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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 $G$ be a one-ended word-hyperbolic group. Does $G$ 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 $\delta$-hyperbolic path metric space — i.e. a path metric space in which there is a constant $\delta$ so that geodesic triangles in the metric space have the property that each side of the triangle is contained in the $\delta$-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 $\mathbb{R}$-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 $G$ that admits a finite $K(G,1)$ is word hyperbolic if and only if it does not contain a copy of a Baumslag-Solitar group $BS(m,n):=\langle x,y \; | \; x^{-1}y^{m}x = y^n \rangle$ for $m,n \ne 0$ (note that the group $\mathbb{Z}\oplus \mathbb{Z}$ is the special case $m=n=1$); in any case, this is a very good heuristic for identifying the word-hyperbolic groups one typically meets in examples.

If $G$ is a finitely generated group, the ends of $G$ really means the ends (as defined by Freudenthal) of the Cayley graph of $G$ with respect to some finite generating set. Given a proper topological space $X$, the set of compact subsets of $X$ gives rise to an inverse system of inclusions, where $X-K'$ includes into $X-K$ whenever $K$ is a subset of $K'$. This inverse system defines an inverse system of maps of discrete spaces $\pi_0(X-K') \to \pi_0(X-K)$, and the inverse limit of this system is a compact, totally disconnected space $\mathcal{E}(X)$, called the space of ends of $X$. A proper topological space is canonically compactified by its set of ends; in fact, the compactification $X \cup \mathcal{E}(X)$ is the “biggest” compactification of $X$ by a totally disconnected space, in the sense that for any other compactification $X \subset Y$ where $Y-X$ is zero dimensional, there is a continuous map $X \cup \mathcal{E}(X) \to Y$ which is the identity on $X$.

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

A group $G$ acts on its own ends $\mathcal{E}(G)$. An elementary argument shows that the cardinality of $\mathcal{E}(G)$ is one of $0,1,2,\infty$ (if a compact set $V$ disconnects $e_1,e_2,e_3$ then infinitely many translates of $V$ converging to $e_1$ separate $e_3$ from infinitely many other ends accumulating on $e_1$). A group has no ends if and only if it is finite. Stallings famously showed that a (finitely generated) group has at least $2$ 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 $G$ is finitely presented. Let $M$ be an $n$-dimensional Riemannian manifold with fundamental group $G$, and let $\tilde{M}$ denote the universal cover. We can identify the ends of $G$ with the ends of $\tilde{M}$. Let $V$ be a least ($n-1$-dimensional) area hypersurface in $\tilde{M}$ amongst all hypersurfaces that separate some end from some other (here the hypothesis that $G$ has at least two ends is used). Then every translate of $V$ by an element of $G$ is either equal to $V$ or disjoint from it, or else one could use the Meeks-Yau “roundoff trick” to find a new $V'$ with strictly lower area than $V$. The translates of $V$ decompose $\tilde{M}$ into pieces, and one can build a tree $T$ whose vertices correspond to to components of $\tilde{M} - G\cdot V$, and whose edges correspond to the translates $G\cdot V$. The group $G$ acts on this tree, with finite edge stabilizers (by the compactness of $V$), exhibiting $G$ either as an HNN extension or an amalgamated product over the edge stabilizers. Note that the special case $|\mathcal{E}(G)|=2$ occurs if and only if $G$ has a finite index subgroup which is isomorphic to $\mathbb{Z}$.

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 $G$ is the double of a free group $F$ along a word $w$; i.e. $G = F *_{\langle w \rangle } F$ (hereafter denoted $D(w)$). Such groups are known to be one-ended if and only if $w$ is not contained in a proper free factor of $F$ (it is clear that this condition is necessary), and to be hyperbolic if and only if $w$ is not a proper power, by a result of Bestvina-Feighn. To see that this condition is necessary, observe that the double $\mathbb{Z} *_{p\mathbb{Z}} \mathbb{Z}$ is isomorphic to the fundamental group of a Seifert fiber space, with base space a disk with two orbifold points of order $p$; such a group contains a $\mathbb{Z}\oplus \mathbb{Z}$. 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 $G$ is word-hyperbolic and one-ended, one can try to find a surface subgroup by first looking for a graph of free groups $H$ in $G$, and then looking for a surface group in $H$. 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 $H_2(G;\mathbb{Q})$ 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 $G$ 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 $G$ be a word-hyperbolic group, and let $\alpha \in H_2(G;\mathbb{Q})$ be nonzero. Then some multiple of $\alpha$ is represented by a norm-minimizing surface (which is necessarily $\pi_1$-injective).

Note that this conjecture does not generalize to wider classes of groups. There are even examples of $\text{CAT}(0)$ groups $G$ with nonzero homology classes $\alpha \in H_2(G;\mathbb{Q})$ with positive, rational Gromov norm, for which there are no $\pi_1$-injective surfaces representing a multiple of $\alpha$ at all.

It is time to define polygonal words in free groups.

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

1. every complementary region is a disk whose boundary is a nontrivial (possibly negative) power of $w$;
2. the (labelled) graph $\Gamma$ immerses in $X$ in a label preserving way;
3. the Euler characteristic of $S$ 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 $w^n$. Notice that it is very important to allow both positive and negative powers of $w$ as boundaries of complementary regions. In fact, if $w$ is not in the commutator subgroup, then the sum of the powers over all complementary regions is necessarily zero (and if $w$ is in the commutator subgroup, then $D(w)$ has nontrivial $H_2$, so one already knows that there is a surface subgroup).

Condition 2. means that at each vertex of $\Gamma$, there is at most one oriented label corresponding to each generator of $F$ or its inverse. This is really the crucial geometric property. If $\Gamma,S$ is a van-Kampen graph as above, then a theorem of Marshall Hall implies that there is a finite cover of $X$ into which $\Gamma$ embeds (in fact, this observation underlies Stallings’s work on foldings of graphs). If we build a $2$-complex $Y$ with $\pi_1(Y)=D(w)$ by attaching two ends of a cylinder to suitable loops in two copies of $X$, then a tubular neighborhood of $\Gamma$ in $S$ (i.e. what is sometimes called a “fatgraph” ) embeds in a finite cover $\tilde{Y}$ of $Y$, and its double — a surface of strictly negative Euler characteristic — embeds as a closed surface in $\tilde{Y}$, and is therefore $\pi_1$-injective. Hence if $w$ is polygonal, $D(w)$ contains a surface subgroup.

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

1. suppose $w$ is a cyclically reduced product of proper powers of the generators or their inverses (e.g a word like $a^3b^7a^{-2}c^{13}$ but not a word like $a^3bc^{-1}$); then $w$ is polygonal;
2. a word of the form $\prod_i a^{p_{2i-1}}(a^{p_{2i}})^b$ is polygonal if $|p_i|>1$ for each $i$;
3. the word $abab^2ab^3$ 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 $3$. Since $w$ is positive, the complementary regions must have boundaries which alternate between positive and negative powers of $w$, so the degree of the vertex must be even. On the other hand, since $\Gamma$ must immerse in a wedge of two circles, the degree of every vertex must be at most $4$, so there is consequently some vertex of degree exactly $4$. Since each $a$ is isolated, at least $2$ edges must be labelled $b$; hence exactly two. Hence exactly two edges are labelled $a$. But one of these must be incoming and one outgoing, and therefore these are adjacent, contrary to the fact that $w$ does not contain a $a^{\pm 2}$.

1 above is quite striking to me. When $w$ 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 $w$ 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 $w \in [F,F]$, and in fact that norm-minimizing surfaces representing powers of the generator of $H_2(D(w))$ 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 $\text{scl}$ norm; in other words, there is a natural correspondence between certain equivalence classes of monotone surfaces for an arbitrary word in $[F,F]$ 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 $D(w)$ contains a surface group for such $w$, 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 $D(w)$ contains a surface subgroup is suffices to show that $D(w')$ contains a surface subgroup, where $w$ and $w'$ differ by an automorphism of $F$. Kim-Wilton conjecture that one can always find an automorphism $\phi$ so that $\phi(w)$ is polygonal. In fact, they make the following:

Conjecture (Kim-Wilton; tiling conjecture): A word $w$ not contained in a proper free factor of shortest length (in a given generating set) in its orbit under $\text{Aut}(F)$ is polygonal.

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

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 $2$-manifold $\Sigma$, 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 $\Sigma$ is a covering space of some other Riemannian surface $S$, 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 $S$ (assuming that the metric on $S$ is sufficiently generic; otherwise, it recovers $S$ modulo its group of isometries). Morally what one is doing is mapping $\Sigma$ into the space $\mathcal{M}$ 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 $\mathcal{M}$ is a pair $(X,p)$ where $X$ is a locally compact metric space, and $p \in X$ is a point. A sequence $X_i,p_i$ converges to $X,p$ if there are metric balls $B_i$ around $p_i$ of diameter going to infinity, metric balls $D_i$ around $p$ also of diameter going to infinity, and isometric inclusions of $B_i,D_i$ into metric spaces $Z_i$ in such a way that the Hausdorff distance between the images of $B_i$ and $D_i$ in $Z_i$ goes to zero as $i \to \infty$. Any locally compact metric space $Y$ has a tautological map to $\mathcal{M}$, where each point $y \in Y$ is sent to the point $(Y,y) \in \mathcal{M}$. Gromov showed (see section 6 of this paper) that the space $\mathcal{M}$ itself is locally compact; in fact, this follows in an obvious way from the Arzela-Ascoli theorem.

If $\Sigma$ 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 $\Sigma$ in $\mathcal{M}$ is precompact, and its closure is a compact metric space $\mathcal{L}$. The path components of $\mathcal{L}$ are exactly the Riemann surfaces which are arbitrarily well approximated (in the metric sense) on every compact subset by compact subsets of $\Sigma$. If you were wandering around on such a component $\Sigma'$, 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 $\Sigma$. Topologically, $\mathcal{L}$ is a Riemann surface lamination; i.e. a locally compact topological space covered by open charts of the form $U \times X$ where $U$ is an open two-dimensional disk, where $X$ is totally disconnected, and where the transition between charts preserves the decomposition into pieces $U \times \text{point}$, and is smooth (in fact, preserves the Riemann surface structure) on the $U \times \text{point}$ slices, in the overlaps. The unions of “surface” slices — i.e. the path components of $\mathcal{L}$ — 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 $K$ in $\Sigma$ and every $\epsilon>0$ there is a $T$ so that every ball in $\Sigma$ of radius $T$ contains a subset $K'$ which is $\epsilon$-close to $K$ in the Gromov-Hausdorff metric. In other words, every “local feature” of $\Sigma$ 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 $\Sigma'$ of $\mathcal{L}$, and therefore $\Sigma$ is in the closure of each $\Sigma'$. Since $\mathcal{L}$ is (in) the closure of $\Sigma$, 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 $\mathcal{L}$. 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 $4$-times punctured sphere. Every other leaf is an infinite cylinder — i.e. it is conformally the punctured plane $\mathbb{C}^*$. 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 $T_1$ be the planar “Greek cross” as in the following figure:

Inductively, if we have defined $T_n$, define $T_{n+1}$ by attaching four copies of $T_n$ to the extremities of $T_1$. The first few examples $T_1,\cdots,T_4$ are illustrated in the following figure:

The limit $T_\infty$ 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 $p_i$ in $T_\infty$ has a subsequence which escapes out one of the ends. Hence every other leaf in the lamination $\mathcal{L}$ 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 $3$-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 $3$-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.

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 $2$-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 $\Sigma$ be a hyperbolic surface with totally geodesic boundary. An orthogeodesic is a geodesic segment properly immersed in $\Sigma$, which is perpendicular to $\partial \Sigma$ at its endpoints. The set of orthogeodesics is countable, and their lengths are proper. Denote these lengths by $l_i$ (with multiplicity). The identity is:

$\sum_i \mathcal{L}(1/\cosh^2{l_i/2}) = -\pi^2\chi(\Sigma)/2$

where $\mathcal{L}$ 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 $\sum_i L(l_i) =$ a term depending only on the topology of $\Sigma$. The proof is very, very short and elegant. By the Gauss-Bonnet theorem, the term on the right is equal to $1/8$ of the volume of the unit tangent bundle of $\Sigma$. Almost every tangent vector on $\Sigma$ can be exponentiated to a geodesic on $\Sigma$ 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 $v$ is such a tangent vector, and $\gamma_v$ is the associated geodesic arc, then $\gamma_v$ is homotopic keeping endpoints on $\partial \Sigma$ to a unique orthogeodesic (which is the unique length minimizer in this relative homotopy class). The volume of the set of $v$ associated to a given orthogeodesic $\alpha$ can be computed as follows. Lift $\alpha$ 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 $\gamma_v$ 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 $v$ depends only on $\text{length}(\alpha)$, giving rise to the explicit formula for $L$. qed.

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

The “ordinary” polylogarithms $\text{Li}_k$ are defined by Taylor series

$\text{Li}_k(z) = \sum_{n=1}^\infty \frac {z^n} {n^k}$

which converges for $|z|<1$, and extends by analytic continuation. Taking derivatives, one sees that they satisfy $\text{Li}_k'(z) = \text{Li}_{k-1}(z)/z$, thereby giving rising to integral formulae. $\text{Li}_0(z)$ is the familiar geometric series $z/(1-z)$, so $\text{Li}_1(z) = -\log(1-z)$ and

$\text{Li}_2(z) = -\int \frac {\log(1-z)} {z} dz$

The Rogers dilogarithm is then given by the formula $\mathcal{L}(z) = \text{Li}_2(z) + \frac 1 2 \log(|z|)\log(1-z)$ for real $z<1$. One sees that the Rogers dilogarithm is obtained by symmetrizing the integrand for the integral expression for $\text{Li}_2$ under the involution $z \to 1-z$:

$\mathcal{L}'(z) = -\frac {1}{2} \left(\frac {\log(1-z)}{z} + \frac {\log(z)}{1-z} \right)$

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 $Q$ (whose unit tangent bundle has area $4\pi^2$) with one pair of opposite sides that are distance $l$ 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 $L$ and $R$) and the other two edges are $U$ and $D$. Similarly, call the four triangles $T_L, T_R, T_U, T_D$ depending on which edge of the quadrilateral they bound. The triangle $T_R$ 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) $0,x,1,\infty$, where $\infty,0$ are the ideal vertices of the gray triangle. Call $\alpha$ the (hyperbolic) angle of the gray triangle at its vertex, so $x = (1+\cos(\alpha))/2$. Moreover, it turns out that $x = 1/\cosh^2(l/2)$ where $l$ is the distance between $L$ and $R$. We will compute $L$ implicitly as a function of $x$, and show that it is a multiple of the Rogers dilogarithm function, thus verifying Bridgeman’s identity.

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

These volumes can be expressed in terms of integrals of harmonic functions. Let $\chi_L$ denote the harmonic function on the disk which is $1$ on the arc of the circle bounded by $L$, and $0$ on the rest of the circle. This function at each point is equal to $1/2\pi$ 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 $\chi_R,\chi_U,\chi_D$ similarly. Then the total volume we need to compute is equal to

$4\pi \left( (\int_{T_L} 2\chi_R) + (\int_{T_U} 1 - 2\chi_U) \right)$

(here we have identified $\int_{T_L} \chi_R = \int_{T_R} \chi_L$ by symmetry, and similarly for the other pair of terms). Let us approach this a bit more systematically. If $\alpha$ denotes the angle at the nonideal vertex of triangle $T_R$, we denote $\int_{T_R} \chi_R = A(\alpha)$, $\int_{T_R} \chi_U = B(\alpha)$ and $\int_{T_R} \chi_L = C(\alpha)$. The integral we want to evaluate can be expressed easily in terms of explicit rational multiples of $\pi$, and the function $A,B,C$. These functions satisfy obvious identities:

$C(\alpha) = \int_{T_R} 1 - A(\alpha) - 2B(\alpha) = \pi-\alpha - A(\alpha) - 2B(\alpha)$

and

$A(\alpha) + B(\pi - \alpha) = \pi/3$

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 $1/3$. In particular, we see that we can express everything in terms of $A$. After some elementary reorganization, we see that the contribution $V(\alpha)$ to the volume of the unit tangent bundle of the surface associated to this particular orthogeodesic is

$V(\alpha) = \pi^2(8 - 16/3) - 4\pi\alpha - 8\pi(A(\alpha) - A(\pi - \alpha))$

To compute $A(\alpha)$, it makes sense to move to the upper half-space model, and move the endpoints of the interval to $0$ and $\infty$. The harmonic function is equal to $1$ on the negative real axis, and $0$ on the positive real axis. It takes the value $\theta/\pi$ on the line $\text{arg}(z) = \theta$. The area form in the hyperbolic metric is proportional to the Euclidean area form, with constant $1/\text{Im}(z)^2$. In other words, we want to integrate $\text{arg}(z)/\pi\text{Im}(z)^2$ over the region indicated in the figure, where the nonideal angle is $\alpha$, and the base point is $0$:

If we normalize so that the circular arc is part of the semicircle from $0$ to $1$, then the real projection of the vertical lines in the figure are $0$ and $x$. There is no elementary way to evaluate this integral, so instead we evaluate its derivative as a function of $x$ where as before, $x = (1+\cos(\alpha))/2$. This is the definite integral

$A'(x) = \int_{y = \sqrt{x-x^2}}^\infty (\tan^{-1}(y/x)/\pi y^2) dy$

Integrating by parts gives $(\alpha/\pi\sin{\alpha}) + 1/\pi \int_{y = \sqrt{x-x^2}}^\infty xdy/y(y^2+x^2)$. This evaluates to

$A'(x) = (\alpha/\pi\sin{\alpha}) - 1/\pi ( \log(1-x)/2x)$

Thinking of $V(\alpha)$ as a function of $x$, we get

$V'(x) = -4\pi d\alpha/dx - 8\pi(A'(x) + A'(1-x)) = 8\mathcal{L}'(x)$

Comparing values at $x=0$ we see that $V=8\mathcal{L}$ 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 $\text{SL}(2,\mathbb{R})$-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 $f:(0,1) \to \mathbb{R}$ be a three times differentiable function satisfying

$f(s_1) - f(s_2) + f(\frac{s_2}{s_1}) - f(\frac{1-s_1^{-1}}{1-s_2^{-1}}) + f(\frac{1-s_1}{1-s_2})=0$

for all $0 < s_2 < s_1 < 1$. Then there is a real constant $\kappa$ such that $f(x) = \kappa L(x)$ where $L(x)$ 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?

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 $\mathbb{R}^3$. 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 $2$-colored, the black vertices are the vertices of a regular tetrahedron. Let $Q$ denote the quadrilateral formed by four edges of this tetrahedron; then a fundamental piece $S$ of Schwarz’s surface is a minimal disk spanning $Q$:

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

The symmetries of $Q$ enable us to give it several descriptions as a Riemann surface. Firstly, we could think of $Q$ as a polygon in the hyperbolic plane with four edges of equal length, and angles $\pi/3$. Twelve copies of $Q$ can be assembled to make a hyperbolic surface $\Sigma$ of genus $3$. Thinking of a surface of genus $3$ as the boundary of a genus $3$ handlebody defines a homomorphism from $\pi_1(\Sigma)$ to $\mathbb{Z}^3$, thought of as $H_1(\text{handlebody})$; the cover $\widetilde{\Sigma}$ associated to the kernel is (conformally) the triply periodic Schwarz surface, and the deck group acts on $\mathbb{R}^3$ as a lattice (of index $2$ in the face-centered cubic lattice).

Another description is as follows. Since the deck group acts by translation, the Gauss map from $\widetilde{\Sigma}$ to $S^2$ factors through a map $\Sigma \to S^2$. The map is injective at each point in the interior or on an edge of a copy of $Q$, but has an order $2$ branch point at each vertex. Thus, the map $\Sigma \to S^2$ is a double-branched cover, with one branch point of order $2$ at each vertex of a regular inscribed cube. This leads one to think (like a late 19th century mathematician) of $\Sigma$ as the Riemann surface on which a certain multi-valued function on $S^2 = \mathbb{C} \cup \infty$ is single-valued. Under stereographic projection, the vertices of the cube map to the eight points $\lbrace \alpha,i\alpha,-\alpha,-i\alpha,1/\alpha,i/\alpha,-1/\alpha,-i/\alpha \rbrace$ where $\alpha = (\sqrt{3}-1)/\sqrt{2}$. These eight points are the roots of the polynomial $w^8 - 14w^4 + 1$, so we may think of $\Sigma$ as the hyperelliptic Riemann surface defined by the equation $v^2 = w^8 - 14w^4 + 1$; equivalently, as the surface on which the multi-valued (on $\mathbb{C} \cup \infty$) function $R(w):= 1/v=1/\sqrt{w^8 - 14w^4 + 1}$ is single-valued.

The function $R(w)$ is known as the Weierstrass function associated to $\Sigma$, and an explicit formula for the co-ordinates of the embedding $\widetilde{\Sigma} \to \mathbb{R}^3$ were found by Enneper and Weierstrass. After picking a basepoint (say $0$) on the sphere, the coordinates are given by integration:

$x = \text{Re} \int_0^{w_0} \frac{1}{2}(1-w^2)R(w)dw$

$y = \text{Re} \int_0^{w_0} \frac{i}{2}(1+w^2)R(w)dw$

$z = \text{Re} \int_0^{w_0} wR(w)dw$

The integral in each case depends on the path, and lifts to a single-valued function precisely on $\widetilde{\Sigma}$.

Geometrically, the three coordinate functions $x,y,z$ are harmonic functions on $\widetilde{\Sigma}$. 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 $1$-forms $\phi_1,\phi_2,\phi_3$ which descend to $\Sigma$ (since the deck group acts by translations). These $1$-forms satisfy the identity $\sum_i \phi_i^2 = 0$ (this identity expresses the fact that the embedding of $\widetilde{\Sigma}$ into $\mathbb{R}^3$ via these functions is conformal). The (composition of the) Gauss map (with stereographic projection) can be read off from the $\phi_i$, and as a meromorphic function on $\Sigma$, it is given by the formula $w = \phi_3/(\phi_1 - i\phi_2)$. Define a function $f$ on $\Sigma$ by the formula $fdw = \phi_1 - i\phi_2$. Then $1/f,w$ are the coordinates of a rational map from $\Sigma$ into $\mathbb{C}^2$ which extends to a map into $\mathbb{CP}^2$, by sending each zero of $f$ to $wf = \phi_3/dw$ in the $\mathbb{CP}^1$ at infinity. Symmetry allows us to identify the image with the hyperelliptic embedding from before, and we deduce that $f=R(w)$. Solving for $\phi_1,\phi_2$ we obtain the integrands in the formulae above.

In fact, any holomorphic function $R(w)$ on a domain in $\mathbb{C}$ 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 $\mathbb{R}^3$. 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 $\pi/n$ the resulting surface closes up smoothly around the vertices, and one obtains a compact abstract Riemann surface $\Sigma$ tiled by copies of the fundamental region, together with a holonomy representation of $\pi_1(\Sigma)$ into $\text{Isom}^+(\mathbb{R}^3)$. Sometimes the image of this representation in the rotational part of $\text{Isom}^+(\mathbb{R}^3)$ is finite, and one obtains an infinitely periodic minimal surface as in the case of Schwarz’s surface. A fundamental tile in $\Sigma$ 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 $\mathbb{R}^3$ 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 $P$ and $Q$ are two such regions, then a conformal map from $P$ to $Q$ can be factored as the product of a map uniformizing $P$ as the upper half-plane, followed by the inverse of a map uniformizing $Q$ 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 $Q$. Near each vertex, $Q$ can be moved by a fractional linear transformation $z \to (az+b)/(cz+d)$ to (part of) a wedge, consisting of complex numbers with argument between $0$ and $\alpha$, where $\alpha$ is the angle at $Q$. The function $f(z) = z^{\alpha/\pi}$ 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: $f' = (f+c)'$. Functions that differ by multiplication by a constant have the same logarithmic derivative: $(\log(f))' = (\log(cf))'$. Putting these two observations together suggest defining the nonlinearity of a function as the composition $N(f):= (\log(f'))' = f''/f'$. This has the property that $N(af+b) = N(f)$ for any constants $a,b$. Under inversion $z \to 1/z$ the nonlinearity transforms by $N(1/f) = N(f) - 2f'/f$. From this, and a simple calculation, one deduces that the operator $N' - N^2/2$ 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 $S(f) = f'''/f' - 3/2(f''/f')^2$. Given the Schwarzian derivative $S(f)$, one may recover the nonlinearity $N(f)$ by solving the Ricatti equation $N' - N^2/2 - S = 0$. 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 $\text{PSL}(2,\mathbb{C})$, acting by fractional linear transformations, as we have just deduced. Once we know the nonlinearity, we can solve for $f$ by $f = \int e^{\int N}$, the usual solution to a first order linear inhomogeneous ODE. The Schwarzian of the function $z^{\alpha/\pi}$ is $(1-\alpha^2/\pi^2)/2z^2$. The advantage of expressing things in these terms is that the Schwarzian of a uniformizing map for a circular polygon $Q$ with angles $\alpha_i$ at the vertices has the form of a rational function, with principal parts $a_i/(z-z_i)^2 + b_i/(z-z_i)$, where the $a_i = (1-\alpha_i^2/\pi^2)/2$ and the $b_i$ and $z_i$ depend (unfortunately in a very complicated way) on the edges of $Q$ (for the ugly truth, see Nehari, chapter 5). To see this, observe that the map has an order two pole near finitely many points $z_i$ (the preimages of the vertices of $Q$ under the uniformizing map) but is otherwise holomorphic. Moreover, it can be analytically continued into the lower half plane across the interval between successive $z_i$, by reflecting the image across each circular edge. After reflecting twice, the image of $Q$ is transformed by a fractional linear transformation, so $S(f)$ 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 $f$ is the uniformizing map from the upper half-plane to a polygon $Q$ with spherical arcs, then $S(f)$ 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.

$S(f) = \sum _i a_i/(z-z_i)^2 + b_i/(z-z_i) + c$

where $a_i$ is as above, and $z_i,b_i,c$ 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.

An amenable group $G$ acting by homeomorphisms on a compact topological space $X$ preserves a probability measure on $X$; in fact, one can given a definition of amenability in such terms. For example, if $G$ is finite, it preserves an atomic measure supported on any orbit. If $G = \mathbb{Z}$, one can take a sequence of almost invariant probability measures, supported on the subset $[-n,n] \cdot p$ (where $p \in X$ is arbitrary), and any weak limit will be invariant. For a general amenable group, in place of the subsets $[-n,n] \subset \mathbb{Z}$, 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 $G$ is not amenable, it is generally not true that there is any probability measure on $X$ invariant under the action of $G$. 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 $G$-action on $X$. To be concrete, suppose $G$ is finitely generated by a symmetric generating set $S$ (symmetric here means that if $s \in S$, then $s^{-1} \in S$). Let $M(X)$ denote the space of probability measures on $X$. One can form an operator $\Delta:M(X) \to M(X)$ defined by the formula

$\Delta(\mu) = \frac {1} {|S|} \sum_{s \in S} s_*\mu$

and then look for a probability measure $\nu$ stationary under $\Delta$, which exists for quite general reasons. This measure $\nu$ is the harmonic measure: the expectation of the $\nu$-measure of $s(A)$ under a randomly chosen $s \in S$ is equal to the $\nu$-measure of $A$. Note for any probability measure $\mu$ that $s_*\mu$ is absolutely continuous with respect to $\Delta(\mu)$; in fact, the Radon-Nikodym derivative satisfies $ds_*\mu/d\Delta(\mu) \le |S|$. Substituting $\nu$ for $\mu$ in this formula, one sees that the measure class of $\nu$ is preserved by $G$, and that for every $g \in G$, we have $dg_*\nu/d\nu \le |S|^{|g|}$, where $|g|$ denotes word length with respect to the given generating set.

The existence of harmonic measure is especially useful when $X$ is one-dimensional, e.g. in the case that $X=S^1$. 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 $G$ be countable is essential; for example, the group $\mathbb{Z}^{\mathbb{Z}}$ acts in a non-bilipschitz way on the interval — see here).

Suppose now that $G = \pi_1(M)$ for some manifold $M$. The action of $G$ on $S^1$ determines a foliated circle bundle $S^1 \to E \to M$; i.e. a circle bundle, together with a codimension one foliation transverse to the circle fibers. To see this, first form the product $\widetilde{M} \times S^1$ with its product foliation by leaves $\widetilde{M} \times \text{point}$, where $\widetilde{M}$ denotes the universal cover of $M$. The group $G = \pi_1(M)$ acts on $\widetilde{M}$ as the deck group of the covering, and on $S^1$ by the given action; the quotient of this diagonal action on the product is the desired circle bundle $E$. The foliation makes $E$ into a “flat” circle bundle with structure group $\text{Homeo}^+(S^1)$. The foliation allows us to associate to each path $\gamma$ in $M$ a homeomorphism from the fiber over $\gamma(0)$ to the fiber over $\gamma(1)$; integrability (or flatness) implies that this homeomorphism only depends on the relative homotopy class of $\gamma$ in $M$. This identification of fibers is called the holonomy of the foliation along the path $\gamma$. If $M$ 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 $M$ 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 $\mathcal{F}$ of a manifold $N$, by finding a fixed point of the leafwise heat flow on the space of probability measures on $N$, 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 $2\pi$ in its harmonic measure. Let $X$ be the vector field on the circle bundle that rotates each circle at unit speed, and let $\alpha$ be the $1$-form on $E$ whose kernel is tangent to the leaves of the foliation. We scale $\alpha$ so that $\alpha(X)=1$ everywhere. The integrability condition for a foliation is expressed in terms of the $1$-form as the identity $\alpha \wedge d\alpha = 0$, and we can write $d\alpha = -\beta \wedge \alpha$ where $\beta(X)=0$. More intrinsically, $\beta$ descends to a $1$-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 $\beta$ 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 $\widetilde{M}$ by projection, the fact that our measure is harmonic means that $\beta$ “is” the gradient of the logarithm of a positive harmonic function on $\widetilde{M}$. As observed by Thurston, the geometry of $M$ then puts constraints on the size of $\beta$. 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 $E$ can be obtained by averaging $\alpha$ under the flow of $X$; i.e. if $\phi_t$ is the diffeomorphism of $E$ which rotates each circle through angle $t$, then

$\omega = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^* \alpha$

is an $X$-invariant $1$-form on $E$, which therefore descends to a $1$-form on $M$, which can be thought of as a connection form for an $\text{SO}(2)$-structure on the bundle $E$. The curvature of the connection (in the usual sense) is the $2$-form $d\omega$, and we have a formula

$d\omega = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^*(d\alpha) = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^*(-\beta \wedge \alpha)$

The action of the $1$-parameter group $\phi_t$ trivializes the cotangent bundle to $E$ over each fiber. After choosing such a trivialization, we can think of the values of $\alpha$ at each point on a fiber as sweeping out a circle $\gamma$ in a fixed vector space $V$. The tangent to this circle is found by taking the Lie derivative

$\mathcal{L}_X(\alpha) = \iota_X d\alpha + d\iota_X \alpha = \alpha(X)\beta = \beta$

In other words, $\beta$ is identified with $d\gamma$ under the identification of $\alpha$ with $\gamma$, and $\int \phi_t^*(-\beta \wedge \alpha) = \int \gamma \wedge d\gamma$; i.e. the absolute value of the curvature of the connection is equal to $1/\pi$ times the area enclosed by $\gamma$.

Now suppose $M$ is a hyperbolic $n$-manifold, i.e. a manifold of dimension $n$ with constant curvature $-1$ everywhere. Equivalently, think of $M$ as a quotient of hyperbolic space $\mathbb{H}^n$ by a discrete group of isometries. A positive harmonic function on $\mathbb{H}^n$ has a logarithmic derivative which is bounded pointwise by $(n-1)$; 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 $e^t$; since the sphere at infinity has dimension $n-1$, the conclusion follows. But this means that the speed of $\gamma$ (i.e. the size of $d\gamma$) is pointwise bounded by $(n-1)$, and the length of the $\gamma$ circle is at most $2\pi(n-1)$. A circle of length $2\pi(n-1)$ can enclose a disk of area at most $\pi (n-1)^2$, so the curvature of the connection has absolute value pointwise bounded by $(n-1)^2$.

One corollary is a new proof of the Milnor-Wood inequality, which says that a foliated circle bundle $E$ over a closed oriented surface $S$ of genus at least $2$ satisfies $|e(E)| \le -\chi(S)$, where $e(E)$ is the Euler number of the bundle (a topological invariant). For, the surface $S$ 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 $1$. The Euler class of the bundle evaluated on the fundamental class of $S$ is the Euler number $e(E)$; we have

$|e(E)| = \frac {1} {2\pi} |\int_S \omega| \le \text{area}(S)/2\pi = -\chi(S)$

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 $S \to M$ is an immersion which is injective on $\pi_1$. There is a cover $\widehat{M}$ of $M$ for which the immersion lifts to a homotopy equivalence, and we get an action of $\pi_1(\widehat{M})$ on the circle at infinity of $S$, and hence a foliated circle bundle as above with $e(E) = -\chi(S)$. Integrating as above over the image of $S$ in $\widehat{M}$, and using the fact that the curvature of $\omega$ is pointwise bounded by $(n-1)^2$, we deduce that the area of $S$ is at least $-2\pi\chi(S)/(n-1)^2$. If $M$ is a $3$-manifold, we obtain $\text{area}(S) \ge -2\pi\chi(S)/4$.

(A somewhat more subtle argument allows one to get better bounds, e.g. replacing $4$ by $(\pi/2)^2$ for $n=3$, and better estimates for higher $n$.)

I was in Stony Brook last week, visiting Moira Chas and Dennis Sullivan, and have been away from blogging for a while; this week I plan to write a few posts about some of the things I discussed with Moira and Dennis. This is an introductory post about the Goldman bracket, an extraordinary mathematical object made out of the combinatorics of immersed curves on surfaces. I don’t have anything original to say about this object, but for my own benefit I thought I would try to explain what it is, and why Goldman was interested in it.

In his study of symplectic structures on character varieties $\text{Hom}(\pi,G)/G$, where $\pi$ is the fundamental group of a closed oriented surface and $G$ is a Lie group satisfying certain (quite general) conditions, Bill Goldman discovered a remarkable Lie algebra structure on the free abelian group generated by conjugacy classes in $\pi$. Let $\hat{\pi}$ denote the set of homotopy classes of closed oriented curves on $S$, where $S$ is itself a compact oriented surface, and let $\mathbb{Z}\hat{\pi}$ denote the free abelian group with generating set $\hat{\pi}$. If $\alpha,\beta$ are immersed oriented closed curves which intersect transversely (i.e. in double points), define the formal sum

$[\alpha,\beta] = \sum_{p \in \alpha \cap \beta} \epsilon(p; \alpha,\beta) |\alpha_p\beta_p| \in \mathbb{Z}\hat{\pi}$

In this formula, $\alpha_p,\beta_p$ are $\alpha,\beta$ thought of as based loops at the point $p$, $\alpha_p\beta_p$ represents their product in $\pi_1(S,p)$, and $|\alpha_p\beta_p|$ represents the resulting conjugacy class in $\pi$. Moreover, $\epsilon(p;\alpha,\beta) = \pm 1$ is the oriented intersection number of $\alpha$ and $\beta$ at $p$.

This operation turns out to depend only on the free homotopy classes of $\alpha$ and $\beta$, and extends by linearity to a bilinear map $[\cdot,\cdot]:\mathbb{Z}\hat{\pi} \times \mathbb{Z}\hat{\pi} \to \mathbb{Z}\hat{\pi}$. Goldman shows that this bracket makes $\mathbb{Z}\hat{\pi}$ into a Lie algebra over $\mathbb{Z}$, and that there are natural Lie algebra homomorphisms from $\mathbb{Z}\hat{\pi}$ to the Lie algebra of functions on $\text{Hom}(\pi,G)/G$ with its Poisson bracket.

The connection with character varieties can be summarized as follows. Let $f:G \to \mathbb{R}$ be a (smooth) class function (i.e. a function which is constant on conjugacy classes) on a Lie group $G$. Define the variation function $F:G \to \mathfrak{g}$ by the formula

$\langle F(A),X\rangle = \frac {d}{dt}|_{t=0} f(A\text{exp}{tX})$

where $\langle \cdot,\cdot\rangle$ is some (fixed) $\text{Ad}$-invariant orthogonal structure on the Lie algebra $\mathfrak{g}$ (for example, if $G$ is reductive (eg if $G$ is semisimple), one can take $\langle X,Y\rangle = \text{tr}(XY)$). The tangent space to the character variety $\text{Hom}(\pi,G)/G$ at $\phi$ is the first cohomology group of $\pi$ with coefficients in $\mathfrak{g}$, thought of as a $G$ module with the $\text{Ad}$ action, and then as a $\pi$ module by the representation $\phi$. Cup product and the pairing $\langle\cdot,\cdot\rangle$ determine a pairing

$H^1(\pi,\mathfrak{g})\times H^1(\pi,\mathfrak{g}) \to H^2(\pi,\mathbb{R}) = \mathbb{R}$

where the last equality uses the fact that $\pi$ is a closed surface group; this pairing defines the symplectic structure on $\text{Hom}(\pi,G)/G$.

Every element $\alpha \in \pi$ determines a function $f_\alpha:\text{Hom}(\pi,G)/G \to \mathbb{R}$ by sending a (conjugacy class of) representation $[\phi]$ to $f(\phi(\alpha))$. Note that $f_\alpha$ only depends on the conjugacy class of $\alpha$ in $\pi$. It is natural to ask: what is the Hamiltonian flow on $\text{Hom}(\pi,G)/G$ generated by the function $f_\alpha$? It turns out that when $\alpha$ is a simple closed curve, it is very easy to describe this Hamiltonian flow. If $\alpha$ is nonseparating, then define a flow $\psi_t$ by $\psi_t\phi(\gamma)=\phi(\gamma)$ when $\gamma$ is represented by a curve disjoint from $\alpha$, and $\psi_t\phi(\gamma)= \text{exp} tF_\alpha(\phi)\phi(\gamma)$ if $\gamma$ intersects $\alpha$ exactly once with a positive orientation (there is a similar formula when $\alpha$ is separating). In other words, the representation is constant on the fundamental group of the surface “cut open” along the curve $\alpha$, and only deforms in the way the two conjugacy classes of $\alpha$ in the cut open surface are identified in $\pi$.

In the important motivating case that $G = \text{PSL}(2,\mathbb{R})$, so that one component of $\text{Hom}(\pi,G)/G$ is the Teichmüller space of hyperbolic structures on the surface $S$, one can take $f = 2\cosh^{-1}\text{tr/2}$, and then $f_\alpha$ is just the length of the geodesic in the free homotopy class of $\alpha$, in the hyperbolic structure on $S$ associated to a representation. In this case, the symplectic structure on the character variety restricts to the Weil-Petersson symplectic structure on Teichmüller space, and the Hamiltonian flow associated to the length function $f_\alpha$ is a family of Fenchel-Nielsen twists, i.e. the deformations of the hyperbolic structure obtained by cutting along the geodesic $\alpha$, rotating through some angle, and regluing. This latter observation recovers a famous theorem of Wolpert, connected in an obvious way to his formula for the symplectic form $\omega = \sum dl_\alpha \wedge d\theta_\alpha$ where $\theta$ is angle and $l$ is length, and the sum is taken over a maximal system of disjoint essential simple curves $\alpha$ for the surface $S$.

The combinatorial nature of the Goldman bracket suggests that it might have applications in combinatorial group theory. Turaev discovered a Lie cobracket on $\mathbb{Z}\hat{\pi}$, and showed that together with the Goldman bracket, one obtains a Lie bialgebra. Motivated by Stallings’ reformulation of the Poincaré conjecture in terms of group theory, Turaev asked whether a free homotopy class contains a power of a simple curve if and only if the cobracket of the class is zero. The answer to this question is negative, as shown by Chas; on the other hand, Chas and Krongold showed that a class $\alpha$ is simple if and only if $[\alpha,\alpha^3]$ is zero. Nevertheless, the full geometric meaning of the Goldman bracket remains mysterious, and a topic worthy of investigation.

If $f$ is a smooth function on a manifold $M$, and $p$ is a critical point of $f$, recall that the Hessian $H_pf$ is the quadratic form $\nabla df$ on $T_pM$ (in local co-ordinates, the coefficients of the Hessian are the second partial derivatives of $f$ at $p$). Since $H_pf$ is symmetric, it has a well-defined index, which is the dimension of the subspace of maximal dimension on which $H_pf$ is negative definite. The Hessian does not depend on a choice of metric. One way to see this is to give an alternate definition $H_pf(X(p),Y(p)) = X(Yf)(p)$ where $X$ and $Y$ are any two vector fields with given values $X(p)$ and $Y(p)$ in $T_pM$. To see that this does not depend on the choice of $X,Y$, observe

$X(Yf)(p) - Y(Xf)(p) = [X,Y]f(p) = df([X,Y])_p = 0$

because of the hypothesis that $df$ vanishes at $p$. This calculation shows that the formula is symmetric in $X$ and $Y$. Furthermore, since $X(Yf)(p)$ only depends on the value of $X$ at $p$, the symmetry shows that the result only depends on $X(p)$ and $Y(p)$ as claimed. A critical point is nondegenerate if $H_pf$ is nondegenerate as a quadratic form.

In Morse theory, one uses a nondegenerate smooth function $f$ (i.e. one with isolated nondegenerate critical points), also called a Morse function, to understand the topology of $M$: the manifold $M$ has a (smooth) handle decomposition with one $i$-handle for each critical point of $f$ of index $i$. In particular, nontrivial homology of $M$ forces any such function $f$ to have critical points (and one can estimate their number of each index from the homology of $M$). Morse in fact applied his construction not to finite dimensional manifolds, but to the infinite dimensional manifold of smooth loops in some finite dimensional manifold, with arc length as a “Morse” function. Critical “points” of this function are closed geodesics. Any closed manifold has a nontrivial homotopy group in some dimension; this gives rise to nontrivial homology in the loop space. Consequently one obtains the theorem of Lyusternik and Fet:

Theorem: Let $M$ be a closed Riemannian manifold. Then $M$ admits at least one closed geodesic.

In higher dimensions, one can study the space of smooth maps from a fixed manifold $S$ to a Riemannian manifold $M$ equipped with various functionals (which might depend on extra data, such as a metric or conformal structure on $S$). One context with many known applications is when $M$ is a Riemannian $3$-manifold, $S$ is a surface, and one studies the area function on the space of smooth maps from $S$ to $M$ (usually in a fixed homotopy class). Critical points of the area function are called minimal surfaces; the name is in some ways misleading: they are not necessarily even local minima of the area function. That depends on the index of the Hessian of the area function at such a point.

Let $\rho(t)$ be a (compactly supported) $1$-parameter family of surfaces in a Riemannian $3$-manifold $M$, for which $\rho(0)$ is smoothly immersed. For small $t$ the surfaces $\rho(t)$ are transverse to the exponentiated normal bundle of $\rho(0)$; hence locally we can assume that $\rho$ takes the form $\rho(t,u,v)$ where $u,v$ are local co-ordinates on $\rho(0)$, and $\rho(\cdot,u,v)$ is contained in the normal geodesic to $\rho(0)$ through the point $\rho(0,u,v)$; we call such a family of surfaces a normal variation of surfaces. For such a variation, one has the following:

Theorem (first variation formula): Let $\rho(t)$ be a normal variation of surfaces, so that $\rho'(0) = f\nu$ where $\nu$ is the unit normal vector field to $\rho(0)$. Then there is a formula:

$\frac d {dt} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle f\nu,\mu\rangle d\text{area}$

where $\mu$ is the mean curvature vector field along $\rho(0)$.

Proof: let $T,U,V$ denote the image under $d\rho$ of the vector fields $\partial_t,\partial_u,\partial_v$. Choose co-ordinates so that $u,v$ are conformal parameters on $\rho(0)$; this means that $\langle U,V\rangle = 0$ and $\|U\|=\|V\|$ at $t=0$.

The infinitesimal area form on $\rho(t)$ is $\sqrt{\|U\|^2\|V\|^2 - \langle U,V \rangle^2} dUdV$ which we abbreviate by $E^{1/2}$, and write

$\frac d {dt} \text{area}(\rho(t)) = \int_{\rho(t)} \frac {dUdV} {2E^{1/2}} (\|U\|^2\langle V,V\rangle' + \|V\|\langle U,U\rangle' - 2\langle U,V\rangle\langle U,V\rangle')$

Since $V,T$ are the pushforward of coordinate vector fields, they commute; hence $[V,T]=0$, so $\nabla_T V = \nabla_V T$ and therefore

$\langle V,V\rangle' = 2\langle \nabla_T V,V\rangle = 2\langle \nabla_V T,V\rangle = 2(V\langle T,V\rangle - \langle T,\nabla_V V\rangle)$

and similarly for $\langle U,U\rangle'$. At $t = 0$ we have $\langle T,V\rangle = 0$, $\langle U,V\rangle = 0$ and $\|U\|^2 = \|V\|^2 = E^{1/2}$ so the calculation reduces to

$\frac d {dt} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle T,\nabla_U U + \nabla_V V\rangle dUdV$

Now, $T|_{t=0} = f\nu$, and $\nabla_U U + \nabla_V V = \mu E^{1/2}$ so the conclusion follows. qed.

As a corollary, one deduces that a surface is a critical point for area under all smooth compactly supported variations if and only if the mean curvature $\mu$ vanishes identically; such a surface is called minimal.

The second variation formula follows by a similar (though more involved) calculation. The statement is:

Theorem (second variation formula): Let $\rho(t)$ be a normal variation of surfaces, so that $\rho'(0)=f\nu$. Suppose $\rho(0)$ is minimal. Then there is a formula:

$\frac {d^2} {dt^2} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle f\nu,L(f)\nu\rangle d\text{area}$

where $L$ is the Jacobi operator (also called the stability operator), given by the formula

$L = \text{Ric}(\nu) + |A|^2 + \Delta_\rho$

where $A$ is the second fundamental form, and $\Delta_\rho = -\nabla^*\nabla$ is the metric Laplacian on $\rho(0)$.

This formula is frankly a bit fiddly to derive (one derivation, with only a few typos, can be found in my Foliations book; a better derivation can be found in the book of Colding-Minicozzi) but it is easy to deduce some significant consequences directly from this formula. The metric Laplacian on a compact surface is negative self-adjoint (being of the form $-X^*X$ for some operator $X$), and $L$ is obtained from it by adding a $0$th order perturbation, the scalar field $|A|^2 + \text{Ric}(\nu)$. Consequently the biggest eigenspace for $L$ is $1$-dimensional, and the eigenvector of largest eigenvalue cannot change sign. Moreover, the spectrum of $L$ is discrete (counted with multiplicity), and therefore the index of $-L$ (thought of as the “Hessian” of the area functional at the critical point $\rho(0)$) is finite.

A surface is said to be stable if the index vanishes. Integrating by parts, one obtains the so-called stability inequality for a stable minimal surface $S$:

$\int_S (\text{Ric}(\nu) + |A|^2)f^2d\text{area} \le \int_S |\nabla f|^2 d\text{area}$

for any reasonable compactly supported function $f$. If $S$ is closed, we can take $f=1$. Consequently if the Ricci curvature of $M$ is positive, $M$ admits no stable minimal surfaces at all. In fact, in the case of a surface in a $3$-manifold, the expression $\text{Ric}(\nu) + |A|^2$ is equal to $R - K + |A|^2/2$ where $K$ is the intrinsic curvature of $S$, and $R$ is the scalar curvature on $M$. If $S$ has positive genus, the integral of $-K$ is non-negative, by Gauss-Bonnet. Consequently, one obtains the following theorem of Schoen-Yau:

Corollary (Schoen-Yau): Let $M$ be a Riemannian $3$-manifold with positive scalar curvature. Then $M$ admits no immersed stable minimal surfaces at all.

On the other hand, one knows that every $\pi_1$-injective map $S \to M$ to a $3$-manifold is homotopic to a stable minimal surface. Consequently one deduces that when $M$ is a $3$-manifold with positive scalar curvature, then $\pi_1(M)$ does not contain a surface subgroup. In fact, the hypothesis that $S \to M$ be $\pi_1$-injective is excessive: if $S \to M$ is merely incompressible, meaning that no essential simple loop in $S$ has a null-homotopic image in $M$, then the map is homotopic to a stable minimal surface. The simple loop conjecture says that a map $S \to M$ from a $2$-sided surface to a $3$-manifold is incompressible in this sense if and only if it is $\pi_1$-injective; but this conjecture is not yet known.

Update 8/26: It is probably worth making a few more remarks about the stability operator.

The first remark is that the three terms $\text{Ric}(\nu)$, $|A|^2$ and $\Delta$ in $L$ have natural geometric interpretations, which give a “heuristic” justification for the second variation formula, which if nothing else, gives a handy way to remember the terms. We describe the meaning of these terms, one by one.

1. Suppose $f \equiv 1$, i.e. consider a variation by flowing points at unit speed in the direction of the normals. In directions in which the surface curves “up”, the normal flow is focussing; in directions in which it curves “down”, the normal flow is expanding. The net first order effect is given by $\langle \nu,\mu\rangle$, the mean curvature in the direction of the flow. For a minimal surface, $\mu = 0$, and only the second order effect remains, which is $|A|^2$ (remember that $A$ is the second fundamental form, which measures the infinitesimal deviation of $S$ from flatness in $M$; the mean curvature is the trace of $A$, which is first order. The norm $|A|^2$ is second order).
2. There is also an effect coming from the ambient geometry of $M$. The second order rate at which a parallel family of normals $\nu$ along a geodesic $\gamma$ diverge is $\langle R(\gamma',\nu)\gamma',\nu\rangle$ where $R$ is the curvature operator. Taking the average over all geodesics $\gamma$ tangent to $S$ at a point gives the Ricci curvature in the direction of $\nu$, i.e. $\text{Ric}(\nu)$. This is the infinitesimal expansion of area of a geodesic plane under the normal flow, and has second order. The interactions between these terms have higher order, so the net contribution when $f \equiv 1$ is $\text{Ric}(\nu) + |A|^2$.
3. Finally, there is the contribution coming from $f$ itself. Imagine that $S$ is a flat plane in Euclidean space, and let $S_\epsilon$ be the graph of $\epsilon f$. The infinitesimal area element on $S_\epsilon$ is $\sqrt{1+\epsilon^2 |\nabla f|^2} \sim 1+\epsilon^2/2 |\nabla f|^2$. If $f$ has compact support, then differentiating twice by $\epsilon$, and integrating by parts, one sees that the (leading) second order term is $\Delta f$. When $S$ is not totally geodesic, and the ambient manifold is not Euclidean space, there is an interaction which has higher order; the leading terms add, and one is left with $L = \text{Ric}(\nu) + |A|^2 + \Delta$.

The second remark to make is that if the support of a variation $f$ is sufficiently small, then necessarily $|\nabla f|$ will be large compared to $f$, and therefore $-L$ will be positive definite. In other words all variations of a (fixed) minimal surface with sufficiently small support are area increasing — i.e. a minimal surface is locally area minimizing (this is local in the surface itself, not in the “space of all surfaces”). This is a generalization of the important fact that a geodesic in a Riemannian manifold is locally length minimizing (though typically not globally length minimizing).

One final remark is that when $|A|^2$ is big enough at some point $p \in S$, and when the injectivity radius of $S$ at $p$ is big enough (depending on bounds on $\text{Ric}(\nu)$ in some neighborhood of  $p$), one can find a variation with support concentrated near $p$ that violates the stability inequality. Contrapositively, as observed by Schoen, knowing that a minimal surface in a $3$-manifold $M$ is stable gives one a priori control on the size of $|A|^2$, depending only on the Ricci curvature of $M$, and the injectivity radius of the surface at the point. Since stability is preserved under passing to covers (for $2$-sided surfaces, by the fact that the largest eigenvalue of $L$ can’t change sign!) one only needs a lower bound on the distance from $p$ to $\partial S$. In particular, if $S$ is a closed stable minimal surface, there is an a priori pointwise bound on $|A|^2$. This fact has many important topological applications in $3$-manifold topology. On the other hand, when $S$ has boundary, the curvature can be arbitrarily large. The following example is due to Thurston (also see here for a discussion):

Example (Thurston): Let $\Delta$ be an ideal simplex in $\mathbb{H}^3$ with ideal simplex parameter imaginary and very large. The four vertices of $\Delta$ come in two pairs which are very close together (as seen from the center of gravity of the simplex); let $P$ be an ideal quadrilateral whose edges join a point in one pair to a point in the other. The simplex $\Delta$ is bisected by a “square” of arbitrarily small area; together with four “cusps” (again, of arbitrarily small area) one makes a (topological) disk spanning $P$ with area as small as desired. Isotoping this disk rel. boundary to a least area (and therefore stable) representative can only decrease the area further. By the Gauss-Bonnet formula, the curvature of such a disk must get arbitrarily large (and negative) at some point in the interior.

Jeremy Kahn kindly sent me a more detailed overview of his argument with Vlad Markovic, that I blogged earlier about here (also see Jesse Johnson’s blog for other commentary). With his permission, this is reproduced below in its entirety.

Editorial note: I have latexified Jeremy’s email; hence “dhat-mu” becomes $\hat{d}\mu$, “boundary-hat” becomes $\hat{d}$, and “boundary-tilde” becomes $\tilde{d}$. I also linkified the link to Caroline Series’ paper.

Hi Danny,

I was busy with the conference on Thursday and Friday, and taking a break on Saturday, and now I’ve finally had a chance to read your blog, and reply to your message. I decided (especially as Jesse had requested it) to write out a complete outline of the theorem. I’m sending a copy of this message to you, Jesse Johnson, Ian Agol, and Francois Labourie: you are all welcome to reproduce it, as long as it is reproduced in its entirety, and states clearly that this is joint work with Vladimir Markovic. Of course, time and energy permitting, I’ll be happy to answer any questions.

Here is an outline of the argument, working backwards to make it clearer:

1. We want to construct a surface made out of skew pants, each of which has complex half-length close to $R$, and which are joined together so that the complex twist-bends are within $o(1/R)$ of $1$. Using a paper of Caroline
Series (published in the Pacific J. of Mathematics) we show that these surfaces are quasi-isometrically embedded in the universal cover of the three-manifold.

2. Consider the following two conditions on two Borel measures $\mu$ and $\nu$ on a metric space $X$ with the same (finite) total measure:

A. For every Borel subset $A$ of $X$, $\mu(A)$ is less than or equal to the $\nu$-measure of an $\epsilon$ neighborhood of $A$.

B. There is a measure space $(Y, \eta)$ and functions $f: Y \to X$ and $g: Y \to X$ such that $\mu$ and $\nu$ are the push-forwards by $f$ and $g$ respectively of the measure $\eta$, and the distance in $X$ between $f(y)$ and $g(y)$ is less than $\epsilon$ for almost every $y \in Y$.

It is easy to show that B implies A (also that A is symmetric in $\mu$ and $\nu$!). In the case where $\mu$ and $\nu$ are discrete and integral measures (the measure of every point is a non-negative integer), we can show that A implies B (and $Y$ will be a finite set with the counting measure) using Hall’s marriage theorem. In fact, the statement that A implies B for discrete and integral measures is easily shown to be equivalent to Hall’s marriage theorem. I don’t know if A implies B in general because I don’t know how to replace the inductive algorithm for Hall’s marriage theorem with a method that works for a relation between two general measure spaces.

We call $\mu$ and $\nu$ $\epsilon$-equivalent if they satisfy condition A, and note that the condition is additively transitive: if $\mu$ is $\epsilon$-equivalent to $\nu$, and $\nu$ is $\delta$-equivalent to $\rho$, then $\mu$ and $\rho$ are $(\epsilon+\delta)$-equivalent.

3. Suppose that $\gamma$ is one boundary component of a pair of skew pants $P$. We can form the common orthogonals in $P$ from $\gamma$ to each of other other two cuffs. For each common orthogonal, at the point where it meets $\gamma$, we can find a unit normal vector to $\gamma$ that points along this common orthogonal. The two resulting normal vectors are related by a translation along the half-length of $\gamma$ (the suitable square root of the loxodromic element for $\gamma$), so we will call them a pair of opposite unit normal vectors (or pounv for short) and they live in the live in the bundle of pounv’s which is conformally equivalent to the complex plane mod the lattice generated by the half-length of $\gamma$ and $2\pi i$. We give the bundle of pounv’s the Euclidean metric inherited from the complex plane, and also the Lebesgue measure.

4. Given a measure on pants we can produce a measure on the union pounv bundles of the boundary geodesics as follows: if the measure is a unit atom on one pair of skew pants, the resulting measure on pounv bundles is a unit atom on the pounv bundle of each the cuffs, at the pounv described in step 3. We extend to a general measure by linearity. This produces a linear operator we will call the $\hat{d}$ operator.

If we are given a positive integral formal sum of pants (or a multi-set of pants) we can think of it as an integral measure on the space of pants.

5. On the pounv bundle for each closed geodesic we can apply a translation of $1 + i \pi$; we will call this translation $\tau$. We can think of $\tau$ as a map from the union of the pounv bundles to itself.

6. Let $\mu$ be an integral measure on pants with cuff half-lengths close to $R$. We can apply the $\hat{d}$ operator described in step 4 to obtain a measure on the union of pounv bundles of all the boundary geodesics; we will call the measure $\hat{d}\mu$. If $\hat{d}\mu$ and the translation of $\hat{d}\mu$ by $\tau$ are $\epsilon/R$ equivalent, then we can take two oriented pants for each pair of pants in our multi-set (taking each of the two possible orientations) and then fit all of these oriented pants into an oriented surface of the type described in step 1. We use Hall’s marriage theorem as described in step 2, and a very small amount of combinatorics.

If the measure $\hat{d}\mu$, restricted to a given pounv bundle, is $\epsilon/R$ equivalent to a rescaling of Lebesgue measure on that torus, then $\hat{d}\mu$ and $\tau$ of $\hat{d}\mu$ are $2\epsilon/R$-equivalent, which is what we wanted.

******************

This is as far as I got in the first talk at Utah, so it would be best to stop and take a breath for a moment. We haven’t really done anything, but we’ve reformulated the problem: the type of surface we want has been well-defined, and the problem of finding this surface has been reformulated as finding a measure on pairs of pants that satisfies a given criterion.

*****************

7. A two-frame for $M$ will comprise a tangent vector and a normal vector both at the same point, unit length and orthogonal. Given a two-frame we can rotate the tangent vector 120 degrees around the normal vector, using the right-hand rule; the orbit of this action is an ordered triple of two-frames, which will call a tripod. We can also rotate 120 degrees in the opposite direction, and obtain an anti-tripod.

8. A connected pair of two-frames is a pair of two frames along with a geodesic segment connecting them. Given $\epsilon$ and $r$, with $r$ large in terms of $\epsilon$, we can find a weighting function on connected two-frames such that the following properties hold whenever the weight is non-zero:

A. The length of the connecting segment is within $\epsilon$ of $r$.

B. If the normal vector of one two-frame is parallel translated along the connecting segment, then it forms an angle of less then $\epsilon$ with the normal vector of the other two-frame.

C. The angle between the the tangent vector of the two frame and (the tangent vector to) the connecting geodesic segment is exponentially small in $r$.

Moreover,

D. Given a pair of two-frames, the sum of the weights of the connecting geodesic segments is exponentially close (in $r$) to 1.

E. The weighting is geometrically natural, in that it depends only the length of the connecting segment, the angle between the parallel translated normal vectors, and the angles between the connecting segment and the tangent vectors.

We will describe the (relatively simple) weighting function in the end; we will use the exponential mixing of geodesic flow to obtain property D.

9. Given a tripod and an anti-tripod, we can form three pairs of two-frames by pairing the frames in order, and then we can measures (or weightings) on the connected pairs of two-frames, and then form the product measure (or weighting) by multiplying the weights of the three connections. This gives us a weighting on “connected pairs of tripods” (really a tripod and an anti-tripod) that is supported on connections that satisfy properties A, B, and C.

10. We call a perfect connection between two two-frames a geodesic segment that has a length of $r$, and angle of zero between the segment and the tangent vectors, and translates one normal vector to the other. If a tripod and an anti-tripod were connected by three perfect connection, then they would be a 1-dimensional retract of a flat pair of pants with three cuffs of equal length $R$, where $R$ is approximately $r + \log \cos \pi/6$ when $r$ is large. If the tripod and anti-tripod are connected by arcs that satisfy properties A and B, then the connected pair of tripods is still a retract of a skew pair of pants, whose cuffs have half-length within $\epsilon$ (or $10\epsilon$) of $R$. Thus there is a map from good connected pairs of tripods to good pairs of pants, which we will denote by $\pi$.

11. We can let $\tilde{\mu}$ be the measure on connected pairs of tripods, given by integrating the weighting of steps 8 and 9 with respect to the Liouville measure on pairs of tripods (or pairs of two-frames). We then push this measure forward by $\pi$ to obtain a measure $\mu$ on pairs of pants; after finding a rational approximation and clearing denominators, it will be the $\mu$ that was asked for in step 6. We will show that $\hat{d}\mu$ (taking the original irrational $\mu$) is $\epsilon/R$-equivalent to a rescaling of Lebesgue measure on each pounv bundle and thereby complete the proof.

12. A partially connected pair of tripods $T$ is a pair of tripods where we have connected two out of the three pairs of two-frames. To a partially connected pair of tripods we can assign a single closed geodesic $\gamma$ that is homotopic to the concatenation (at both ends) of the two connecting segments. If we connect the third pair of two-frames and apply $\pi$ we obtain a pair of pants $P$, and we can then find a pair of opposite unit normal vectors for gamma pointing to the two cuffs of $P$ (as described in step 3). We will describe a method for predicting the pounv for $\gamma$ and $P$ knowing only the partially connected tripod $T$: First, lift $T$ to the solid torus cover of $M$ determined by $\gamma$, and then follow geodesic segments from the tangent vectors of the two unconnected two frames of (the lift of) $T$ to the ideal boundary of this $\gamma$-cover. We can connect these two points in the boundary by two geodesics, each of which goes about half-way around this solid torus cover. We can then find the common orthogonals from each of these geodesics to (the lift of) $\gamma$, and then obtain two normal vectors to $\gamma$ pointing along these common orthogonals; it is easy to verify that these are half-way along $\gamma$ from each other (in the complex sense) and hence form a pounv. Property C of the connections between two-frames (and hence tripods) implies that this predicted pounv will be exponentially close (in $r$) to the actually pounv of any pair of pants $P$.

To summarize: given a good connected pair of tripods, we get a good pair of pants $P$, and taking one cuff gamma of $P$, we get a pounv for $\gamma$ as described in step 3. But we only need two out of the three connecting segments to get $\gamma$, and using the third pair of two frames, without even knowing the third connecting segment, we can predict the pounv for $\gamma$ and $P$ to very high accuracy.

13. We can then define the $\tilde{d}$ operator from measures on partially connected pairs of tripods to measures on the pounv bundles for the associated geodesics; this operator is just the linear extension of the operation in step 12. Given a connected pair of tripods, we can get three partially connected pairs of tripods in the obvious way; we can thereby extend $\tilde{d}$ to map measures on connected pairs of tripods to measures on the bundles of pounv’s; because the predicted pounv described in step 12 is exponentially close to the actual pounv described in step 3, the two measures $\tilde{d} \tilde{\mu}$ and $\hat{d}\mu$ are $\exp(-\alpha r)$-equivalent, by the B => A of step 2.

14. For each closed geodesic $\gamma$, we can lift all the partially connected tripods that give $\gamma$ to the $\gamma$ cover of $M$ described in step 12. There is a natural torus action on the normal bundle of $\gamma$, and this extends to an action on all of the solid torus cover associated to $\gamma$. Moreover, it acts on the (lifts of) partially connected tripods, and it does not change the weightings of the two established connecting segments, because of property E of the weighting function.

This is the crucial point: the effective weighting on a partially connected pair of tripods is not just the product of the weights of the two established connections, but that product times the sum of the weights of all possible third connections. By property D of the weighting function, this sum, while not constant, is exponentially close to being constant, so the effective weighting is exponentially close to being invariant under the torus action. Because the predicted pounv for a partially connected pair of tripods is equivariant for the torus action, the measure $\tilde{d} \tilde{\mu}$ is exponentially close to a torus invariant measure on the pounv bundle (which is necessary a rescaling of Lebesgue measure), in the sense that the Radon-Nikodym derivative is exponentially close to 1. It is then an easy lemma that the two measures are exponentially close in the sense of step 2. And then we’re finished: $\hat{d}\mu$ is exponentially close to $\tilde{d} \tilde{\mu}$, which is exponentially close to a rescaling of Lebesgue measure, which is what we wanted (with
overkill) in step 6.

15. It remains only to define the weighting function described in step 8, which is surprisingly simple: We take some left-invariant metric on $\text{PSL}_2(\bf{C})$, and hence on the two-frame bundle for $M$ and its universal cover. Given a connected pair of two-frames in $M$, we lift to the universal cover, to obtain two two-frames $v$ and $w$. We then flow $v$ and $w$ forward by the frame flow for time $r/4$ to obtain $v'$ and $w'$. We let $V$ be the $\epsilon$ neighborhood of $v'$, and $W$ be the $\epsilon$ neighborhood of $w'$, with the tangent vector of $w'$ replaced by its negation. Then the weighting of the connection is the volume of the intersection of $W$ with the image of $V$ under the frame flow for time $r/2$.

Properties A, B, and C are not difficult to verify. Property D follows immediately from exponential mixing: If we have $v$ and $w$ downstairs without any connection, and similarly define $v'$, $w'$, $V$ and $W$, then the sum of the weights of the possible connections will just be the volume of the intersection of the downstairs $W$ with the frame flow of $V$. By exponential mixing, this converges at the rate $\exp(-\alpha r)$ to the square of the volume of an $\epsilon$ neighborhood, divided by the volume of $M$.

We can normalize the weights by dividing by this constant.

Jeremy

One obvious comment to make is that the argument is remarkably short, and does not depend on any very delicate or complicated analytic estimates (maybe the argument that the glued up surfaces are quasi-geodesic is the most delicate part). It is fair to say that it defies the conventional wisdom in that respect — I was personally very surprised that the general method could be made to work, especially in light of the failure of Bowen’s program. Kudos to Jeremy and Vlad for their boldness and ingenuity.

Another comment to make is that the matching argument is surprisingly robust and general, and I expect it to have many broader applications. One thing I was confused about in my last post seems to be resolved by Jeremy’s sketch above — if I understand it correctly, one first (almost) pairs continuous measures, and only then approximates them by discrete integral measures (with a little bit of combinatorics at the end). And one really does need exponential mixing rather than just mixing.

Incidentally, apropos the matching argument, there are some interesting and well-known variations where things go haywire. For example, papers by Burago-Kleiner and (Curt) McMullen show that there are examples of separated nets in Euclidean space which are not bilipschitz to a lattice (though, interestingly, Curt shows that they are Holder equivalent). No such examples exist in hyperbolic space, because of — nonamenability and Hall’s marriage theorem! Roughly, when trying to match up points in two nets in hyperbolic space, one doesn’t need to look very far because the number of options grows exponentially. This is one reason why Kahn-Markovic need to control the matchings of their measures carefully, because it must be done on a very small scale (where the exponential growth does not kick in).

I thought I would also mention that in case my previous comments lead one to believe otherwise, exponential mixing of the geodesic flow on a hyperbolic manifold is somewhat delicate. Exponential mixing under a flow $g_t$ on a space $X$ preserving a probability measure $\mu$ means that for all (sufficiently nice) functions $f$ and $h$ on $X$, the correlations $\rho(h,f,t):= \int_X h(x)f(g_tx) d\mu - \int_X h(x) d\mu \int_X f(x) d\mu$ are bounded in absolute value by an expression of the form $C_1e^{-tC_2}$ for suitable constants $C_1,C_2$ (which might depend on the analytic quality of $f$ and $h$). For example, one takes $X$ to be the unit tangent bundle of a hyperbolic manifold, and $g_t$ the geodesic flow (i.e. the flow which pushes vectors along the geodesics they are tangent to, at constant speed). Exponential mixing should be contrasted with the much slower mixing of the horocycle flow on a hyperbolic surface, for which the correlation is bounded by an expression like $C_1(\log t)^{C_2}t^{-1}$. The geodesic flow on a hyperbolic manifold is an example of what is called an Anosov flow; i.e. the tangent bundle $TM$ splits equivariantly under the flow into three subbundles $E^0, E^s, E^u$ where $E^0$ is $1$-dimensional and tangent to the flow, $E^s$ is contracted uniformly exponentially by the flow, and $E^u$ is expanded uniformly exponentially by the flow. The best one knows for (certain) Anosov flows (by Chernov) is that the flow is stretched exponentially mixing, i.e. with an estimate of the form $C_1e^{-\sqrt{t}C_2}$. One knows exponential mixing for the geodesic flow on variable negative curvature surfaces by Dolgopyat, and on certain locally symmetric spaces, using representation theory. See Pollicott’s lecture notes here for more details. I don’t know if exponential mixing for geodesic flows is known on manifolds of variable negative curvature in high dimensions. Also I’d appreciate it if any reader who knows some ergodic theory can confirm/deny/clarify this paragraph . . .

(Update 8/12): Jeremy tells me that he and Vladimir only need “sufficiently high degree polynomial” mixing, so perhaps there is a decent chance the methods can be extended to variable negative curvature.

(Update 10/29): The paper is now available from the arXiv.