Quasigeodesic flows on hyperbolic 3-manifolds

My student Steven Frankel has just posted his paper Quasigeodesic flows and Mobius-like groups on the arXiv. This heartbreaking work of staggering genius interesting paper makes a deep connection between dynamics, hyperbolic geometry, and group theory, and represents the first significant progress that I know of on a conjectural program I formulated a few years ago.

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.

Laying train tracks

This morning I was playing trains with my son Felix. At the moment he is much more interested in laying the tracks than putting the trains on and moving them around, but he doesn’t tend to get concerned about whether the track closes up to make a loop. The pieces of track are all roughly the following shape:

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

The Hall-Witt identity

The purpose of this blog post is to try to give some insight into the “meaning” of the Hall-Witt identity in group theory. This identity can look quite mysterious in its algebraic form, but there are several ways of describing it geometrically which are more natural and easier to understand.

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.

Ziggurats and the Slippery Conjecture

A couple of months ago I discussed a method to reduce a dynamical problem (computing the maximal rotation number of a prescribed element  in a free group given the rotation numbers of the generators) to a purely combinatorial one. Now Alden Walker and I have uploaded our paper, entitled “Ziggurats and rotation numbers”, to the arXiv.

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 )

Zonohedra and the Sylvester-Gallai theorem

When I was in Melbourne recently, I spent some time browsing through a copy of “Twelve Geometric Essays” by Harold Coxeter in the (small) library at AMSI. One of these essays was entitled “The classification of zonohedra by means of projective diagrams”, and it contained a very cute proof of the Sylvester-Gallai theorem, which I thought would make a nice (short!) blog post.

The Sylvester-Gallai theorem says that a finite collection of points in a projective plane are either all on a line, or else there is some line that contains exactly two of the points. Coxeter’s proof of this theorem falls out incidentally from an apparently unrelated study of certain polyhedra known as zonohedra.

For subsets and of a vector space , the Minkowski sum  is the set of points of the form for and . If and are polyhedra, so is , and the vertices of are sums of vertices of and . One natural way to think of is that it is the projection of the product under the affine map .

The simplest definition of a zonohedron (in any dimension) is that it is the Minkowski sum of finitely many intervals. Thus the faces of a zonohedra are themselves zonohedra. In 2 dimensions a zonohedron is a centrally symmetric polygon, and therefore has an even number of edges which come in parallel pairs of the same length. A zonohedron is convex, being the Minkowski sum of convex sets. Thus it is topologically a ball, and its boundary is topologically a sphere. A parallelepiped is an example of a 3-dimensional zonohedron; so is the rhombic dodecahedron and the rhombic triacontahedron. One can think of a zonohedron as a projection to a low dimensional space of a high dimensional parallelepiped; one can use this observation to produce interesting aperiodic tilings from zonohedra.

Here is Coxeter’s proof of the Sylvester-Gallai theorem. Let be a 3-dimensional zonohedron, expressed as the Minkowski sum of some collection of intervals . Each determines a point in the projective plane; conversely, a collection of points in the projective plane determines a family of zonohedra, where each element of the family is determined by the edge lengths of the . The faces of the zonohedra correspond to the colinear collections of . A decomposition of the sphere into polygons meeting at least 3 to a vertex must contain at least one polygon with sides, by Euler’s formula; hence every 3 dimensional zonohedron has at least one face with exactly sides. This corresponds to a line containing exactly 2 of the ; qed.