Thurston’s inductive proof of geometrization for Haken 3-manifolds.

One can also think of the circle packing as a discrete version of a conformal structure; at a talk at the conference in 1985 celebrating de Branges’ proof of the Bieberbach conjecture, Thurston proposed using circle packings to approximate conformal mappings. One starts with a region U in the complex plane, and packs it nearly tightly with a hexagonal packing of small round circles. Together with the boundary of U one obtains a topological circle packing; the round circle packing with the same combinatorics can be thought of as a packing of the unit disk. One therefore obtains a coarse “map” from U to the disk, taking each round circle in the packing to a round circle in the disk, and one should think of this as a discrete version of a conformal map. As the mesh size goes to zero, Thurston conjectured these maps should converge to the uniformizing map. This conjecture was proved by Rodin-Sullivan.

In the context of our Basic Question we can try to find our desired rational map as follows. Starting with the collection of points V in the Riemann sphere, we build an (almost) round circle packing in such a way that one of the centers is at 0 and infinity and at each point of V. One should probably choose the mesh size quite small near these special points, since the derivative of is going to blow up near V. This determines a topological graph (the 1-skeleton of the triangulation described above). The branching data defines a new graph in an obvious way, and we can find the circle packing associated to that graph. The “preimages” of 0 and infinity are given as the centers of d circles in this new circle packing, and we can take these as our approximate zeros and poles for .

Anyway, this is all preamble to explaining that I wrote a little code to perform circle packing with prescribed combinatorial data, and in case I don’t do anything else with it (which is quite likely) I thought it might be amusing to post the code and some of the pictures it produces. Note that very sophisticated and highly optimized code for circle packing is already available from many other places; for example, Ken Stephenson has an amazing collection of resources (both theoretical and computational) on his website.

The latest code is, and will continue to be, available at github here:

(link to github repository)

Download the source as a zip file, then unzip and type “make” to make. The program is written in C++, and outputs graphics to the screen using Xlib, and to an .eps file. It can either be run interactively (without an argument) or non-interactively, taking a file name (containing a topological circle packing) as a parameter.

A topological circle packing is written in a (ASCII text) file with the following structure:

number of vertices

valence of vertex 0; list of adjacent vertices in circular order

valence of vertex 1; list of adjacent vertices in circular order (etc.)

valence of vertex n-1; list of adjacent vertices in circular order

initial radius of vertex 0

initial radius of vertex 1 (etc)

initial radius of vertex n-1.

By convention, vertex 0 is the “outer” circle (centered at infinity). The program doesn’t check that the adjacency data that it is given is consistent, or that it gives rise to a topological sphere.

If you try this out, please let me know what you think could be improved or fixed. Feel free to modify or change the code however you like (subject to the usual GPL license conditions). A wishlist would include to add the functionality I sketched above, i.e. to find approximate rational maps with prescribed critical values and branching data; any reader with some time on their hands is warmly invited to do this!

Some screen shots of the X-windows output:

and some sample .eps output (converted to .jpg for wordpress):

The effect of taking a double branched cover over two circles:

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.

Agol works with a characterization of virtually special groups, due to Wise, which is more closely tied to the notion of hierarchies.

Definition: A hyperbolic group G is QVH if it is obtained inductively by the following procedures:

1. the trivial group is QVH;
2. If G splits as an amalgam where A and C are QVH, and B is quasiconvex in G, then G is QVH;
3. similarly for an HNN extension ; and
4. If H is QVH and is contained in G with finite index, then G is QVH.

Wise shows the following:

Theorem (Wise): A group is hyperbolic and acts cocompactly on a CAT(0) cube complex with special quotient if and only if it is QVH.

Thus the virtually special groups are the analogue in the world of hyperbolic groups of virtually Haken hyperbolic 3-manifolds in topology; we can say they are the fundamental groups of hyperbolic NPC cube complexes that admit a “quasiconvex virtual hierarchy”. So Ian’s argument works by exhibiting a finite cover of X/G with such a hierarchy.

The groups along which we would like to split in this hierarchy are (finite index in) the fundamental groups of the hyperplanes in X/G. Finding the cover amounts to separating these subgroups in finite covers. We don’t know how to do this directly, but the Weak Separation Theorem of Agol-Groves-Manning (discussed yesterday) shows that these subgroups can be “separated” in infinite covers (in a certain sense). The separation refers to the fact that a finite index subgroup of the hyperplane groups lifts to the cover, so that the hyperplanes in the infinite cover will be compact, 2-sided and embedded. In algebraic language, the Weak Separation Theorem guarantees the existence of a normal subgroup G’ of G so that if we write and then has locally 2-sided embedded compact acylindrical hyperplanes (the acylindricity implies that the fundamental groups of the hyperplanes are malnormal). We can split along these hyperplanes to produce a kind of hierarchy, so if were compact we would be done. The idea is to take the (infinitely many) pieces that are created by the cutting, separate them into finitely many classes, and glue them together in such a way as to create a finite covering of X with a hierarchy of its own.

Ian calls the pieces into which is decomposed by its hyperplanes cubical polyhedra, although they are not really polyhedra, but rather cubical subcomplexes of the cubical barycentric subdivision of . The combinatorics of the system of (compact!) hyperplanes in the (noncompact!) is encoded by the so-called crossing graph . This graph has as vertex set equal to , the set of hyperplanes of (there is a slightly unfortunate point that Ian uses the terminology “hyperplane” for what Wise calls “walls” when they are embedded; the letter W is supposed to represent the word “walls”; anyway, Ian used the terms hyperplane and walls synonymously in his talk, so if I sometimes accidentally use one word instead of the other, that’s the reason). A pair of hyperplanes (i.e. vertices) share an edge in in two cases, if either:

1. the hyperplanes intersect; or
2. conjugates of their fundamental groups have an infinite intersection (i.e. they are not acylindrical as a pair)

The group acts on , and moreover the maximum degree of is finite, because hyperplanes are compact and fall into finitely many -orbits, and is locally compact (the hyperbolicity of G guarantees that there are no arbitrarily long essential cylinders running between distinct hyperplanes, so hyperplanes which are sufficiently far away from each other will not contribute an edge as in case 2 above) . Let k be the maximum degree of the vertices of .

Suppose we could color the graph with finitely many colors in such a way that adjacent vertices have different colors, and the coloring is invariant under some finite index subgroup of . Then the quotient would be compact with a quasiconvex hierarchy, and we would be done. Another way of saying that the coloring should be invariant under a finite index subgroup of is to say that we have a finite set of colorings of which are permuted by the action of . Let denote the set of all colorings of by the numbers such that adjacent vertices get different colors. The set  can be topologized with the topology of convergence of colorings on finite subgraphs, making it into a compact (totally disconnected) space, and the group acts on this space by homeomorphisms. Translating the statement above into this language, if we could find an invariant finite set on this space, we would be done. Instead, Ian finds an invariant probability measure. This is a completely general statement, and applies to all locally finite graphs with cocompact group actions.

Theorem: Let be a graph with bounded valence k, and G a group acting cocompactly on . Then there is a G-invariant probability measure on the space of (k+1)-colorings of .

Ian gave a very elegant proof of this theorem; after working it out, Lewis Bowen informed him that the theorem is a consequence of known work of Kechris-Solecki-Todorocevic on Borel colorings of Borel graphs (the kind that arise in the theory of measure equivalence of group actions). But Ian’s proof is so elegant that I can’t resist reproducing it here.

Pick orbit classes of the edges of . We call an assignment of colors to the vertices which does not necessarily assign different colors to adjacent vertices a labeling. The weight of a labeling is the number of whose vertices get the same color. Weight extends linearly to the space of probability measures on labelings; a G-invariant probability measure on labelings of weight 0 will be a G-invariant probability measure on colorings.

Now, a random labeling with n colors will have weight m/n. There is a function from labelings with n colors to labelings with (n-1) colors whenever that takes each vertex labeled n to the smallest number which is not a label on an adjacent vertex. Extend this operation by linearity to probability measures on labelings. This operation is G-equivariant and does not increase weight. So start with a random G-invariant probability measure on labelings with n colors (call it ) and reduce the number of colors one by one to . This gives a measure on G-invariant labelings with colors whose weight is at most m/n (the weight of the random labeling). Take a weak limit of the as ; this is a G-invariant probability measure on -labelings whose weight is 0; that is, it is a probability measure on -colorings, as desired. qed

OK, we now have a -invariant probability measure on colorings of . The colors 1 to (k+1) correspond to the order in which to cut along the hyperplanes in a hierarchy, so we can think of this probability measure as a kind of “superposition” of hierarchies of , which want to be pulled back from some hierarchy on a finite cover of . When we cut along hyperplanes and then try to glue them back up, we need to remember the labels on all the cut open facets meeting the hyperplanes we are gluing. So it is important not just to remember the colors on vertices, but also all the colors on adjacent vertices we have cut up earlier, and the colors on vertices adjacent to them that we have cut up earlier, and so on.

Given a graph and a coloring of the vertices by numbers from 1 to (k+1), each vertex determines a “descending link”, which is the union of all simplicial paths emanating from that vertex along which the numbers decrease. The supercolor of a vertex is the structure of its descending link as a colored graph. Since the valence is bounded, there are only finitely many supercolors. Supercoloring determines an equivalence relation finer than coloring, and therefore an equivalence relation on where if there is some g in so that gv=w and the supercolor of v with respect to c is equal to the supercolor of w with respect to (remember that denotes the set of hyperplanes). Similarly we can define equivalence relations on where denotes the collection of faces of in the barycentric subdivision dual to edges, by saying that if where is the hyperplane containing , and is the hyperplane containing . Finally we can define an equivalence relation on where denotes cubical polyhedra, by if for corresponding faces F,E of P,Q.

A non-negative -invariant real valued function

satisfies the gluing equations if for every face in the boundary of polyhedra , and for every equivalence class we have where the sum is taken over equivalence classes restricting on F to equivalence classes which are equivalent to (i.e. the supercolors agree on the given face). Note that this is a finite sum, since there are only finitely many supercolors and orbits of faces or cubical polyhedra.

The point of the measure is that it defines a solution to the gluing equations, by setting to be equal to the measure of the set of colors inducing the given supercolor on P. Since there are only finitely many gluing equations, and they have integer coefficients, the existence of one nontrivial solution implies the existence of a nontrivial integer solution; i.e. we can find an taking integer values.

I think it’s time for me to take a break now, so I’m posting this with the intention of coming back to add more details about how to use this integer solution to the gluing equations to get a hierarchy. Let me just say cryptically that this solution lets one glue up a finite collection of pieces which immerses into our (partially glued up) hierarchy in such a way that at the next step what needs to be glued are a finite collection of pieces which cover the same compact hyperplane with equal total degrees. It is at this point that Wise’s Malnormal Special Quotient Theorem lets one find finite covers in which the pieces can be matched in pairs and glued up along the hyperplane in question. More later (I hope). For the moment, here’s Ian explaining a detail to some doofus.

OK, after a break and another full day of lectures, I’m suitably rested, and ready to (briefly!) describe the endgame.

We imagine that we have already glued everything up to get a quasiconvex hierarchy, and then we inductively split along hyperplanes in the order of their colors . The result will be a collection of cubical polyhedra whose boundaries are decorated with what is known as a boundary pattern, which keeps track of where the cuts where made.

If V is a finite cube complex obtained by (perhaps partially) cutting open along a quasiconvex hierarchy, we will denote its boundary pattern by . It is glued back to a hierarchy by first gluing up ; this gives a new finite cube complex V’ with a new boundary pattern which are the image of the boundary pattern in V.

So Ian’s inductive gluing procedure starts with with boundary pattern which consists simply of a collection of cubical polyhedra. The number of colored polyhedra of each type is the corresponding coefficient of our (integer) solution to the gluing equations, and the facets in the given boundary pattern are those with the corresponding color.

There is no obstruction to gluing up the facets in pairs, since this is exactly what the gluing equation guarantees we can do. This gives rise to with boundary pattern . Now the components of are more complicated, and it is not immediately clear how to glue them up. It will turn out that the components of all cover certain boundary components (with respect to a particular boundary pattern) of a particular compact cube complex in such a way that the sum of the degrees of the covers on either side of the component agrees. We would therefore like to take a finite cover of in which the components of map to corresponding components of the boundary of in a way which can be matched up and then glued. In Ian’s paper he describes a method to find such a finite cover inductively, using a method in an appendix of Agol-Groves-Manning; however he pointed out in his talk on Wednesday that the cover can be found in one step by the MSQT. Anyway, I won’t say any more about it here.

OK, in this way we glue up in a finite cover to get , and then the components of immerse into covering the same object from two sides with the same degree, so we can pass to a further cover (by the MSQT) where they can be paired, and glued up to get , and so on. Eventually we have glued up everything in a finite cover, obtained a hierarchy, applied Wise’s theorem that QVH is equivalent to virtually special, and then it’s time to break out the champagne.

So what’s ? In fact there is a for each j; it is a disjoint union of hyperplanes of the original complex split open along the hyperplanes they intersect of smaller color, quotiented out by the action of the stabilizer in of the associated equivalence class. This complex has the property that each equivalence class of face for which the color of F in the coloring c is j has a unique representative in the complex . It is this fact, together with the fact that the set of polyhedra in satisfies the gluing equations (because inductively it is a cover of a partially glued union of polyhedra which as a set satisfy the gluing equations) which implies that the map is an immersion, and then the gluing equations say that the degrees on either side have the same sum, and now the previous paragraph makes sense (ahem!).

Well, that’s it for my summary. I will post a few photos of the blackboard taken by Patrick Massot and Alden Walker when I get a chance (one such photo by Patrick is above). If you want more details, then I believe Ian intends to post his preprint before too long, so keep watching the skies arXiv.

Update (April 13): Ian’s preprint is now available on the arXiv here.

Photos by Patrick Massot:       

Photos by Alden Walker:  

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 so that

1. is hyperbolic;
2. is finite; and
3. is not contained in .

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.

Recall from my previous post that an NPC (non-positively curved) cube complex is a compact quotient of a CAT(0) cube complex by a group (in this case H) acting properly discontinuously, and that the complex is virtually special if it has a finite (orbi-)cover satisfying the Haglund-Wise conditions (i.e. hyperplanes are embedded, two-sided, and there are no self- or interosculations). The fundamental group of a virtually special cube complex is a linear group (i.e. embeds in for some n), and is therefore residually finite; this means that the intersection of all finite index normal subgroups consists only of the identity element. The fact that all finitely generated linear groups are residually finite is known (to topologists, anyway) as Selberg’s Lemma. Roughly, the idea is to consider the ring A of matrix entries in a faithful linear representation, and then the desired finite index subgroups are the kernels of maps to for suitable prime ideals in (see e.g. here for more details). More generally, if G is a group, a subgroup H is said to be separable if for any g not in H there is a homomorphism from G to a finite group so that the image of g is disjoint from the image of H. Groups in which every finitely generated subgroup is separable are said to be LERF; it is a consequence of Agol’s proof of the VHC that every hyperbolic 3-manifold group is LERF, but we are getting ahead of ourselves here.

The Weak Separation Theorem is in the direction of showing that the subgroup H is separable; if the quotient could be taken to be finite, that is exactly what it would show. But finding a quotient where is finite turns out to be good enough for Agol’s purposes.

An important special case of the theorem is when H is almost malnormal. A subgroup H of G is normal if conjugation in G sends H to itself. H is malnormal if for all , where superscript denotes conjugation, and H is almost malnormal if is finite for all .

Bowditch showed that if G is hyperbolic and H is quasiconvex and almost malnormal in G, then the pair is relatively hyperbolic. The concept of relative hyperbolicity generalizes the fundamental group of a noncompact complete negatively curved manifold of finite volume; the fundamental group is not (necessarily) a hyperbolic group (although it is in some cases!) but the lack of hyperbolicity is concentrated in the noncompact cusp of the manifold; the fundamental group of the cusp itself might or might not be hyperbolic. It is a parabolic subgroup of the fundamental group, and the pair is relatively hyperbolic. Abstractly, a pair is relatively hyperbolic, where is a collection of conjugacy classes of subgroups of , if the space obtained by attaching “horoballs” to the conjugates of the subgroups in in (the Cayley graph of) G is hyperbolic (see here for more details). Note that if the subgroups are themselves hyperbolic, then is also hyperbolic (in the absolute sense).

Relatively hyperbolic groups invite relatively hyperbolic Dehn filling, by analogy with Thurston’s hyperbolic Dehn surgery for (noncompact) hyperbolic 3-manifolds with cusps. Suppose is relatively hyperbolic, where . For each i choose some normal subgroup of . The quotient of by the normal closure (in G) of all the is called a filling of G by the , and is denoted . If each of the is finite index in , we call it a peripherally finite filling. The fundamental theorem of hyperbolic Dehn surgery, due (in this form) originally to Osin, is as follows:

Theorem (Osin): Let F be a finite subset of G, and let be relatively hyperbolic. Then there is a finite subset B of G, so that for any filling for which B does not intersect any of the , one has the following:

1. for each i;
2. the image pair in the quotient is relatively hyperbolic
3. the restriction of to is injective.
4. for all i

Actually, the fourth condition is not proved by Osin, but can be deduced with “a bit of work”, according to Jason. Notice that if the filling is peripherally finite, so that all are finite, then these quotients of the parabolic groups are trivially hyperbolic (since every finite group is hyperbolic), and therefore the quotient group is also hyperbolic.

If H is almost malnormal, Osin’s surgery theorem implies the Weak Separability Theorem, as follows. Since H is almost malnormal and quasiconvex, and G hyperbolic, by Bowditch the pair is relatively hyperbolic. We let F (as in the statement of Osin’s theorem) consist only of the element , and let be the finite set of “bad” fillings the theorem guarantees. Since H is assumed to be virtually special, it is residually finite, and therefore contains a finite index normal subgroup N missing B. Taking the quotient of G by the normal closure of N gives a surjection to a group in which the image of H is finite, and disjoint from the image of g, as desired.

What if H is not malnormal? Then one must induct on an invariant called the height, which measures the failure of the group to be almost malnormal. The idea of height was introduced by Gitik-Mitra-Rips-Sageev.

Definition: If H is quasiconvex in a hyperbolic group G, the height of H is the least integer n so that if there are elements so that are distinct, then the intersection of conjugates is finite.

H has height 0 if and only if it is finite. It has height 1 if and only if it is almost malnormal and infinite. One can define a complex whose k-simplices are the -fold infinite intersections of distinct conjugates of H, and height is then the dimension of this complex plus one. Gitik-Mitra-Rips-Sageev prove that every quasiconvex subgroup of a hyperbolic group has finite height. They also show that for any k there are only finitely many H-conjugacy classes of infinite groups of the form (this is vacuous for k as big as the height or bigger). The minimal such infinite intersections are not far from being malnormal, and after a suitable modification, will give rise to a suitable relatively hyperbolic family.

Start with , the collection of H-conjugacy classes of minimal infinite intersections of the form (these are conjugacy classes of subgroups in H). These subgroups are intersections of quasiconvex subgroups, and are easily seen to be quasiconvex themselves. Replace each D in by its commensurator in H (note that each D is finite index in ), and then choose one such subgroup per H-conjugacy class. This produces a new collection of conjugacy classes of subgroups in H. Observe that the elements of are almost malnormal — in fact is an almost malnormal collection, meaning that nontrivial conjugates of one subgroup intersect any other subgroup in the collection in a finite set  — so that is relatively hyperbolic (Bowditch’s criterion for relative hyperbolicity generalizes to almost malnormal collections of quasiconvex subgroups).

Now, replace each D in by its commensurator in G; again each D is finite index in . This produces a collection of subgroups of G; choose one subgroup for each G-conjugacy class to arrive finally at an almost malnormal collection of quasiconvex subgroups of G, and deduce (by Bowditch again) that is relatively hyperbolic.

Definition: A filling with each in some in as above is an H-filling if whenever is infinite for some in , then is contained in D.

An H-filling by definition induces a filling of H, i.e. a quotient . With this terminology, Agol-Groves-Manning prove

Theorem (Agol-Groves-Manning): Let G be hyperbolic, let H be quasiconvex in G of height at least 1, and let and be as above, and let g be in . Then for all “sufficiently long” peripherally finite H-fillings ,

1. is isomorphic to the induced filling of H;
2. is quasiconvex in ;
3. is not contained in ; and
4. the height of is strictly less than the height of H.

Let’s not worry too much about what the condition “sufficiently long” means here; suffice it to say that such fillings can be found if H is residually finite.

Now, in the setup of the Weak Separation Theorem, the subgroup H is the fundamental group of a virtually special NPC complex, and is therefore residually finite. So we can apply this filling theorem of AGM to reduce the height of H. But now one is stuck, because the resulting image , while of strictly lower height, might not be residually finite. Here is where Wise’s Malnormal Special Quotient Theorem (alluded to at the end of my previous post) comes in. The statement of the MSQT is as follows:

 Malnormal Special Quotient Theorem (Wise): Let H be hyperbolic, let be relatively hyperbolic, where the in are almost malnormal and quasiconvex. Suppose H is the fundamental group of a virtually special NPC complex. Then there are finite index subgroups in the so that if is any peripherally finite filling with contained in the , then is the fundamental group of a virtually special NPC complex.

The MSQT implies that, providing we are careful for our H-filling to kill subgroups contained in the , the image of H will be virtually compact special, and the induction can be continued, reducing the height of (the image of) H until the Weak Separation Theorem is proved.

We make the assumption in what follows that all manifolds under discussion are smooth (this is not a problem in 3-manifold topology, where the categories TOP, PL and DIFF are all equivalent) and orientable (which can be achieved by passing to double covers, if necessary).

A compact 3-manifold M, possibly with boundary, is said to be irreducible if every embedded 2-sphere in M bounds a 3-ball in M. A compact 3-manifold is said to be Haken (the terminology sufficiently large is also standard) if it is irreducible, and if it contains a closed, 2-sided properly embedded surface S (other than a 2-sphere) which is incompressible and boundary incompressible; incompressibility means that if an embedded disk intersects S only in an embedded loop, then this loop is (homotopically) inessential in S, and boundary incompressibility means that if an embedded disk intersects S only in a proper arc (with the rest of the boundary of the disk on the boundary of M) then the arc on S is (homotopically) inessential in S. Incompressibility and boundary incompressibility mean roughly that the surface can’t be simplified by a “local” move. Such a surface is also said to be essential. Wolfgang Haken used such surfaces to solve the homeomorphism problem for the 3-manifolds that contain them. This includes as a very important special case manifolds obtained as complements of (open tubular neighborhoods of) knots in the 3-sphere; hence Haken’s methods solve the knot recognition problem — the problem of deciding when two different diagrams determine the same knot type. The key point of Haken’s approach is that once one has an essential surface, one can cut along it to produce a simpler 3-manifold with boundary.  Every irreducible 3-manifold with nonempty boundary is Haken, so if this decomposition process can be begun, it can be continued. (One way to see this is to observe that every nontrivial relative 2-dimensional homology class in a 3-manifold contains an essential surface. Now, 2-dimensional relative homology is dual to 1-dimensional absolute (co-)homology in a 3-manifold. Moreover, exactly half the rank of 1-dimensional (co-)homology of the boundary of a 3-manifold is killed under the inclusion; hence an irreducible 3-manifold with boundary — other than the 3-ball — has nontrivial relative 2-dimensional homology and is therefore Haken). After decomposing along finitely many such pieces, one obtains at the end a finite collection of 3-balls; turning this procedure upside down, a Haken 3-manifold is one that can be obtained from 3-balls by gluing in certain special controlled ways. This opens the possibility of inductive proofs of theorems about Haken 3-manifolds; Thurston’s famous geometrization theorem for Haken 3-manifolds is such an example. The Virtual Haken Conjecture, formulated by Waldhausen in 1968, says that every aspherical (i.e. with trivial for ) closed 3-manifold has a finite cover which is Haken.

Among the most beautiful and interesting examples of Haken 3-manifolds are those that fiber over the circle. These are obtained by taking a surface S (usually assumed by convention to have non-positive Euler characteristic) and a self homeomorphism and forming the mapping torus . The fibers are the slices ; these are all essential, so 3-manifolds that fiber over the circle are Haken. Thurston’s Virtual Fibration Conjecture says that every hyperbolic 3-manifold M has a finite cover which fibers over the circle. Since the “generic” closed 3-manifold (in a suitable sense) is now known to be hyperbolic (by Perelman), this says that essentially all aspherical irreducible 3-manifolds — with some very well-understood simple exceptions — are virtually fibered. Ian already showed that Haken hyperbolic 3-manifolds whose fundamental group satisfies a certain technical condition (called “RFRS”, or “reefers”) satisfy the VFC. Wise’s work shows that Haken 3-manifolds have fundamental group satisfying this condition, so Ian’s work proves the VFC too.

There are many other amazing corollaries of Ian’s theorem, including the fact that the fundamental group of every hyperbolic 3-manifold has a finite index subgroup which surjects onto a free group of rank 2 (i.e. it is “large”), has a finite index subgroup which is bi-orderable, injects in for some , and so on. It is hard to think of a question about fundamental groups of hyperbolic 3-manifolds that it doesn’t answer.

Ian’s preprint is only 31 pages long (although some of it is quite dense), but it rests on a considerable amount of prior work by many people (including Ian himself), of which the biggest breakthrough is of course Perelman’s proof of the geometrization theorem (see here, here and here). Other substantial pieces of the puzzle are the Kahn-Markovic proof of the surface subgroup conjecture, and Wise’s work on special cube complexes. In fact, the technical content of Ian’s preprint is the proof of a conjecture in Wise’s paper about groups acting on CAT(0) cube complexes. So let’s start with the definition of a CAT(0) cube complex.

Definition: A cube complex is a space obtained by gluing Euclidean cubes of edge length 1 along subcubes. It is CAT(0) if it is non-positively curved (in the sense of comparison geometry).

Gromov showed that a cube complex is CAT(0) if and only if it is simply connected, and if the link of every vertex is a flag complex (i.e. every complete subgraph of the 1-skeleton are the edges of a simplex in the complex). A cube complex satisfying the flag condition without necessarily being simply connected is said to be nonpositively curved, or NPC. The universal cover of an NPC complex is a CAT(0) cube complex (actually, Ian must consider slightly more general “orbi-NPC complexes”, which can be thought of as locally the quotient of a CAT(0) complex by a finite group acting cellularly, but not freely; we ignore this issue for simplicity).

The edges in a cube complex fall into equivalence classes, where two edges are equivalent if they are on opposite sides of a square in the complex. Dual to an equivalence class is a hyperplane, which can be thought of as a collection of codimension 1 faces in each cube dual to the edges in that cube in the given equivalence class.

Haglund and Wise singled out a particular class of NPC cube complexes, which they call special cube complexes.

Definition: A special cube complex is an NPC cube complex C satisfying the following conditions:

(i) hyperplanes are embedded;

(ii) hyperplanes are 2-sided (this can be arranged by passing to a 2-fold cover if necessary, given (i))

(iii) there are no self-osculating hyperplanes; and

(iv) there is no interosculation.

Self-osculation rules out the following kind of picture, where a hyperplane comes back and bounds a “monogon” with itself:

while interosculation rules out the condition that two hyperplanes osculate somewhere and are transverse elsewhere:

Haglund-Wise show (and this is the main reason to introduce the class of special NPC complexes) that the fundamental group of a special NPC cube complex embeds into a right-angled Artin group (RAAG) with one generator for each hyperplane , with the relation that and commute whenever hyperplanes cross, and no other relations. (Artin groups are a kind of “complexification” of Coxeter groups; if one takes a real hyperplane arrangement which are the fixed sets of the generators of a Coxeter group and complexifies it to obtain a complex hyperplane arrangement, and then takes the quotient of the complement of the complex hyperplane arrangement by the (complexified action of the) Coxeter group, the fundamental group of the quotient is an Artin group.) RAAGs are very nice groups, with many very useful properties. For Agol’s purposes, the most important property is that if G is a (word) hyperbolic RAAG, and H is a quasiconvex subgroup (i.e. if we think of G as a metric space via the Cayley graph construction, then every geodesic in G which starts and ends in H stays within a bounded distance of H) then H is subgroup separable. This means that for every g in G but not in H there is some homomorphism from G to a finite group in which the image of g is not contained in the image of H.

With this terminology, the main technical theorem Agol proves is 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 H of G so that the quotient of X by H is special.

One concludes from Haglund-Wise that if G is a hyperbolic group acting properly discontinuously and cocompactly on a CAT(0) cube complex X, then quasiconvex subgroups of G are separable.

What does this have to do with 3-manifolds? This is where the work of Kahn-Markovic (discussed on this blog a couple of years ago) comes in. Recall that Kahn-Markovic proved the famous Surface Subgroup Conjecture by showing that every closed hyperbolic 3-manifold contains many immersed surfaces S which are very nearly totally geodesic, and whose fundamental group therefore injects into that of M. If one could find a finite cover of M in which one of these surfaces became embedded, one could deduce easily from its geometry (and from Gauss-Bonnet) that the surface was incompressible and boundary incompressible, and therefore one would deduce that M was virtually Haken. Now, the universal cover of the surface certainly embeds in the universal cover of M (in fact, it is very close to a flat hyperbolic plane), but the problem is that different lifts of the surface might intersect each other transversely. These different lifts correspond to elements of the fundamental group of M that are not conjugate into the fundamental group of S. If one knew that the fundamental group of S was separable (in the sense above), one could pass to finite covers in which these “bad” elements failed to lift, one by one, until some finite cover of S became embedded; so if one knew that the fundamental group of M acts properly discontinuously and cocompactly on a CAT(0) cube complex, one could apply the theorem of Agol (above) and prove that M is virtually Haken. But where to find such an action?

The key is that Kahn-Markovic produce not just one immersed surface S, but many such surfaces — so many that one can be found arbitrarily close to any immersed totally geodesic plane on any compact subset. The circle at infinity of the universal cover of such a surface is an almost-round quasicircle in the sphere at infinity of hyperbolic 3-space, and this collection of circles crisscrosses the sphere all over the place. In fact, so many surfaces can be found that we can find one separating any pair of distinct points in the sphere at infinity. This collection of almost-flat planes crashing through hyperbolic 3-space should make one think of a hyperplane arrangement decomposing Euclidean space into cells. The fundamental group of M acts by permuting this cell structure. The idea is that one should think of these (3-dimensional) cells as the “shadows” of higher dimensional cubes — the cubes in the sought-after cube complex on which the fundamental group will act.

(Note: the next two paragraphs are my partial summary of the second talk this morning by Nicolas Bergeron). The idea of building a cube complex from a collection of (suitable) immersed submanifolds goes back to a remarkable construction of Sageev from the early 90′s (also see here). Sageev’s construction starts with a hyperbolic group G and a collection of codimension 1 subgroups . A subgroup H of G is said to be codimension 1 if for some the neighborhood of H in the Cayley graph of G separates the Cayley graph into at least 2 distinct (non cofinite) H-orbits (to say these orbits are not cofinite means that their quotient by H is not compact). In the case in question, G will be the fundamental group of a hyperbolic 3-manifold, and the will be the fundamental groups of lots of Kahn-Markovic surfaces — enough to separate pairs of points at infinity. The collection of universal covers of the surfaces in the universal cover of M will become the hyperplanes of the cube complex. Each surface decomposes into two halfspaces; a 0-cube of the cube complex corresponds to a choice of one halfspace for each surface satisfying the following two properties:

(1) if are two halfspaces, and is one of the chosen halfspaces, then B must also be a chosen halfspace; and

(2) every point in lies in all but finitely many of the chosen halfspaces.

In this correspondence, the halfspaces “corresponding” to a given vertex will be exactly the sides of the associated hyperplanes facing the vertex in the cube complex. A 1-cube of the cube complex corresponds to a pair of 0-cubes which differ in the choice of halfspace associated to exactly one surface; in the cube complex, this is the unique hyperplane crossing the associated edge. It turns out that, having specified the 0 and 1 cubes, there is a unique way to complete the result to be CAT(0).

Sageev showed that G as above acts cocompactly on this cube complex (which is finite dimensional). Bergeron-Wise showed that the fact that the surfaces separate pairs of points at infinity is enough to guarantee that the action is proper, and therefore one deduces that the fundamental group of a hyperbolic 3-manifold M acts properly discontinuously and cocompactly on a CAT(0) cube complex, and (by Agol), is virtually special. So one concludes that the surface subgroups are separable after all, and therefore M is virtually Haken!

None of this even touches on the new ingredients in Ian’s argument, or Wise’s key work on quasiconvex virtual hierarchies and his malnormal quotient theorem; nor does it get to the work of Agol-Manning-Groves which combines with the malnormal quotient theorem to prove a “weak” separability theorem which allows Ian to begin his inductive proof that NPC groups are virtually special. But that’s for the next post (oof!)

(edited March 27: corrected typos, and changed pictures of inter/self osculating complexes to make them more generic)

I thought it would be a nice idea to discuss some pieces of the theory in a blog post (note that I have made no attempt to bring the material “up to date”). There are two starting points for the theory; the first is the work of Gelfand-Kazhdan on formal vector fields, which establishes the existence of a natural homomorphism where is the frame bundle (i.e. the bundle of 1-jets) of a p-dimensional manifold , and is the Lie algebra of formal vector fields on . The second is the work of Godbillon-Vey who discovered a 3-dimensional characteristic class associated to a codimension 1 foliation on a manifold, which is a kind of transgression of a characteristic class of the normal bundle . These ideas were synthesized by the work of Bernstein-Rosenfeld, who showed how to construct a homomorphism . Classes in the image can be integrated over the fiber to produce characteristic classes on , of which the simplest is the Godbillon-Vey invariant.

The Godbillon-Vey invariant of a codimension 1 foliation on a 3-manifold can be described using an auxiliary Riemannian metric. Under holonomy transport, leaves spread apart from each other infinitesimally; the logarithmic derivative of a transverse measure (i.e. the multiplicative rate of spreading apart of leaves) defines a vector field tangent to . The Godbillon-Vey form measures the infinitesimal rate at which spins as one moves transverse to ; Thurston famously called this “helical wobble”. If one uses the metric to make dual to a 1-form then the Godbillon-Vey form measures how non-integrable is, and the Godbillon-Vey invariant is the integral of this form over . For example, if , the unit tangent bundle of a hyperbolic surface with its stable foliation (see this post), then is the geodesic flow itself (up to a change of orientation), and the Godbillon-Vey invariant is the volume of (up to a nonzero constant). The figure below (taken from Thurston’s paper) shows such an example of constant helical wobble locally.

The differential-geometric approach to constructing foliated characteristic classes goes via connections and curvature. Let’s fix a manifold and a codimension p foliation . The issue of smoothness is very important for foliations, so for simplicity assume that the tangent field to the foliation is a distribution , and let denote the normal bundle. Dual to is , the collection of 1-forms on whose kernel contains (pointwise). Working with in place of makes it easier to use the language of differential algebra. The crucial property of the sections is that they generate a differential ideal; i.e. if are forms which locally give a basis for at each point in an open neighborhood , then each can be expressed as a linear combination for certain 1-forms . This statement is equivalent (and dual) to Frobenius’s theorem, which characterizes the integrability of a distribution (i.e. the property that it should be tangent to a foliation) precisely by saying that sections form a Lie algebra: i.e. for sections we have . This property of enables one to construct a certain connection on which is said to be torsion-free. Recall that a connection on defines a map

and it is said to be torsion-free if the composition with the antisymmetrizing map coincides with exterior d. In local coordinates therefore one can define a connection on by the formula

and observe that integrability implies that this connection is torsion-free. Taking convex combinations of connections defined on open neighborhoods (by using a partition of unity) preserves the torsion-free property (since both and exterior d satisfy the Leibniz formula) and one thereby obtains a torsion-free connection on . Differentiating the equation gives

where is the entry in the curvature of the connection . This last equation implies that is in the (differential) ideal generated by the , and therefore any homogeneous polynomial in the of degree is identically zero. This observation is due to Bott, and implies (for example) that the (rational) Pontriagin classes of the normal bundle of a smooth foliation of codimension p vanish in degrees .

On the other hand, we can choose a Riemannian connection on (this does not make any use of integrability at all), and then the associated curvature matrix will be skew-symmetric. In particular, the invariant homogeneous polynomials in of odd degree will vanish identically (this is just the usual observation that the odd rational Chern classes of a real vector bundle vanish). If we let and denote the differential forms on of dimension representing the Chern classes associated to the connections and respectively (i.e. they are, up to a constant, pointwise the th coefficients of the characteristic polynomial of the and respectively), then is identically zero for i odd, and every polynomial in the of total degree is also identically zero. Now, Chern showed that for any two connections on a bundle, the difference of the associated Chern forms is exact, and is exterior d of a canonical form of one dimension lower. To see this in our context, let be a connection on the pullback of to restricting to on , let be the associated Chern class, and let be the integral of along the fibers point . Then is a form on satisfying .

We define to be the following differential graded algebra:

where , and where has degree , and has degree , and the differential is given by and . A choice of a pair of connections determines a map of dgas from to , and the induced map on cohomology is independent of all choices. The images are the characteristic classes of the foliation. For example, if then the Godbillon-Vey class is the image of .

The algebro-geometric approach goes via formal vector fields, thought of as living on the local “space of leaves”. In every sufficiently small open ball on , there is a submersion for which the kernel is precisely . So we can identify with forms on locally. Consider the principal (frame) bundle whose fiber at each point is a basis for at that point. There is a canonical trivialization of the pullback ; for each , a point is a frame for , and the fiber of over is itself a copy of , so one can trivialize it by the tautological frame . Dualizing, we obtain p canonical sections of . Since exterior d commutes with projection, these generate a differential ideal in so there are forms with . The form is not unique, but there is a canonical choice if we first pull back to a further bundle over , namely the “bundle of 2-jets”. In fact, one can reinterpret as , the bundle of 1-jets, and consider it as the first step in a tower of bundles

where the fiber of over keeps track of the derivatives of order of a local submersion to sending to . The conclusion is that we obtain canonical 1-forms on satisfying , canonical 1-forms on satisfying and so on (each form on pulls back to a form of the same name on all with which is where these formulae hold). Let denote the Lie algebra, which is a module on p generators over the ring of formal power series on , with Lie bracket defined (formally) in the obvious way. We can think of as the Lie algebra of formal vector fields on . The continuous dual (with respect to the obvious topology) has a basis consisting of the forms , and there is a differential graded algebra obtained by dualizing the Lie bracket. From the discussion above, there is a map of dgas and thereby a map on cohomology . Now, topologically, the fiber of each fibration is contractible for , so at the level of cohomology we may identify with . Dual to projection there is a map identifying the (de Rham) cohomology of with the cohomology of the complex of -invariant forms on . Up to homotopy we can replace by ; the Lie algebra of sits inside in an obvious way (by thinking of elements of as vector fields on and thence as formal vector fields), and we obtain a map . The relation to the discussion above is that there is a canonical isomorphism of with defined above.

This (highly abbreviated) discussion brings us roughly to the end of the third chapter of Harsh’s book. A fourth chapter discusses how to measure the variation of the characteristic classes in families of foliations. There is also an appendix, giving a short exposition of the Chern-Weil theory of (ordinary) characteristic classes, and another appendix on the cohomology of Lie algebras. Composing this blog post gave me an excuse to read Harsh’s book again (for the first time in quite a few years), and I must say it was every bit as good as I remember. Mathematics is a conversation in which the participants might be separated by unbridgeable distances in space or in time, but it is some consolation to know that we will still have the opportunity — through our work — to take part in this conversation once we are gone.

This idea of using surgery to modify Anosov flows goes back at least to Fried, Goodman and Handel-Thurston in the early 80′s. In the 90′s Fenley analyzed the geometry and topology of the resulting manifolds obtained by surgery, obtaining strong results. This was in the days before Perelman, when perhaps the main goal of 3-manifold topology was to prove Thurston’s Geometrization Conjecture, and one of the main avenues of attack was to prove the conjecture under the hypothesis of some extra structure, for example the existence of a certain kind of foliation or flow. Barbot showed that the stable/unstable foliations of contact Anosov flows are -covered (i.e. their leaf space in the universal cover is Hausdorff, and therefore homeomorphic to the real line); meanwhile, Fenley and I obtained essentially a strong classification theorem for manifolds with -covered foliations, showing (before Perelman!) that they satisfy a weak version of the geometrization conjecture (that their fundamental groups either contain a or are word-hyperbolic), and that they contain quasigeodesic pseudo-Anosov flows. Thus they can be understood and analyzed in many ways, and we have an essentially complete picture of their geometry and topology.

So we have known for quite some time that Anosov flows are quite flexible, and there are many known constructions. By contrast, the only known contact Anosov flows were very special — essentially the only general example available before this paper was the geodesic flow on a Riemannian (or Finsler) manifold of negative curvature.

The key example is a hyperbolic surface . A geodesic in lifts to a knot in the unit tangent bundle , by associating to each the unit vector in perpendicular to and on the positive side. This knot is Legendrian, and is transverse to both the stable and unstable foliations, so Foulon-Hasselblatt show that one can do surgery on it to produce interesting new contact Anosov flows on new manifolds. If is hyperbolic, the result of a sufficiently big surgery will be a hyperbolic manifold. Consequently, Foulon-Hasselblatt raise the natural question of what conditions on ensure that is hyperbolic.

In fact, one obvious necessary condition is that should be filling in , that is, the complementary regions to should be polygons. For, otherwise, an essential embedded loop in suspends to an essential embedded (non-boundary parallel) torus in , which is an obstruction to hyperbolicity. In fact, Foulon-Hasselblatt ask explicitly whether this filling condition is sufficient.

Anyway, when I read this, I immediately felt that this should be the only obstruction. I have been out of 3-manifold theory for a while, but the statement seemed vaguely familiar, and I’m reasonably confident that this fact is somewhere in the literature (though who knows; I’d be grateful to any reader that can point me to a specific reference). It is also vaguely reminiscent of the well-known theorem of Menasco that if is a nonsplit prime alternating link which is not a torus link, then is hyperbolic. On the other hand, it turns out to be simple enough to prove directly, so the purpose of this blog post (apart from to break my record of only blogging in odd numbered years) is to give a short proof of this fact.

Let’s be a bit more precise. Let be a closed, oriented hyperbolic surface, let be a finite union of immersed, oriented, primitive geodesics in . Assume that no two components of are the same geodesic with opposite orientation.

Associated to is the link (or just if is understood) in the unit tangent bundle consisting of unit vectors based at points of (the image of) for which the ordered pair make an oriented orthonormal basis for . Note that it is just as easy to let and be unoriented, and work in the projective unit tangent bundle instead of .

Theorem. If is filling in then is hyperbolic.

Note that we could just as well take to be the set of tangent vectors to , since this is isotopic (as a link) to .

Proof: Thurston famously showed (also see here for a detailed proof) that a 3-manifold with boundary is hyperbolic if and only if it is irreducible (i.e. every embedded sphere bounds a ball), has infinite fundamental group, and contains no essential embedded torus which is not parallel to a boundary component. Since is irreducible, and no component of is contained in a ball, so is . Furthermore, surjects onto which is infinite. It follows that either is hyperbolic, or it contains an embedded essential non-boundary parallel torus . We show that no such can exist.

The inclusion induces a map on fundamental groups. We let . This is a free abelian group (because is torsion free) of rank at most 2. The proof reduces to a case-by-case analysis depending on the rank of .

Case . In this case, since is embedded in , the map on fundamental groups is injective, and therefore (by the classification of essential embedded tori in Seifert fibered spaces) is vertical — i.e. it is the union of circle fibers over an embedded essential loop in . Since is filling, some has nontrivial (geometric) intersection number with . Evidently, the lift intersects .

Case . In this case, and lifts to an embedded torus in the universal cover . The flowlines of the geodesic flow make into a topological product , and sits in this as for some discrete set . Hence any embedded torus in the complement is compressible; this shows that was already compressible in .

Case . In this case is cyclic and is equal to for some . A component of the preimage is proper, homeomorphic to a cylinder, and of uniformly bounded thickness (i.e. it is foliated by circles of uniformly bounded diameter). To see this, first foliate by circles in the homotopy class of the kernel of , and lift the foliation to . Note also that the foliation is -equivariant. The idea of the proof is now straightforward: must separate some of the components of from others. Since it has bounded thickness, it can only separate finitely many. But if is not boundary parallel, it must separate at least two. These two must be a finite Hausdorff distance apart in ; this will readily imply that they are both lifts of the same geodesic in , which will give a contradiction. We now flesh out this argument.

Let’s consider the geometry of . There is a canonical 1-Lipschitz projection with fibers homeomorphic to (these are lifted flowlines of the circle flow on ). Moreover, there is a foliation of by stable leaves of the geodesic flow (on ). If we pick one stable leaf and let denote the generator of the center, then the “slab” between and (a fundamental domain for the deck action of on ) is quasi-isometric to , and every component of in this slab is quasi-isometric to a geodesic in .

Now consider how sits in . Since is essential in , it follows that is essential in ; the latter is homeomorphic to for some discrete set , so must be properly isotopic to a proper annulus of the form for some embedded in (warning: this product structure is purely topological, and not quasi-metrical). Since is assumed not to be boundary parallel, must enclose at least two distinct points of , corresponding to components and of ; i.e. must separate these components from other components. Recall that is foliated (-equivariantly if we like) by circles of uniformly bounded diameter. Each such circle bounds a disk in of uniformly bounded diameter, and moreover each such disk intersects and in at least two points. The set of such disks is proper; in particular, it follows that and contain a pair of proper sequences of points which are a uniformly bounded distance apart; in particular, and are a finite Hausdorff distance apart. It follows that the have the same projection to under , and therefore they are lifts of the same component of .

But now one readily obtains a contradiction. Pick an arc disjoint from from one component to the other. This can be chosen to project to a closed essential loop in , and we deduce that encloses infinitely many lifts of once it encloses two. But this set of lifts is discrete, so only finitely many have uniformly bounded Hausdorff distance from any one of them. This contradiction completes the proof. qed.

This method of proof is a little bit specific and possibly not as simple as possible, but I believe it does generalize to the case that is a filling union of round circles in — i.e. curves of constant geodesic curvature (which might vary from component to component). Some of these circles (those with extrinsic curvature ) will be essential in and some (those with extrinsic curvature ) will be inessential; but their lifts to will all be essential, and if the union is filling in , the complement of the lifts will be atoroidal. The main issues to deal with are to show that the preimage in looks like a collection of straight lines in (i.e. it is globally unknotted) and to deal with the fact that there are now distinct circles whose lifts are a finite Hausdorff distance apart (any two circles with extrinsic curvature ). Casson-Jungreis‘s criterion for unknottedness could be used to deal with the first issue, I think.

This is now no longer relevant to contact Anosov surgery, but rather to “regulating surgery”, of the kind considered in another paper I wrote some time ago. (Idle remark: this was the third paper I ever wrote, and I was very pleased with myself for disproving a conjecture of Thurston. Naturally the silence from the mathematical world was deafening. Finally a review appeared on MathSciNet, and I thought: finally I’ll get some feedback! Needless to say I found the actual review a little disappointing . . . C’est la vie)

One of the main results of the paper is to show that every quasigeodesic flow on a closed hyperbolic 3-manifold either has a closed orbit, or the fundamental group of the manifold admits an action on a circle with some very peculiar properties, namely that it is Mobius-like but not Mobius. The problem of giving necessary and sufficient conditions on a vector field on a 3-manifold to guarantee the existence of a closed orbit is a long and interesting one, and the introduction to the paper gives a brief sketch of this history as follows:

In 1950, Seifert asked whether every nonsingular flow on the 3-sphere has a closed orbit. Schweitzer gave a counterexample in 1974 and showed more generally that every homotopy class of nonsingular flows on a 3-manifold contains a representative with no closed orbits. Schweitzer’s examples were generalized considerably and it is known that the flows can be taken to be smooth or volume-preserving.

On the other hand, Taubes’ 2007 proof of the 3-dimensional Weinstein conjecture shows that flows satisfying certain geometric constraints must have closed orbits. Explicitly, Taubes showed that every Reeb vector field on a closed 3-manifold has a closed orbit. Reeb flows are geodesible, i.e. there is a Riemannian metric in which the flowlines are geodesics. Complementary to this result, though by different methods, Rechtman showed in 2010 that the only geodesible real analytic flows on closed 3-manifolds that contain no closed orbits are on torus bundles over the circle with reducible monodromy.

Geodesibility is a local condition, and furthermore one that is not stable under perturbations. By contrast, a nonsingular flow is said to be quasigeodesic if the flowlines of the flow pulled back to the universal cover are quasigeodesics. This is a macroscopic condition, and when the ambient 3-manifold is hyperbolic it is a stable condition under perturbations; this stability is for global topological reasons and not because the flow itself is structurally stable (which it will not typically be).

Calegari conjectured in 2006 that quasigeodesic flows on closed hyperbolic 3- manifolds should all have closed orbits, and moreover that every homotopy class of quasigeodesic flow should contain a pseudo-Anosov representative that is unique up to isotopy. Pseudo-Anosov flows are hyperbolic and therefore structurally stable, so this conjecture implies that one should be able to deduce the existence of closed orbits from the dynamics of the fundamental group on the orbit space in the universal cover.

Our paper is devoted to fleshing out some aspects of Calegari’s conjectural program. We are able to find conditions that guarantee the existence of a closed orbit for a quasigeodesic flow on a closed hyperbolic 3-manifold expressed in terms of the action of the fundamental group on an associated “universal circle”.

To go more deeply into this, let me start with some basic definitions. We are concerned always with a closed hyperbolic 3-manifold with a 1-dimensional foliation (the leaves of the foliation are the flowlines of the flow). The universal cover of is isometric to hyperbolic 3-space, and the foliation lifts to a 1-dimensional foliation of the universal cover. To say that the flow (foliation) is quasigeodesic is to say that the leaves of are quasigeodesics in hyperbolic 3-space.

The fact that the flowlines are quasigeodesics easily implies that the leaf space of (i.e. the quotient of hyperbolic 3-space by the equivalence relation that collapses every leaf to a point) is Hausdorff; since it is simply-connected and noncompact, it is homeomorphic to the plane. Notice that the plane comes together with an action by the fundamental group ; a closed orbit of the flow corresponds precisely to a fixed point for some nontrivial element of .

Now, every oriented quasigeodesic in hyperbolic 3-space is asymptotic to two distinct points in the sphere at infinity. It follows that we can define two equivariant endpoint maps . The point preimages of under (say) decompose the plane into closed, connected sets. It turns out that each of these sets is unbounded and therefore has a nonempty collection of ends. The nice thing about a collection of disjoint, closed, connected, unbounded subsets of the plane is that the set of ends of such subsets can be circularly ordered in a canonical way, and one therefore obtains a natural action of on a circularly ordered set, which can be bootstrapped to a (faithful) action of on a so-called universal circle  by homeomorphisms. This much of the story is contained in my 2006 paper.

Some hyperbolic 3-manifolds have fundamental groups which do not act faithfully on a circle; from this one deduces that there are hyperbolic 3-manifolds with no quasigeodesic flow, which answered a long-standing question of Thurston. We’ll return to this question in a minute.

Now, from the discussion above, we see that the existence of a quasigeodesic flow gives rise to a natural action of on a plane and a circle . It is natural to wonder if (and in fact I conjectured that) there is a natural topology on , compatible with the actions, for which the union is homeomorphic to a closed disk. This is the first main theorem Steven proves:

Compactification Theorem (Frankel): There is a natural compactification of homeomorphic to the closed disc so that . The action of on extends to and restricts to the universal circle action on .

The proof of this is quite deep and involved. One of the main difficulties is that a priori, the point preimages under the endpoint maps are arbitrary closed subsets of the plane, so dealing with their separation properties is very involved. Steven develops his theory in quite some generality. An unbounded decomposition of the plane is a partition of the plane into unbounded continua; Steven’s main theorem is that any such decomposition with uncountably many elements gives rise to a canonical compactification of the plane, homeomorphic to the disk. Applying this to the special case arising in the context of a quasigeodesic flow, gives the Compactification Theorem above.

The story can be repeated with in place of , and one gets another universal circle compactifying . In fact, one can work with both and simultaneously, and obtain a “master” compactification, obtained by adding a “master” universal circle , with canonical monotone surjections to the positive and negative universal circles constructed as above. One must deal with generalized unbounded decompositions to achieve this result; this is Theorem 7.9 in Steven’s paper. Using this, one can build a “universal sphere” from two copies of glued together along . From the construction, the following conjecture seems quite plausible:

Conjecture 1: The maps extend to a monotone map .

Note that since the image of under such a hypothetical map should be both closed and invariant, it should be equal to all of ; i.e. it would be a group-invariant Peano curve. Examples of such curves arise very naturally by the Cannon-Thurston construction associated to surface bundles. A proof of Conjecture 1 would give a new proof (and considerable generalization) of the Cannon-Thurston theorem. The connection between quasigeodesic flows and surface bundles is the simple fact that any 1-dimensional foliation of a hyperbolic 3-manifold transverse to the surfaces of a surface fibration is quasigeodesic, and the universal circle in this case should be the circle at infinity of the universal cover of a surface fiber.

Let’s return to the question of closed orbits. Now, any homeomorphism of a closed disk has a fixed point, by the Brouwer fixed point theorem. So one deduces either that has a closed leaf, or that every nontrivial element of has at least one fixed point in . To make more progress, one must understand the relationship between the dynamics of on and the dynamics on . A positive answer to Conjecture 1 above would simplify things, but even without it, Steven is able to get a great deal of traction.

Let’s consider the following definition:

Definition: A group of homeomorphisms of the circle is Mobius-like if every element is conjugate to an element of . It is rotationless if every element is conjugate to a hyperbolic or parabolic element. It is Mobius if the entire group is conjugate into .

With this definition, Steven’s next main theorem is the following:

Mobius-like Theorem (Frankel): Let be a quasigeodesic flow on a closed hyperbolic 3- manifold . Suppose that the action of on the universal circle is not a rotationless Mobius-like group. Then has a closed orbit.

This is nicely complemented by:

Conjugacy Theorem (Frankel): Let be a quasigeodesic flow on a closed hyperbolic 3- manifold . Then the action of on is not conjugate into .

Steven conjectures that the action of on should never be Mobius-like; this would imply that every quasigeodesic flow on a hyperbolic 3-manifold should have a closed orbit.

While we’re being speculative, let’s imagine how far such a program could go. Quasigeodesicity persists under perturbations, even though a quasigeodesic flow need not be structurally stable (for example, it could contain a solid torus foliated by closed orbits). We can create closed orbits by a small perturbation, and these give rise to fixed points in for the perturbed actions. The connected preimage under containing the fixed point must itself be fixed, and so must its set of ends. If this set is finite, some power fixes the ends pointwise; a similar picture holds for , and we should obtain a collection of fixed points in the master circle which one expects to have alternating source-sink dynamics. In this way, we expect to be able to produce a pair of invariant stable/unstable laminations. These should give rise in turn to a pseudo-Anosov quasigeodesic flow, whose closed orbits should correspond to Nielsen classes of closed orbits of the original flow . Hence (conjecturally), not only should have one closed orbit, it should have infinitely many! Explicitly:

Conjecture 2: Let be a quasigeodesic flow on a hyperbolic 3-manifold . Then should be homotopic to a pseudo-Anosov quasigeodesic flow whose closed orbits should be in bijection to free homotopy classes of closed orbits of .

The relationship between and should be like the relationship between a surface homeomorphism, and its pseudo-Anosov representative. Interestingly enough, the “stable/unstable laminations” we would like to find are already actually known to exist; they are constructed in Theorem B of my paper. What is missing is the interpretation of these laminations as the residue on the universal circle of a pair of stable/unstable laminations of the flow space of a homotopic flow.

How canonical should be? As far as I know, there is no known obstruction to the following conjecture:

Conjecture 3: Every connected component of the space of quasigeodesic flows on a hyperbolic 3-manifold should contain a unique pseudo-Anosov quasigeodesic flow, up to isotopy.

Well, this picture is all very nice, if true. But it raises the significant problem of constructing quasigeodesic flows, or understanding exactly which hyperbolic 3-manifolds do or don’t have them. As remarked above, the existence of a quasigeodesic flow implies that the fundamental group is circularly orderable, and therefore that some finite index subgroup is left orderable. In fact, if is an integral homology sphere, the fundamental group is circularly orderable if and only if it is left orderable. The condition of left orderability is quite interesting in its own right; there are many known examples of hyperbolic 3-manifolds whose fundamental groups are not left orderable (e.g. double branched covers of alternating knots in the 3-sphere), and some people are trying to connect up this condition to the concept of a (Heegaard Floer Homology) L-space.

But I prefer to be a bit more optimistic, and look at a quasigeodesic flow as a potentially quite flexible structure. Suppose is a hyperbolic 3-manifold with a cusp. Such a 3-manifold has nontrivial 2-dimensional (relative) homology, and combined work of Fenley-Gabai-Mosher shows that it admits a pseudo-Anosov flow, which persists (and is quasigeodesic) in “most” Dehn fillings (see e.g. Fenley-Mosher or my foliations book for a discussion of this). Now, if we have a hyperbolic 3-manifold with an embedded geodesic with a sufficiently thick embedded tube around it, we know is hyperbolic, and has such a nice flow. We can try to extend this flow over by spinning it around . It is plausible that the resulting flow on should be quasigeodesic: far from , it should be quasigeodesic because the geometry should be close to the geometry of , and quasigeodesity is stable. Close to , it should be quasigeodesic, because it wraps around and around . Anyway, I think it is worth making another conjecture:

Conjecture 4: For any there is a so that if is a hyperbolic 3-manifold with an embedded geodesic of length contained in an embedded tube of radius at least , then admits a quasigeodesic flow.

If is an arbitrary hyperbolic 3-manifold, one can find a finite cover satisfying the hypotheses of this conjecture, by using the fact that cyclic groups in hyperbolic 3-manifold groups are subgroup separable.

Some elements of this program are more approachable than others, but Steven’s work definitely represents a big step forward.

Eight of them fit together to make a circle; but the pieces of track can also be picked up and turned over, and connected in more complicated ways:

Or even more complicated:

(I love postscript!)

The last example illustrates the fundamental problem of laying train tracks: after laying a long stretch of track, it’s hard to get the two ends to match up precisely. The reason for this is quite straightforward: how close the ends are together doesn’t give a good indication of how many segments are needed to join them up.

A more “sophisticated” way of saying this has to do with holonomy and indiscrete representations. Let’s suppose we are laying the pieces of our track one by one, always attaching the next piece to the same end of what has been built so far. We have two choices of how to lay each subsequent piece of track; call these R and L depending on whether the piece bends to the right or to the left. A sequence of tracks is therefore encoded by a string in the 2-letter alphabet; for example, the nice complete track as above is encoded by the cyclic string RRRRRLRRRRRL (cyclic, because the track closes up to make a loop). There are some highly non-obvious constraints on the strings that are allowed if we insist that the tracks are embedded in the plane, but if we are happy with “immersed” train tracks, then any string in R and L is allowed, and determines a unique track, up to isometry. We would like to give some simple criterion to tell when a string corresponds to a closed track, and more generally, given a string, to give a simple method to determine the shortest string that can be added to it to close up the track.

This is a problem in group theory. Each R and L can be thought of as applying a 1/8 twist to the plane (clockwise or anticlockwise) centered at a point which can be determined from the current location of the track. In other words, we can think of appending an R or L as a right multiplication by an isometry of the plane, and we want to know exactly when a given composition represents the trivial isometry. If we let G denote the group generated by R and L, then the first and most significant observation is that G is indiscrete. That is, an element in G might move points a very small distance (i.e. the track might “almost” close up), but the shortest length of added track needed to close it completely might be arbitrarily long.

Let’s try to be a bit more systematic. The group of isometries of the Euclidean plane is a semidirect product; it contains a normal subgroup consisting of the group of translations, and the quotient is the orthogonal group of rotations. Each of the elements R and L corresponds to a 1/8 and -1/8 rotation respectively, so the first and most obvious condition to get the track to close up is that the number of Rs minus the number of Ls must be 8 times an integer. Incidentally, this integer is the winding number of the immersed track; a necessary (but not sufficient) for the track to be embedded is that this integer must be 1 or -1. If this condition is satisfied, the two ends of the track will be aligned in the correct way for them to join up, but their positions might differ by a translation. Evidently each R or L translates the “lock” in one of the eight directions NNE, ENE, ESE, SSE, SSW, WSW, WNW, NNW; after an order 1/16 rotation we could relabel these directions as N,NE,E,SE,S,SW,W,NW. If we identify the Euclidean plane with the complex plane, then (after a change of scale), the relative position of any two “locks” is an element of the additive group A generated by the 1/8 roots of unity. This group is an indiscrete subgroup of the translations of the plane, isomorphic to . But now the answer to our problem is obvious: instead of measuring distance in the plane, measure distance instead in A. On way to do this is to use the fact that the natural embedding is discrete, where maps an 1/8th root of unity to its cube. In terms of RL sequences, this means building two sequences of train tracks simultaneously; if the first corresponds to some sequence RRLRL etc., then the second corresponds to the sequence where each R is replaced with RRR and each L with LLL; i.e. RRRRRRLLLRRRLLL etc. in this example. If S is the first sequence, let’s call the second sequence S’. A track corresponding to S can be closed up by a small number of pieces providing both that track, and the track corresponding to S’, have ends which are close together in the usual sense. A very nice illustration of a closely related example is Rich Schwartz’s game “Lucy and Lily”.

How many closed tracks are there of a given length? Giving an exact formula is possible although a little tricky; it’s a bit easier to give an asymptotic formula, using some probability theory. Build a finite directed graph whose eight vertices correspond to the eight possible orientations of the end of the track, and put a directed edge from one vertex to another with the label R or L when orientations differ by an 1/8 turn in the positive or negative sense:

There is a function T from the 16 edges to the set of 1/8th roots of unity, and the value on a given edge corresponds to the way in which appending an R or L in a given orientation translates the lock. An RL string determines a walk in the graph by proceeding at every stage along the edge with the label corresponding to the letter in the string. Then the total translation associated to a string is the sum of the function T on the edges visited in the corresponding path. We take this sum in the abstract group for simplicity. A path closes up if and only if it starts and ends at the same vertex, and if the T sum coming from the edges is zero.

A random string of Rs and Ls thereby corresponds to a random walk on the directed graph ; this is an example of a stationary Markov chain, and the value of the function on a random walk of length satisfies a central limit theorem. We want more precise information, namely the chance that a random walk returns to its initial vertex, and satisfies ; such a result is called a local limit theorem. The ergodic theorem says that the chance of returning to the origin is 1/4 if is even, and 0 if is odd. Similarly, can only be zero if is even, in which case the chance that satisfies as , where is a particular algebraic number approximately equal to 1.1024. Since the number of RL words of length n is , this means that the number of closed tracks of (even) length has order .

There are many obvious directions one can take these ideas. There is an obvious relation to Conway’s tiling groups, as explained by Thurston. The phenomenon of an indiscrete finitely generated group of isometries becoming discrete in a suitable (Galois twisted) product lies behind the construction of what are known as arithmetic lattices. One can also try to generalize this discussion to other geometries; e.g. to study train track configurations on the sphere, or in the hyperbolic plane. Finally, one can try to attack the much harder problem of enumerating the number of embedded closed tracks of given length (or finding an asymptotic formula). But we’ll save that for another post.

(Added December 7)

I thought it might be instructive to give an example of a “complicated” pair of closed (immersed) tracks corresponding to the pair of RL strings

LLRRLRRRLRLRRRRLLRRRLRLRLLLRLLLLLLRRRLRRRLRRRRRR

and

LLLLLLRRRRRRLLLRLLLRRRLLLRRRRLLLLLLRLLLRRRLLLRRRLRRRLLRLLLRLLLRR.

Each string is obtained from the other by substituting RRR for each R and LLL for each L (and then removing strings of 8 consecutive Rs or Ls, which just remove a little closed loop from the corresponding track). These strings were generated by the method described above: after laying down some random initial string, I added bits to each simultaneously in an effort to get both tracks to close up. I was quite pleased that this worked out nicely in practice. In case you want to have a play with this yourself, here is the postscript code to generate this figure. Fiddle with the RL strings at the ends to lay a different track. And if you come up with a nice pattern, please email me!

%!PS-Adobe-2.0 EPSF-2.0

%%BoundingBox: 0 0 540 300

gsave
5 5 scale
1 4 div setlinewidth

20 30 translate

/socket {
 newpath
 2 0 moveto
 1 -0.2 1 0 1.2 0.2 curveto
 1 0.4 0.2 2 sqrt mul -45 225 arc
 1 0 1 -0.2 0 0 curveto
 stroke
} def

/Rtrack {
 gsave
 -1 0 translate
 newpath
 5 0 3 135 180 arc
 stroke
 newpath
 5 0 5 135 180 arc
 stroke
 socket
 gsave
 5 0 translate
 -45 rotate
 -5 0 translate
 socket
 grestore
 grestore
} def

/Ltrack {
 gsave
 -1 1 scale
 Rtrack
 grestore
} def

/R {
 Rtrack
 4 0 translate
 -45 rotate
 -4 0 translate
} def

/L {
 Ltrack
 -4 0 translate
 45 rotate
 4 0 translate
} def

gsave
L L R R L R R R L R L R R R R L L R R R L R L R L L L R L L L L L L

R R R L R R R L R R R R R R
grestore

70 20 translate

L L L L L L R R R R R R L L L R L L L R R R L L L R R R R L L L L L

L R L L L R R R L L L R R R L R R R L L R L L L R L L L R R
grestore
%eof

If is a group, and are elements of , the commutator of and (denoted ) is the expression (note: algebraists tend to use the convention that instead). Commutators (as their name suggests) measure the failure of a pair of elements to commute, in the sense that . Since , the property of being a commutator is invariant under conjugation (here the superscript means conjugation by ; i.e. ; again, the algebraists use the opposite convention).

If is a space with fundamental group , conjugacy classes of elements in correspond to free homotopy classes of loops in . So let be some conjugacy class, and let  be in the corresponding free homotopy class. The element is a commutator in if and only if there is a genus 1 surface (i.e. a torus) with one boundary component, and a map  for which the restriction of to  factors as for some homeomorphism . In words, an element in a group is a commutator if and only if the corresponding loop in a space bounds a genus 1 surface.

If then the loops representing and can be thought of as the meridian and the longitude of the bounding torus. There’s some very nice pictures of this (and loads of other stuff) at the blog Sketches of Topology.

Now, the Hall-Witt identity is the identity , valid in any group. To prove this identity it suffices to prove it in a free group, where it follows just by expanding the expressions (we use the convention that  and so on).

First, the expression just means  which simplifies to . The other two expressions are all obtained from the first by cyclic permutation of . Using the notation , and  we see that the three expressions expand to , proving the identity.

Incidentally, some people write the term slightly differently. Taking conjugation by outside the brackets shows that this expression is equal to which in turn is equal to , which itself is equal to . In a group in which three-fold commutators are trivial (i.e. a “nilpotent group of class 3”) this is just and the Hall-Witt identity becomes a little simpler.

A slightly more geometric way to see this identity is to think about words in a free group as directed paths in a graph, where two words represent the same element if the corresponding paths are the same “after eliminating backtracks”. It is convenient to work in the graph whose vertices are the lattice and whose edges are parallel to the coordinate axes and labeled depending on their alignment. This graph is the fundamental group of the commutator subgroup of the free group ; one way to see this is to observe that the deck group is equal to the homology group , and to remember that this first homology group is just the abelianization. In this graph, the magic word is a kind of “bent letter S”; see figure:

and the composition is a kind of dumbell, made by tracing around the boundary of two opposite squares in a cube together with an edge joining them:

The boundary of the cube can be decomposed into three such dumbells in a symmetric way, and this decomposition “explains” the Hall-Witt identity (pardon the lack of hidden line removal; I write figures in .eps):

Higher dimensional generalizations of this picture (where the loops going around squares are replaced with spheres going around cubes of various dimensions) explain why the Whitehead product in homotopy theory makes the rational homotopy groups of a space into a graded Lie algebra (this is still approximately true over the integers, except that one needs to be a bit careful about 2-torsion).

A more geometric way still is to think about maps of surfaces to spaces, and what are called gropes. An expression like can be thought of geometrically as follows. The elements and are the meridian and longitude of a once-punctured torus with boundary on . But the meridian is itself the boundary of another once-puncture torus , whose meridian and longitude (in turn) are and . Geometrically, we can think of as bounding a certain kind of grope: a once-punctured torus with another once-punctured torus glued onto its meridian.

This grope can be embedded in 3-dimensional space, and thickening it slightly we obtain a genus 3 handlebody whose fundamental group is . The boundary is a genus 3 surface, and the loop divides it into a genus 1 surface and a genus 2 surface. We can think of the genus 1 surface as the “inside” of , and the genus 2 surface as the “outside” of cut open along with two copies of attached. One copy of is tucked inside the other; we can fold it out as in the figure to lay it flat.

The genus 1 surface represents in an obvious way, in the sense that there is a choice of meridian and longitude corresponding to and respectively. The genus 2 surface can be expressed as a product of 2 commutators in many ways; a pair of embedded loops intersecting transversely once gives one commutator, and a disjoint pair intersecting in the same way gives the other. The figure indicates a choice for which one meridian-longitude pair is and up to conjugacy, and the other is and (note that is not represented by a loop in the genus 2 surface, but rather as a path between the two loops where was cut open).

So this expresses as a product of something of the form and , up to suitably conjugating the entries. Keeping track of basepoints determines the correct conjugations, giving the Hall-Witt identity.

The purpose of this blog post (aside from continuing the trend of posts titles containing the letter “Z”) is to discuss a very interesting conjecture that arose in the course of writing this paper. The conjecture does not need many prerequisites to appreciate or to attack, and it is my hope that some smart undergrad somewhere will crack it. The context is as follows.

We let denote the group of orientation-preserving homeomorphisms of the circle, and let denote its universal cover, which is the group of orientation-preserving homeomorphisms of the real line which commute with integer translation. Poincaré’s rotation number is a class function  which descends to . The function is a kind of “average translation distance”, defined by .

Let be a free group of rank 2 with generators  and . An element is positive if it is a product of positive powers of the generators. Given a word and real numbers we let denote the supremum of  under all
representations of into for which and .

The main theorems we prove are the following:

Rationality Theorem: If and are rational, and is positive, then is rational with denominator no bigger than the denominators of or .

Stability Theorem: If and are rational with denominators at most , and
is positive, there is some positive so that for all .

Both theorems can be proved rather easily by the combinatorial method described in my previous post. Roughly speaking, to compute look at all cyclic words in the alphabet with s and s, and for each one, compute a “combinatorial” rotation number associated to a discrete dynamical system. Then is the maximum of this finite list of rational numbers. A nice aspect of this proof is that it is effective, and gives the means to actually compute and draw a graph of it.

The graph of R(abaab,r,s) for r,s in

Now, although the function is nondecreasing as a function of  it is discontinuous, and can jump up at a limit. We define  to be the supremum of over . It is not hard to prove the following:

Lemma: is the supremum of under all representations of into for which and are conjugate to rigid rotations  respectively.

Here the notation means the rotation . If we denote by the number of ‘s in , and by the number of ‘s in , then it is always true that , since we always have the representation for which and .

In contrast to the Stability Theorem, it turns out that there are words and points for which there is a strict inequality  for all . We call such a point a slippery point for . The Slippery Conjecture is then the following:

Slippery Conjecture: If is positive, and is a slippery point for , then

How should one interpret this conjecture? One should think of the Rationality and Stability theorems as a kind of nonlinear analog of the phenomenon of Arnol’d tongues: when we perturb a linear system of circle rotations by adding nonlinear noise, phase locking tends to produce periodic orbits and therefore rational rotation numbers. In our context, the representation which is “maximally nonlinear” (i.e. for which  differs from the most) tends to have a small denominator. If nonlinearity produces “rigidity”, then slippery phenomena should be associated with linearity.

The point (1/2,1/2) is slippery for abaab

Notice if is slippery for that  must have arbitrarily large denominators as and . We can make a quantitative refinement of the Slippery Conjecture as follows:

Refined Slippery Conjecture: Let  be positive, and suppose . Then

This conjecture says that the bigger the denominator of  — i.e. the rotation number associated to the “maximally nonlinear” representation — the less nonlinear this maximal representation is. The Refined Slippery Conjecture implies the Slippery Conjecture.

Computer experiments support the Refined Slippery Conjecture, but we don’t have a good feel for why it might be true. But it can be translated into a purely combinatorial question, using cyclic -words, and maybe there is a clever combinatorial way to obtain the desired estimate.

Plot of against (the denominator of )