Agol’s Virtual Haken Theorem (part 1)

I am in Paris attending a workshop at the IHP where Ian Agol has just given the first of three talks outlining his proof of the Virtual Haken Conjecture and Virtual Fibration Conjecture in 3-manifold topology (hat tip to Henry Wilton at the Low Dimensional Topology blog from whom I first learned about Ian’s announcement last week). I think it is no under overstatement to say that this marks the end of an era in 3-manifold topology, since the proof ties up just about every loose end left over on the list of problems in 3-manifold topology from Thurston’s famous Bulletin article (with the exception of problem 23 — to show that volumes of closed hyperbolic 3-manifolds are not rationally related — which is very close to some famous open problems in number theory). The purpose of this blog post is to say what the Virtual Haken Conjecture is, and some of the background that goes into Ian’s argument. I hope to follow this up with more details in another post (after Agol gives talks 2 and 3 this coming Wednesday). Needless to say this post has been written in a bit of a hurry, and I have probably messed up some crucial details; but if that caveat is not enough to dissuade you, then read on.

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).

Hyperbolic Geometry Notes #2 – Triangles and Gauss Bonnet

In this post, I will cover triangles and area in spaces of constant (nonzero) curvature. We are focused on hyperbolic space, but we will talk about spheres and the Gauss-Bonnet theorem.

1. Triangles in Hyperbolic Space

Suppose we are given 3 points in hyperbolic space . A triangle with these points as vertices is a set of three geodesic segments with these three points as endpoints. The fact that there is a unique triangle requires a (brief) proof. Consider the hyperboloid model: three points on the hyperboloid determine a unique 3-dimensional real subspace of which contains these three points plus the origin. Intersecting this subspace with the hyperboloid gives a copy of , so we only have to check there is a unique triangle in . For this, consider the Klein model: triangles are euclidean triangles, so there is only one with a given three vertices.

In hyperbolic space, it is still true that knowing enough side lengths and/or angles of a triangles determines it. For example, knowing two side lengths and the angle between them determines the triangle. Similarly, knowing all the angles determines it. However, not every set of angles can be realized (in euclidean space, for example, the angles must add to ), and the inequalities which must be satisfied are more complicated for hyperbolic space.

2. Ideal Triangles and Area Theorems

We can think about moving one (or more) of the points of a hyperbolic triangle off to infinity (the boundary of the disk). An ideal triangle is one with all three “vertices” (the vertices do not exist in hyperbolic space) on the boundary. Using a conformal map of the disk (which is an isometry of hyperbolic space), we can move any three points on the boundary to any other three points, so up to isometry, there is only one ideal triangle. We have fixed our metric, so we can find the area of this triangle. The logically consistent way to find this is with an integral since we will use this fact in our proof sketch of Gauss-Bonnet, but as a remark, suppose we know Gauss-Bonnet. Imagine a triangle very close to ideal. The curvature is , and the euler characteristic is . The sum of the exterior angles is just slightly under , so using Gauss-Bonnet, the area is very close to , and goes to as we push the vertices off to infinity.

One note is that suppose we know what the geodesics are, and we know what the area of an ideal triangle is (suppose we just defined it to be without knowing the curvature). Then by pasting together ideal triangles, as we will see, we could find the area of any triangle. That is, really the key to understanding area is knowing the area of an ideal triangle.

As mentioned above, there is a single triangle, up to isometry, with given angles, so denote the triangle with angles by .

2.1. Area

Knowing the area of an ideal triangle allows us to calculate the area of any triangle. In fact:

Theorem 1 (Gauss)

This geometric proof relies on the fact that the angles in the Poincare model are the euclidean angles in the model. Consider the generic picture:

We have extended the sides of and drawn the ideal triangle containing these geodesics. Since the angles are what they look like, we know that the area of is the area of the ideal triangle (), minus the sum of the areas of the smaller triangles with two points at infinity:

Thus it suffices to show that .

For this fact, we need another picture:

Define . The picture shows that the area of the left triangle (with two vertices at infinity and one near the origin) plus the area of the right triangle is the area of the top triangle plus the area of the (ideal) bottom triangle:

We also know some boundary conditions on : we know (this is a degenerate triangle) and (this is an ideal triangle). We therefore conclude that

Similarly,

And we can find by observing that

Similarly, if we know , then

And by subtracting , we find that . By induction, then, if is a dyadic rational times . This is a dense set, so we know for all by continuity. This proves the theorem.

3. Triangles On Spheres

We can find a similar formula for triangles on spheres. A lune is a wedge of a sphere:

A lune.

Since the area of a lune is proportional to the angle at the peak, and the lune with angle has area , the lune with angle has area . Now consider the following picture:

Notice that each corner of the triangle gives us two lunes (the lunes for are shown) and that there is an identical triangle on the rear of the sphere. If we add up the area of all 6 lunes associated with the corners, we get the total area of the sphere, plus twice the area of both triangles since we have triple-counted them. In other words:

Solving,

4. Gauss-Bonnet

If we encouter a triangle of constant curvature , then we can scale the problem to one of the two formulas we just computed, so

This formula allows us to give a slightly handwavy, but accurate, proof of the Gauss-Bonnet theorem, which relates topological information (Euler characteristic) to geometric information (area and curvature). The proof will precede the statement, since this is really a discussion.

Suppose we have any closed Riemannian manifold (surface) . The surface need not have constant curvature. Suppose for the time being it has no boundary. Triangulate it with very small triangles such that and . Then since the deviation between the curvature and the curvature at the midpoint is times the distance from the midpoint,

For each triangle , we can form a comparison triangle with the same edge lengths and constant curvature . Using the formula from the beginning of this section, we can rewrite the right hand side of the formula above, so

Now since the curvature deviates by times the distance from the midpoint, the angles in deviate from those in just slightly:

So we have

Therefore, summing over all triangles,

The right hand side is just the total angle sum. Since the angle sum around each vertex in the triangulation is ,

Where is the number of vertices, and is the number of triangles. The number of edges, , can be calculated from the number of triangles, since there are edges for each triangle, and they are each double counted, so . Rewriting the equation,

Taking the mesh size to zero, we get the Gauss-Bonnet theorem .

4.1. Variants of Gauss-Bonnet

• If is compact with totally geodesic boundary, then the formula still holds, which can be shown by doubling the surface, applying the theorem to the doubled surface, and finding that euler characteristic also doubles.
• If has geodesic boundary with corners, then

  

  Where the turning angle is the angle you would turn tracing the shape from the outside. That is, it is , where is the interior angle.

• Most generally, if has smooth boundary with corners, then we can approximate the boundary with totally geodesic segments; taking the length of these segments to zero gives us geodesic curvature ():

  

4.2. Examples

• The Euler characteristic of the round disk in the plane is , and the disk has zero curvature, so . The geodesic curvature is constant, and the circumference is , so , so .
• A polygon in the plane has no curvature nor geodesic curvature, so .

The Gauss-Bonnet theorem constrains the geometry in any space with nonzero curvature. This the “reason” similarities which don’t preserve length and/or area exist in euclidean space; it has curvature zero.

Polygonal words

Last Friday, Henry Wilton gave a talk at Caltech about his recent joint work with Sang-hyun Kim on polygonal words in free groups. Their work is motivated by the following well-known question of Gromov:

Question(Gromov): Let be a one-ended word-hyperbolic group. Does contain a subgroup isomorphic to the fundamental group of a closed hyperbolic surface?

Let me briefly say what “one-ended” and “word-hyperbolic” mean.

A group is said to be word-hyperbolic if it acts properly and cocompactly by isometries on a proper -hyperbolic path metric space — i.e. a path metric space in which there is a constant so that geodesic triangles in the metric space have the property that each side of the triangle is contained in the -neighborhood of the union of the other two sides (colloquially, triangles are thin). This condition distills the essence of negative curvature in the large, and was shown by Gromov to be equivalent to several other conditions (eg. that the group satisfies a linear isoperimetric inequality; that every ultralimit of the group is an -tree). Free groups are hyperbolic; fundamental groups of closed manifolds with negative sectional curvature (eg surfaces with negative Euler characteristic) are word-hyperbolic; “random” groups are hyperbolic — and so on. In fact, it is an open question whether a group that admits a finite is word hyperbolic if and only if it does not contain a copy of a Baumslag-Solitar group for (note that the group is the special case ); in any case, this is a very good heuristic for identifying the word-hyperbolic groups one typically meets in examples.

If is a finitely generated group, the ends of really means the ends (as defined by Freudenthal) of the Cayley graph of with respect to some finite generating set. Given a proper topological space , the set of compact subsets of gives rise to an inverse system of inclusions, where includes into whenever is a subset of . This inverse system defines an inverse system of maps of discrete spaces , and the inverse limit of this system is a compact, totally disconnected space , called the space of ends of . A proper topological space is canonically compactified by its set of ends; in fact, the compactification is the “biggest” compactification of by a totally disconnected space, in the sense that for any other compactification where is zero dimensional, there is a continuous map which is the identity on .

For a word-hyperbolic group , the Cayley graph can be compactified by adding the ideal boundary , but this is typically not totally disconnected. In this case, the ends of can be recovered as the components of .

A group acts on its own ends . An elementary argument shows that the cardinality of is one of (if a compact set disconnects then infinitely many translates of converging to separate from infinitely many other ends accumulating on ). A group has no ends if and only if it is finite. Stallings famously showed that a (finitely generated) group has at least ends if and only if it admits a nontrivial description as an HNN extension or amalgamated free product over a finite group. One version of the argument proceeds more or less as follows, at least when is finitely presented. Let be an -dimensional Riemannian manifold with fundamental group , and let denote the universal cover. We can identify the ends of with the ends of . Let be a least (-dimensional) area hypersurface in amongst all hypersurfaces that separate some end from some other (here the hypothesis that has at least two ends is used). Then every translate of by an element of is either equal to or disjoint from it, or else one could use the Meeks-Yau “roundoff trick” to find a new with strictly lower area than . The translates of decompose into pieces, and one can build a tree whose vertices correspond to to components of , and whose edges correspond to the translates . The group acts on this tree, with finite edge stabilizers (by the compactness of ), exhibiting either as an HNN extension or an amalgamated product over the edge stabilizers. Note that the special case occurs if and only if has a finite index subgroup which is isomorphic to .

Free groups and virtually free groups do not contain closed surface subgroups; Gromov’s question more or less asks whether these are the only examples of word-hyperbolic groups with this property.

Kim and Wilton study Gromov’s question in a very, very concrete case, namely that case that is the double of a free group along a word ; i.e. (hereafter denoted ). Such groups are known to be one-ended if and only if is not contained in a proper free factor of (it is clear that this condition is necessary), and to be hyperbolic if and only if is not a proper power, by a result of Bestvina-Feighn. To see that this condition is necessary, observe that the double is isomorphic to the fundamental group of a Seifert fiber space, with base space a disk with two orbifold points of order ; such a group contains a . One might think that such groups are too simple to give an insight into Gromov’s question. However, these groups (or perhaps the slightly larger class of graphs of free groups with cyclic edge groups) are a critical case for at least two reasons:

1. The “smaller” a group is, the less room there is inside it for a surface group; thus the “simplest” groups should have the best chance of being a counterexample to Gromov’s question.
2. If is word-hyperbolic and one-ended, one can try to find a surface subgroup by first looking for a graph of free groups in , and then looking for a surface group in . Since a closed surface group is itself a graph of free groups, one cannot “miss” any surface groups this way.

Not too long ago, I found an interesting construction of surface groups in certain graphs of free groups with cyclic edge groups. In fact, I showed that every nontrivial element of in such a group is virtually represented by a sum of surface subgroups. Such surface subgroups are obtained by finding maps of surface groups into which minimize the Gromov norm in their (projective) homology class. I think it is useful to extend Gromov’s question by making the following

Conjecture: Let be a word-hyperbolic group, and let be nonzero. Then some multiple of is represented by a norm-minimizing surface (which is necessarily -injective).

Note that this conjecture does not generalize to wider classes of groups. There are even examples of groups with nonzero homology classes with positive, rational Gromov norm, for which there are no -injective surfaces representing a multiple of at all.

It is time to define polygonal words in free groups.

Definition: Let be free. Let be a wedge of circles whose edges are free generators for . A cyclically reduced word in these generators is polygonal if there exists a van-Kampen graph on a surface such that:

1. every complementary region is a disk whose boundary is a nontrivial (possibly negative) power of ;
2. the (labelled) graph immerses in in a label preserving way;
3. the Euler characteristic of is strictly less than the number of disks.

The last condition rules out trivial examples; for example, the double of a single disk whose boundary is labeled by . Notice that it is very important to allow both positive and negative powers of as boundaries of complementary regions. In fact, if is not in the commutator subgroup, then the sum of the powers over all complementary regions is necessarily zero (and if is in the commutator subgroup, then has nontrivial , so one already knows that there is a surface subgroup).

Condition 2. means that at each vertex of , there is at most one oriented label corresponding to each generator of or its inverse. This is really the crucial geometric property. If is a van-Kampen graph as above, then a theorem of Marshall Hall implies that there is a finite cover of into which embeds (in fact, this observation underlies Stallings’s work on foldings of graphs). If we build a -complex with by attaching two ends of a cylinder to suitable loops in two copies of , then a tubular neighborhood of in (i.e. what is sometimes called a “fatgraph” ) embeds in a finite cover of , and its double — a surface of strictly negative Euler characteristic — embeds as a closed surface in , and is therefore -injective. Hence if is polygonal, contains a surface subgroup.

Not every word is polygonal. Kim-Wilton discuss some interesting examples in their paper, including:

1. suppose is a cyclically reduced product of proper powers of the generators or their inverses (e.g a word like but not a word like ); then is polygonal;
2. a word of the form is polygonal if for each ;
3. the word is not polygonal.

To see 3, suppose there were a van-Kampen diagram with more disks than Euler characteristic. Then there must be some vertex of valence at least . Since is positive, the complementary regions must have boundaries which alternate between positive and negative powers of , so the degree of the vertex must be even. On the other hand, since must immerse in a wedge of two circles, the degree of every vertex must be at most , so there is consequently some vertex of degree exactly . Since each is isolated, at least edges must be labelled ; hence exactly two. Hence exactly two edges are labelled . But one of these must be incoming and one outgoing, and therefore these are adjacent, contrary to the fact that does not contain a .

1 above is quite striking to me. When is in the commutator subgroup, one can consider van-Kampen diagrams as above without the injectivity property, but with the property that every power of on the boundary of a disk is positive; call such a van-Kampen diagram monotone. It turns out that monotone van-Kampen diagrams always exist when , and in fact that norm-minimizing surfaces representing powers of the generator of are associated to certain monotone diagrams. The construction of such surfaces is an important step in the argument that stable commutator length (a kind of relative Gromov norm) is rational in free groups. In my paper scl, sails and surgery I showed that monomorphisms of free groups that send every generator to a power of that generator induce isometries of the norm; in other words, there is a natural correspondence between certain equivalence classes of monotone surfaces for an arbitrary word in and for a word of the kind that Kim-Wilton show is polygonal (Note: Henry Wilton tells me that Brady, Forester and Martinez-Pedroza have independently shown that contains a surface group for such , but I have not seen their preprint (though I would be very grateful to get a copy!)).

In any case, if not every word is polygonal, all is not lost. To show that contains a surface subgroup is suffices to show that contains a surface subgroup, where and differ by an automorphism of . Kim-Wilton conjecture that one can always find an automorphism so that is polygonal. In fact, they make the following:

Conjecture (Kim-Wilton; tiling conjecture): A word not contained in a proper free factor of shortest length (in a given generating set) in its orbit under is polygonal.

If true, this would give a positive answer to Gromov’s question for groups of the form .

Minimal laminations with leaves of different conformal types

The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it comes from.

The example comes from the idea of a Riemann surface lamination. This is an object that geometrizes some ideas in 1-dimensional complex analysis. The basic idea is simple: given a noncompact infinite Riemannian -manifold , one gives it a new topology by declaring that two points on the surface are “close” in the new topology if there are balls of big radius in the surface centered at the two points which are “almost isometric”. Points that were close in the old topology are close in the new topology, but points that might have been far away in the old topology can become close in the new. For example, if is a covering space of some other Riemannian surface , then points in the orbit of the deck group are “infinitely close” in the new topology. This means that the resulting topological space is not Hausdorff; one “Hausdorffifies” by identifying pairs of points that are not contained in disjoint open sets, and the quotient recovers the surface (assuming that the metric on is sufficiently generic; otherwise, it recovers modulo its group of isometries). Morally what one is doing is mapping into the space of pointed locally compact metric spaces (which is itself a locally compact topological space), and giving it the subspace topology. In more detail, a point in is a pair where is a locally compact metric space, and is a point. A sequence converges to if there are metric balls around of diameter going to infinity, metric balls around also of diameter going to infinity, and isometric inclusions of into metric spaces in such a way that the Hausdorff distance between the images of and in goes to zero as . Any locally compact metric space has a tautological map to , where each point is sent to the point . Gromov showed (see section 6 of this paper) that the space itself is locally compact; in fact, this follows in an obvious way from the Arzela-Ascoli theorem.

If has bounded geometry — i.e. if the injectivity radius is uniformly bounded below, and the curvature is bounded above and below — then the image of in is precompact, and its closure is a compact metric space . The path components of are exactly the Riemann surfaces which are arbitrarily well approximated (in the metric sense) on every compact subset by compact subsets of . If you were wandering around on such a component , and you wandered over a compact region, and were only able to measure the geometry up to some (arbitrarily fine) definite precision, you could never rule out the possibility that you were actually wandering around on . Topologically, is a Riemann surface lamination; i.e. a locally compact topological space covered by open charts of the form where is an open two-dimensional disk, where is totally disconnected, and where the transition between charts preserves the decomposition into pieces , and is smooth (in fact, preserves the Riemann surface structure) on the slices, in the overlaps. The unions of “surface” slices — i.e. the path components of — piece together to make the leaves of the lamination, which are (complete) Riemann surfaces. In our case, the leaves have Riemannian metrics, which vary continuously in the direction transverse to the leaves. (Surface) laminations occur in other areas of mathematics, for example as inverse limits of sequences of finite covers of a fixed compact surface, or as objects obtained by inductively splitting open sheets in a branched surface (the latter can easily occur as attractors of certain kinds of partially hyperbolic dynamical systems). One well-known example is sometimes called the (punctured) solenoid; its Teichmüller theory is studied by Penner and Šarić  (question: does anyone know how to do a “\acute c” in wordpress? update 11/6: thanks Ian for the unicode hint).

A lamination is said to be minimal if every leaf is dense. In our context this means that for every compact region in and every there is a so that every ball in of radius contains a subset which is -close to in the Gromov-Hausdorff metric. In other words, every “local feature” of that appears somewhere, appears with definite density to within any desired degree of accuracy. Consequently, such features will “almost” appear, with the same definite density, in every other leaf of , and therefore is in the closure of each . Since is (in) the closure of , this implies that every leaf is dense, as claimed.

In a Riemann surface lamination, the conformal type of every leaf is well-defined. If some leaf is elliptic, then necessarily that leaf is a sphere. So if the lamination is minimal, it is equal to a single closed surface. If every leaf is hyperbolic, then each leaf admits a unique hyperbolic metric in its conformal class (i.e. each leaf can be uniformized), and Candel showed that this family of hyperbolic metrics varies continuously in . Étienne Ghys asked whether there is an example of a minimal Riemann surface lamination in which some leaves are conformally parabolic, and others are conformally hyperbolic. It turns out that the answer to this question is yes; Richard Kenyon found an example, which I will now describe.

The lamination in question has exactly one hyperbolic leaf, which is topologically a -times punctured sphere. Every other leaf is an infinite cylinder — i.e. it is conformally the punctured plane . Since the lamination is minimal, to describe the lamination, one just needs to describe one leaf. This leaf will be obtained as the boundary of a thickened neighborhood of an infinite planar graph, which is defined inductively, as follows.

Let be the planar “Greek cross” as in the following figure:

Inductively, if we have defined , define by attaching four copies of to the extremities of . The first few examples are illustrated in the following figure:

The limit is a planar tree with exactly four ends; the boundary of a thickened tubular neighborhood is conformally equivalent to a sphere with four points removed, which is hyperbolic. Every unbounded sequence of points in has a subsequence which escapes out one of the ends. Hence every other leaf in the lamination this defines has exactly two ends, and is conformally equivalent to a punctured plane, which is parabolic.

The header image is a very similar construction in -dimensional space, where the initial seed has six legs along the coordinate axes instead of four; some (quite large) approximation was then rendered in povray.

When I was in graduate school, I was very interested in the (complex) geometry of Riemann surface laminations, and wanted to understand their deformation theory, perhaps with the aim of using structures like taut foliations and essential laminations to hyperbolize -manifolds, as an intermediate step in an approach to the geometrization conjecture (now a theorem of Perelman). I know that at one point Sullivan was quite interested in such objects, as a tool in the study of Julia sets of rational functions. I have the impression that they are not studied so much these days, but I would be happy to be corrected.

Bridgeman’s orthospectrum identity

Martin Bridgeman gave a nice talk at Caltech recently on his discovery of a beautiful identity concerning orthospectra of hyperbolic surfaces (and manifolds of higher dimension) with totally geodesic boundary. The -dimensional case is (in my opinion) the most beautiful, and I would like to take a post to explain the identity, and give a derivation which is slightly different from the one Martin gives in his paper. There are many other things one could say about this identity, and its relation to other identities that turn up in the theory of hyperbolic manifolds (and elsewhere); I hope to get to this in a later post.

Let be a hyperbolic surface with totally geodesic boundary. An orthogeodesic is a geodesic segment properly immersed in , which is perpendicular to at its endpoints. The set of orthogeodesics is countable, and their lengths are proper. Denote these lengths by (with multiplicity). The identity is:

where is the Rogers’ dilogarithm function (to be defined in a minute). Treating this function as a black box for the moment, the identity has the form a term depending only on the topology of . The proof is very, very short and elegant. By the Gauss-Bonnet theorem, the term on the right is equal to of the volume of the unit tangent bundle of . Almost every tangent vector on can be exponentiated to a geodesic on which intersects the boundary in finite forward and backward time (eg. by ergodicity of the geodesic flow on a closed hyperbolic surface obtained by doubling). If is such a tangent vector, and is the associated geodesic arc, then is homotopic keeping endpoints on to a unique orthogeodesic (which is the unique length minimizer in this relative homotopy class). The volume of the set of associated to a given orthogeodesic can be computed as follows. Lift to the universal cover, where it is the crossbar of a letter “H” whose vertical lines are lifts of the geodesics it ends on. Any lifts to a unique geodesic segment in the universal cover with endpoints on the edges of the H. So the volume of the set of such depends only on , giving rise to the explicit formula for . qed.

That’s it — that’s the whole proof! . . . modulo some calculations, which we now discuss.

The “ordinary” polylogarithms are defined by Taylor series

which converges for , and extends by analytic continuation. Taking derivatives, one sees that they satisfy , thereby giving rising to integral formulae. is the familiar geometric series , so and

The Rogers dilogarithm is then given by the formula for real . One sees that the Rogers dilogarithm is obtained by symmetrizing the integrand for the integral expression for under the involution :

Martin derives his identity by direct calculation, but in fact this calculation can be simplified a bit by some hyperbolic geometry. Consider an ideal quadrilateral (whose unit tangent bundle has area ) with one pair of opposite sides that are distance apart. Join opposite vertices in pairs to decompose the quadrilateral into four triangles, each with one non-ideal point:

In the (schematic) picture, suppose the two edges of the H are the left and right side (call them and ) and the other two edges are and . Similarly, call the four triangles depending on which edge of the quadrilateral they bound. The triangle is colored gray in the figure. We secretly identify this figure with the upper half-plane, in such a way that the ideal vertices are (in circular order) , where are the ideal vertices of the gray triangle. Call the (hyperbolic) angle of the gray triangle at its vertex, so . Moreover, it turns out that where is the distance between and . We will compute implicitly as a function of , and show that it is a multiple of the Rogers dilogarithm function, thus verifying Bridgeman’s identity.

Every vector in exponentiates to a (bi-infinite) geodesic , and we want to compute the volume of the set of vectors for which the corresponding geodesic intersects both and . The point of the decomposition is that for in (say), the geodesic intersects whenever it intersects , so we only need to compute the volume of the in for which intersects . Similarly, we only need to compute the volume of the in for which intersects . For in , we compute the volume of the which do not intersect (since these are exactly the ones that intersect both and ), and similarly for .

These volumes can be expressed in terms of integrals of harmonic functions. Let denote the harmonic function on the disk which is on the arc of the circle bounded by , and on the rest of the circle. This function at each point is equal to times the visual angle (i.e. the length in the unit tangent circle) subtended by the given arc of the circle, as seen from the given point in the hyperbolic plane. Define similarly. Then the total volume we need to compute is equal to

(here we have identified by symmetry, and similarly for the other pair of terms). Let us approach this a bit more systematically. If denotes the angle at the nonideal vertex of triangle , we denote , and . The integral we want to evaluate can be expressed easily in terms of explicit rational multiples of , and the function . These functions satisfy obvious identities:

and

where the last identity comes by observing that we are integrating a certain function over an ideal triangle, and observing that the average of this function under the symmetries of the ideal triangle is equal to the constant function . In particular, we see that we can express everything in terms of . After some elementary reorganization, we see that the contribution to the volume of the unit tangent bundle of the surface associated to this particular orthogeodesic is

To compute , it makes sense to move to the upper half-space model, and move the endpoints of the interval to and . The harmonic function is equal to on the negative real axis, and on the positive real axis. It takes the value on the line . The area form in the hyperbolic metric is proportional to the Euclidean area form, with constant . In other words, we want to integrate over the region indicated in the figure, where the nonideal angle is , and the base point is :

If we normalize so that the circular arc is part of the semicircle from to , then the real projection of the vertical lines in the figure are and . There is no elementary way to evaluate this integral, so instead we evaluate its derivative as a function of where as before, . This is the definite integral

Integrating by parts gives . This evaluates to

Thinking of as a function of , we get

Comparing values at we see that and the identity is proved.

Well, OK, this is not terribly simple, but a posteriori it gives a way to express the Rogers dilogarithm as a sum of integrals of very simple harmonic functions over hyperbolic triangles, which is a nice geometric way to think of it.

(Update 10/30): This paper by Dupont and Sah relates Rogers dilogarithm to volumes of -simplices, and discusses some interesting connections to conformal field theory and lattice model calculations. I feel like a bit of a dope, since I read this paper while I was in graduate school more than a dozen years ago, but forgot all about it until I was cleaning out my filing cabinet this morning. They cite an older paper of Dupont for the explicit calculations; these are somewhat tedious and unenlightening; however, he does manage to show that the Rogers dilogarithm is characterized by the Abel identity. In other words,

Lemma A.1 (Dupont): Let be a three times differentiable function satisfying

for all . Then there is a real constant such that where is the Rogers dilogarithm (up to an additive constant).

Nevertheless, they don’t seem to have noticed the formula in terms of integrals of harmonic functions over hyperbolic triangles. Perhaps this is also well-known. Do any readers know?

Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface

Hermann Amandus Schwarz (1843-1921) was a student of Kummer and Weierstrass, and made many significant contributions to geometry, especially to the fields of minimal surfaces and complex analysis. His mathematical creations are both highly abstract and flexible, and at the same time intimately tied to explicit and practical calculation.

I learned about Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface as three quite separate mathematical objects, and I was very surprised to discover firstly that they had all been discovered by the same person, and secondly that they form parts of a consistent mathematical narrative, which I will try to explain in this post to the best of my ability. There is an instructive lesson in this example (for me), that we tend to mine the past for nuggets, examples, tricks, formulae etc. while forgetting the points of view and organizing principles that made their discovery possible. Another teachable example is that of Dehn’s “invention” of combinatorial (infinite) group theory, as a natural branch of geometry; several generations of followers went about the task of reformulating Dehn’s insights and ideas in the language of algebra, “generalizing” them and stripping them of their context, before geometric and topological methods were reintroduced by Milnor, Schwarz (a different one this time), Stallings, Thurston, Gromov and others to spectacular effect (note: I have the second-hand impression that the geometric point of view in group theory (and every other subject) was never abandoned in the Soviet Union).

Schwarz’s minimal surface (also called “Schwarz’s D surface”, and sometimes “Schwarz’s H surface”) is an extraordinarily beautiful triply-periodic minimal surface of infinite genus that is properly embedded in . According to Nitsche’s excellent book (p.240), this minimal surface closely resembles the separating wall between inorganic and organic materials in the skeleton of a starfish. The basic building block of the surface can be described as follows. If the vertices of a cube are -colored, the black vertices are the vertices of a regular tetrahedron. Let denote the quadrilateral formed by four edges of this tetrahedron; then a fundamental piece of Schwarz’s surface is a minimal disk spanning :

The surface may be “analytically continued” by rotating through an angle around each boundary edge. Six copies of fit smoothly around each vertex, and the resulting surface extends (triply) periodically throughout space.

The symmetries of enable us to give it several descriptions as a Riemann surface. Firstly, we could think of as a polygon in the hyperbolic plane with four edges of equal length, and angles . Twelve copies of can be assembled to make a hyperbolic surface of genus . Thinking of a surface of genus as the boundary of a genus handlebody defines a homomorphism from to , thought of as ; the cover associated to the kernel is (conformally) the triply periodic Schwarz surface, and the deck group acts on as a lattice (of index in the face-centered cubic lattice).

Another description is as follows. Since the deck group acts by translation, the Gauss map from to factors through a map . The map is injective at each point in the interior or on an edge of a copy of , but has an order branch point at each vertex. Thus, the map is a double-branched cover, with one branch point of order at each vertex of a regular inscribed cube. This leads one to think (like a late 19th century mathematician) of as the Riemann surface on which a certain multi-valued function on is single-valued. Under stereographic projection, the vertices of the cube map to the eight points where . These eight points are the roots of the polynomial , so we may think of as the hyperelliptic Riemann surface defined by the equation ; equivalently, as the surface on which the multi-valued (on ) function is single-valued.

The function is known as the Weierstrass function associated to , and an explicit formula for the co-ordinates of the embedding were found by Enneper and Weierstrass. After picking a basepoint (say ) on the sphere, the coordinates are given by integration:

The integral in each case depends on the path, and lifts to a single-valued function precisely on .

Geometrically, the three coordinate functions are harmonic functions on . This corresponds to the fact that minimal surfaces are precisely those with vanishing mean curvature, and the fact that the Laplacian of the coordinate functions (in terms of isothermal parameters on the underlying Riemann surface) can be expressed as a nonzero multiple of the mean curvature vector. A harmonic function on a Riemann surface is the real part of a holomorphic function, unique up to a constant; the holomorphic derivative of the (complexified) coordinate functions are therefore well-defined, and give holomorphic -forms which descend to (since the deck group acts by translations). These -forms satisfy the identity (this identity expresses the fact that the embedding of into via these functions is conformal). The (composition of the) Gauss map (with stereographic projection) can be read off from the , and as a meromorphic function on , it is given by the formula . Define a function on by the formula . Then are the coordinates of a rational map from into which extends to a map into , by sending each zero of to in the at infinity. Symmetry allows us to identify the image with the hyperelliptic embedding from before, and we deduce that . Solving for we obtain the integrands in the formulae above.

In fact, any holomorphic function on a domain in defines a (typically immersed with branch points) minimal surface, by the integral formulae of Enneper-Weierstrass above. Suppose we want to use this fact to produce an explicit description of a minimal surface bounded by some explicit polygonal loop in . Any minimal surface so obtained can be continued across the boundary edges by rotation; if the angles at the vertices are all of the form the resulting surface closes up smoothly around the vertices, and one obtains a compact abstract Riemann surface tiled by copies of the fundamental region, together with a holonomy representation of into . Sometimes the image of this representation in the rotational part of is finite, and one obtains an infinitely periodic minimal surface as in the case of Schwarz’s surface. A fundamental tile in can be uniformized as a hyperbolic polygon; equivalently, as a region in the upper half-plane bounded by arcs of semicircles perpendicular to the real axis. Since the edges of the loop are straight lines, the image of this hyperbolic polygon under the Gauss map is a region in also bounded by arcs of round circles; thus Schwarz’s study of minimal surfaces naturally led him to the problem of how to explicitly describe conformal maps between regions in the plane bounded by circular arcs. This problem is solved by the Schwarz-Christoffel transformation, and its generalizations, with help from the Schwarzian derivative.

Note that if and are two such regions, then a conformal map from to can be factored as the product of a map uniformizing as the upper half-plane, followed by the inverse of a map uniformizing as the upper half-plane. So it suffices to find a conformal map when the domain is the upper half plane, decomposed into intervals and rays that are mapped to the edges of a circular polygon . Near each vertex, can be moved by a fractional linear transformation to (part of) a wedge, consisting of complex numbers with argument between and , where is the angle at . The function uniformizes the upper half-plane as such a wedge; however it is not clear how to combine the contributions from each vertex, because of the complicated interaction with the fractional linear transformation. The fundamental observation is that there are certain natural holomorphic differential operators which are insensitive to the composition of a holomorphic function with groups of fractional linear transformations, and the uniformizing map can be expressed much more simply in terms of such operators.

For example, two functions that differ by addition of a constant have the same derivative: . Functions that differ by multiplication by a constant have the same logarithmic derivative: . Putting these two observations together suggest defining the nonlinearity of a function as the composition . This has the property that for any constants . Under inversion the nonlinearity transforms by . From this, and a simple calculation, one deduces that the operator is invariant under inversion, and since it is also invariant under addition and multiplication by constants, it is invariant under the full group of fractional linear transformations. This combination is called the Schwarzian derivative; explicitly, it is given by the formula . Given the Schwarzian derivative , one may recover the nonlinearity by solving the Ricatti equation . As explained in this post, solutions of the Ricatti equation preserve the projective structure on the line; in this case, it is a complex projective structure on the complex line. Equivalently, different solutions differ by an element of , acting by fractional linear transformations, as we have just deduced. Once we know the nonlinearity, we can solve for by , the usual solution to a first order linear inhomogeneous ODE. The Schwarzian of the function is . The advantage of expressing things in these terms is that the Schwarzian of a uniformizing map for a circular polygon with angles at the vertices has the form of a rational function, with principal parts , where the and the and depend (unfortunately in a very complicated way) on the edges of (for the ugly truth, see Nehari, chapter 5). To see this, observe that the map has an order two pole near finitely many points (the preimages of the vertices of under the uniformizing map) but is otherwise holomorphic. Moreover, it can be analytically continued into the lower half plane across the interval between successive , by reflecting the image across each circular edge. After reflecting twice, the image of is transformed by a fractional linear transformation, so has an analytic continuation which is single valued on the entire Riemann sphere, with finitely many isolated poles, and is therefore a rational function! When the edges of the polygon are straight, a simpler formula involving the nonlinearity specializes to the “familiar” Schwarz-Christoffel formula.

(Update 10/22): In fact, I went to the library to refresh myself on the contents of Nehari, chapter 5. The first thing I noticed — which I had forgotten — was that if is the uniformizing map from the upper half-plane to a polygon with spherical arcs, then is real-valued on the real axis. Since it is a rational function, this implies that its nonsingular part is actually a constant; i.e.

where is as above, and are real constants (which satisfy some further conditions — really see Nehari this time for more details).

The other thing that struck me was the first paragraph of the preface, which touches on some of the issues I alluded to above:

In the preface to the first edition of Courant-Hilbert’s “Methoden der mathematischen Physik”, R. Courant warned against a trend discernible in modern mathematics in which he saw a menace to the future development of mathematical analysis. He was referring to the tendency of many workers in this field to lose sight of the roots of mathematical analysis in physical and geometric intuition and to concentrate their efforts on the refinement and the extreme generalization of existing concepts.

Instead of using a word like “menace”, I would rather take this as a lesson about the value of returning to the points of view that led to the creation of the mathematical objects we study every day; which was (to some approximation) the point I was trying to illustrate in this post.

Harmonic measure

An amenable group acting by homeomorphisms on a compact topological space preserves a probability measure on ; in fact, one can given a definition of amenability in such terms. For example, if is finite, it preserves an atomic measure supported on any orbit. If , one can take a sequence of almost invariant probability measures, supported on the subset (where is arbitrary), and any weak limit will be invariant. For a general amenable group, in place of the subsets , one works with a sequence of Folner sets; i.e. subsets with the property that the ratio of their size to the size of their boundary goes to zero (so to speak).

But if is not amenable, it is generally not true that there is any probability measure on invariant under the action of . The best one can expect is a probability measure which is invariant on average. Such a measure is called a harmonic measure (or a stationary measure) for the -action on . To be concrete, suppose is finitely generated by a symmetric generating set (symmetric here means that if , then ). Let denote the space of probability measures on . One can form an operator defined by the formula

and then look for a probability measure stationary under , which exists for quite general reasons. This measure is the harmonic measure: the expectation of the -measure of under a randomly chosen is equal to the -measure of . Note for any probability measure that is absolutely continuous with respect to ; in fact, the Radon-Nikodym derivative satisfies . Substituting for in this formula, one sees that the measure class of is preserved by , and that for every , we have , where denotes word length with respect to the given generating set.

The existence of harmonic measure is especially useful when is one-dimensional, e.g. in the case that . In one dimension, a measure (at least one of full support without atoms) can be “integrated” to a path metric. Consequently, any finitely generated group of homeomorphisms of the circle is conjugate to a group of bilipschitz homeomorphisms (if the harmonic measure associated to the original action does not have full support, or has atoms, one can “throw in” another random generator to the group; the resulting action can be assumed to have a harmonic measure of full support without atoms, which can be integrated to give a structure with respect to which the group action is bilipschitz). In fact, Deroin-Kleptsyn-Navas showed that any countable group of homeomorphisms of the circle (or interval) is conjugate to a group of bilipschitz homeomorphisms (the hypothesis that be countable is essential; for example, the group acts in a non-bilipschitz way on the interval — see here).

Suppose now that for some manifold . The action of on determines a foliated circle bundle ; i.e. a circle bundle, together with a codimension one foliation transverse to the circle fibers. To see this, first form the product with its product foliation by leaves , where denotes the universal cover of . The group acts on as the deck group of the covering, and on by the given action; the quotient of this diagonal action on the product is the desired circle bundle . The foliation makes into a “flat” circle bundle with structure group . The foliation allows us to associate to each path in a homeomorphism from the fiber over to the fiber over ; integrability (or flatness) implies that this homeomorphism only depends on the relative homotopy class of in . This identification of fibers is called the holonomy of the foliation along the path . If is a Riemannian manifold, there is another kind of harmonic measure on the circle bundle; in other words, a probability measure on each circle with the property that the holonomy associated to an infinitesimal random walk on preserves the expected value of the measure. This is (very closely related to) a special case of a construction due to Lucy Garnett which associates a harmonic transverse measure to any foliation of a manifold , by finding a fixed point of the leafwise heat flow on the space of probability measures on , and disintegrating this measure into the product of the leafwise area measure, and a “harmonic” transverse measure.

In any case, we normalize our foliated circle bundle so that each circle has length in its harmonic measure. Let be the vector field on the circle bundle that rotates each circle at unit speed, and let be the -form on whose kernel is tangent to the leaves of the foliation. We scale so that everywhere. The integrability condition for a foliation is expressed in terms of the -form as the identity , and we can write where . More intrinsically, descends to a -form on the leaves of the foliation which measures the logarithm of the rate at which the transverse measure expands under holonomy in a given direction (the leafwise form is sometimes called the Godbillon class, since it is “half” of the Godbillon-Vey class associated to a codimension one foliation; see e.g. Candel-Conlon volume 2, Chapter 7). Identifying the universal cover of each leaf with by projection, the fact that our measure is harmonic means that “is” the gradient of the logarithm of a positive harmonic function on . As observed by Thurston, the geometry of then puts constraints on the size of . The following discussion is taken largely from Thurston’s paper “Three-manifolds, foliations and circles II” (unfortunately this mostly unwritten paper is not publicly available; some details can be found in my foliations book, example 4.6).

An orthogonal connection on can be obtained by averaging under the flow of ; i.e. if is the diffeomorphism of which rotates each circle through angle , then

is an -invariant -form on , which therefore descends to a -form on , which can be thought of as a connection form for an -structure on the bundle . The curvature of the connection (in the usual sense) is the -form , and we have a formula

The action of the -parameter group trivializes the cotangent bundle to over each fiber. After choosing such a trivialization, we can think of the values of at each point on a fiber as sweeping out a circle in a fixed vector space . The tangent to this circle is found by taking the Lie derivative

In other words, is identified with under the identification of with , and ; i.e. the absolute value of the curvature of the connection is equal to times the area enclosed by .

Now suppose is a hyperbolic -manifold, i.e. a manifold of dimension with constant curvature everywhere. Equivalently, think of as a quotient of hyperbolic space by a discrete group of isometries. A positive harmonic function on has a logarithmic derivative which is bounded pointwise by ; identifying positive harmonic functions on hyperbolic space with distributions on the sphere at infinity, one sees that the  “worst case” is the harmonic extension of an atomic measure concentrated at a single point at infinity, since every other positive harmonic function is the weighted average of such examples. As one moves towards or away from a blob at infinity concentrated near this point, the radius of the blob expands like ; since the sphere at infinity has dimension , the conclusion follows. But this means that the speed of (i.e. the size of ) is pointwise bounded by , and the length of the circle is at most . A circle of length can enclose a disk of area at most , so the curvature of the connection has absolute value pointwise bounded by .

One corollary is a new proof of the Milnor-Wood inequality, which says that a foliated circle bundle over a closed oriented surface of genus at least satisfies , where is the Euler number of the bundle (a topological invariant). For, the surface can be given a hyperbolic metric, and the bundle a harmonic connection whose average is an orthogonal connection with pointwise curvature of absolute value at most . The Euler class of the bundle evaluated on the fundamental class of is the Euler number ; we have

where the first equality is the Chern-Weil formula for the Euler class of a bundle in terms of the curvature of a connection, and the last equality is the Gauss-Bonnet theorem for a hyperbolic surface. Another corollary gives lower bounds on the area of an incompressible surface in a hyperbolic manifold. Suppose is an immersion which is injective on . There is a cover of for which the immersion lifts to a homotopy equivalence, and we get an action of on the circle at infinity of , and hence a foliated circle bundle as above with . Integrating as above over the image of in , and using the fact that the curvature of is pointwise bounded by , we deduce that the area of is at least . If is a -manifold, we obtain .

(A somewhat more subtle argument allows one to get better bounds, e.g. replacing by for , and better estimates for higher .)