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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)


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.

Last week, Michael Brandenbursky from the Technion gave a talk at Caltech on an interesting connection between knot theory and quasimorphisms. Michael’s paper on this subject may be obtained from the arXiv. Recall that given a group G, a quasimorphism is a function \phi:G \to \mathbb{R} for which there is some least real number D(\phi) \ge 0 (called the defect) such that for all pairs of elements g,h \in G there is an inequality |\phi(gh) - \phi(g) - \phi(h)| \le D(\phi). Bounded functions are quasimorphisms, although in an uninteresting way, so one is usually only interested in quasimorphisms up to the equivalence relation that \phi \sim \psi if the difference |\phi - \psi| is bounded. It turns out that each equivalence class of quasimorphism contains a unique representative which has the extra property that \phi(g^n) = n\phi(g) for all g\in G and n \in \mathbb{Z}. Such quasimorphisms are said to be homogeneous. Any quasimorphism may be homogenized by defining \overline{\phi}(g) = \lim_{n \to \infty} \phi(g^n)/n (see e.g. this post for more about quasimorphisms, and their relation to stable commutator length).

Many groups that do not admit many homomorphisms to \mathbb{R} nevertheless admit rich families of homogeneous quasimorphisms. For example, groups that act weakly properly discontinuously on word-hyperbolic spaces admit infinite dimensional families of homogeneous quasimorphisms; see e.g. Bestvina-Fujiwara. This includes hyperbolic groups, but also mapping class groups and braid groups, which act on the complex of curves.

Michael discussed another source of quasimorphisms on braid groups, those coming from knot theory. Let I be a knot invariant. Then one can extend I to an invariant of pure braids on n strands by I(\alpha) = I(\widehat{\alpha \Delta}) where \Delta = \sigma_1 \cdots \sigma_{n-1}, and the “hat” denotes plat closure. It is an interesting question to ask: under what conditions on I is the resulting function on braid groups a quasimorphism?

In the abstract, such a question is probably very hard to answer, so one should narrow the question by concentrating on knot invariants of a certain kind. Since one wants the resulting invariants to have some relation to the algebraic structure of braid groups, it is natural to look for functions which factor through certain algebraic structures on knots; Michael was interested in certain homomorphisms from the knot concordance group to \mathbb{R}. We briefly describe this group, and a natural class of homomorphisms.

Two oriented knots K_1,K_2 in the 3-sphere are said to be concordant if there is a (locally flat) properly embedded annulus A in S^3 \times [0,1] with A \cap S^3 \times 0 = K_1 and A \cap S^3 \times 1 = K_2. Concordance is an equivalence relation, and the equivalence classes form a group, with connect sum as the group operation, and orientation-reversed mirror image as inverse. The only subtle aspect of this is the existence of inverses, which we briefly explain. Let K be an arbitrary knot, and let K^! denote the mirror image of K with the opposite orientation. Arrange K \cup K^! in space so that they are symmetric with respect to reflection in a dividing plane. There is an immersed annulus A in S^3 which connects each point on K to its mirror image on K^!, and the self-intersections of this annulus are all disjoint embedded arcs, corresponding to the crossings of K in the projection to the mirror. This annulus is an example of what is called a ribbon surface. Connect summing K to K^! by pushing out a finger of each into an arc in the mirror connects the ribbon annulus to a ribbon disk spanning K \# K^!. A ribbon surface (in particular, a ribbon disk) can be pushed into a (smoothly) embedded surface in a 4-ball bounding S^3. Puncturing the 4-ball at some point on this smooth surface, one obtains a concordance from K\#K^! to the unknot, as claimed.

The resulting group is known as the concordance group \mathcal{C} of knots. Since connect sum is commutative, this group is abelian. Notice as above that a slice knot — i.e. a knot bounding a locally flat disk in the 4-ball — is concordant to the unknot. Ribbon knots (those bounding ribbon disks) are smoothly slice, and therefore slice, and therefore concordant to the trivial knot. Concordance makes sense for codimension two knots in any dimension. In higher even dimensions, knots are always slice, and in higher odd dimensions, Levine found an algebraic description of the concordance groups in terms of (Witt) equivalence classes of linking pairings on a Seifert surface; (some of) this information is contained in the signature of a knot.

Let K be a knot (in S^3 for simplicity) with Seifert surface \Sigma of genus g. If \alpha,\beta are loops in \Sigma, define f(\alpha,\beta) to be the linking number of \alpha with \beta^+, which is obtained from \beta by pushing it to the positive side of \Sigma. The function f is a bilinear form on H_1(\Sigma), and after choosing generators, it can be expressed in terms of a matrix V (called the Seifert matrix of K). The signature of K, denoted \sigma(K), is the signature (in the usual sense) of the symmetric matrix V + V^T. Changing the orientation of a knot does not affect the signature, whereas taking mirror image multiplies it by -1. Moreover, if \Sigma_1,\Sigma_2 are Seifert surfaces for K_1,K_2, one can form a Seifert surface \Sigma for K_1 \# K_2 for which there is some sphere S^2 \in S^3 that intersects \Sigma in a separating arc, so that the pieces on either side of the sphere are isotopic to the \Sigma_i, and therefore the Seifert matrix of K_1 \# K_2 can be chosen to be block diagonal, with one block for each of the Seifert matrices of the K_i; it follows that \sigma(K_1 \# K_2) = \sigma(K_1) + \sigma(K_2). In fact it turns out that \sigma is a homomorphism from \mathcal{C} to \mathbb{Z}; equivalently (by the arguments above), it is zero on knots which are topologically slice. To see this, suppose K bounds a locally flat disk \Delta in the 4-ball. The union \Sigma':=\Sigma \cup \Delta is an embedded bicollared surface in the 4-ball, which bounds a 3-dimensional Seifert “surface” W whose interior may be taken to be disjoint from S^3. Now, it is a well-known fact that for any oriented 3-manifold W, the inclusion \partial W \to W induces a map H_1(\partial W) \to H_1(W) whose kernel is Lagrangian (with respect to the usual symplectic pairing on H_1 of an oriented surface). Geometrically, this means we can find a basis for the homology of \Sigma' (which is equal to the homology of \Sigma) for which half of the basis elements bound 2-chains in W. Let W^+ be obtained by pushing off W in the positive direction. Then chains in W and chains in W^+ are disjoint (since W and W^+ are disjoint) and therefore the Seifert matrix V of K has a block form for which the lower right g \times g block is identically zero. It follows that V+V^T also has a zero g\times g lower right block, and therefore its signature is zero.

The Seifert matrix (and therefore the signature), like the Alexander polynomial, is sensitive to the structure of the first homology of the universal abelian cover of S^3 - K; equivalently, to the structure of the maximal metabelian quotient of \pi_1(S^3 - K). More sophisticated “twisted” and L^2 signatures can be obtained by studying further derived subgroups of \pi_1(S^3 - K) as modules over group rings of certain solvable groups with torsion-free abelian factors (the so-called poly-torsion-free-abelian groups). This was accomplished by Cochran-Orr-Teichner, who used these methods to construct infinitely many new concordance invariants.

The end result of this discussion is the existence of many, many interesting homomorphisms from the knot concordance group to the reals, and by plat closure, many interesting invariants of braids. The connection with quasimorphisms is the following:

Theorem(Brandenbursky): A homomorphism I:\mathcal{C} \to \mathbb{R} gives rise to a quasimorphism on braid groups if there is a constant C so that |I([K])| \le C\cdot\|K\|_g, where \|\cdot\|_g denotes 4-ball genus.

The proof is roughly the following: given pure braids \alpha,\beta one forms the knots \widehat{\alpha\Delta}, \widehat{\beta\Delta} and \widehat{\alpha\beta\Delta}. It is shown that the connect sum L:= \widehat{\alpha \Delta} \# \widehat{\beta\Delta} \# \widehat{\alpha\beta\Delta}^! bounds a Seifert surface whose genus may be universally bounded in terms of the number of strands in the braid group. Pushing this Seifert surface into the 4-ball, the hypothesis of the theorem says that I is uniformly bounded on L. Properties of I then give an estimate for the defect; qed.

It would be interesting to connect these observations up to other “natural” chiral, homogeneous invariants on mapping class groups. For example, associated to a braid or mapping class \phi \in \text{MCG}(S) one can (usually) form a hyperbolic 3-manifold M_\phi which fibers over the circle, with fiber S and monodromy \phi. The \eta-invariant of M_\phi is the signature defect \eta(M_\phi) = \int_Y p_1/3 - \text{sign}(Y) where Y is a 4-manifold with \partial Y = M_\phi with a product metric near the boundary, and p_1 is the first Pontriagin form on Y (expressed in terms of the curvature of the metric). Is \eta a quasimorphism on some subgroup of \text{MCG}(S) (eg on a subgroup consisting entirely of pseudo-Anosov elements)?

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.)


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