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Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3-manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flow-invariant way into stable, unstable and flow directions) and contact (i.e. they preserve a contact form — that is, a 1-form \alpha for which \alpha \wedge d\alpha is a volume form). Their preprint gives some very interesting new constructions of such flows, obtained by surgery along a Legendrian knot (one tangent to the kernel of the contact form) which is transverse to the stable/unstable foliations of the Anosov flow.

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

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

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

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

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I’m in Melbourne right now, where I recently attended the Hyamfest and the preceding workshop. There were many excellent talks at both the workshop and the conference (more on that in another post), but one thing that I found very interesting is that both Michel Boileau and Cameron Gordon gave talks on the relationships between taut foliations, left-orderable groups, and L-spaces. I haven’t thought seriously about taut foliations in almost ten years, but the subject has been revitalized by its relationship to the theory of Heegaard Floer homology. The relationship tends to be one-way: the existence of a taut foliation on a manifold M implies that the Heegard Floer homology of M is nontrivial. It would be very interesting if Heegaard Floer homology could be used to decide whether a given manifold M admits a taut foliation or not, but for the moment this seems to be out of reach.

Anyway, both Michel and Cameron made use of the (by now 20 year old) classification of taut foliations on Seifert fibered 3-manifolds. The last step of this classification concerns the case when the base orbifold is a sphere; the precise answer was formulated in terms of a conjecture by Jankins and Neumann, proved by Naimi, about rotation numbers. I am ashamed to say that I never actually read Naimi’s argument, although it is not long. The point of this post is to give a new, short, combinatorial proof of the conjecture which I think is “conceptual” enough to digest easily.

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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 am in Melbourne at the moment, in the middle of giving a lecture series, as part of the 2009 Clay-Mahler lectures (also see here). Yesterday I gave a lecture with the title “faces of the scl norm ball”, and I thought I would try to give a sense of what it was all about. This also gives me an excuse to fiddle around with images in wordpress.

One starts with a basic question: given an immersion of a circle in the plane, when is there an immersion of the disk in the plane that bounds the given immersion of a circle? I.e., given a immersion \gamma:S^1 \to \bf{R}^2, when is there an immersion f:D^2 \to \bf{R}^2 for which \partial f factors through \gamma? Obviously this depends on \gamma. Consider the following examples:

immersed_circlesThe first immersed circle obviously bounds an immersed disk; in fact, an embedded disk.

The second circle does not bound such a disk. One way to see this is to use the Gauss map, i.e. the map \gamma'/|\gamma'|:S^1 \to S^1 that takes each point on the circle to the unit tangent to its image under the immersion. The degree of the Gauss map for an embedded circle is \pm 1 (depending on a choice of orientation). If an immersed circle bounds an immersed disk, one can use this immersed disk to define a 1-parameter family of immersions, connecting the initial immersed circle to an embedded immersed circle; hence the degree of the Gauss map is aso \pm 1 for an immersed circle bounding an immersed disk; this rules out the second example.

The third example maps under the Gauss map with degree 1, and yet it does not bound an immersed disk. One must use a slightly more sophisticated invariant to see this. The immersed circle divides the plane up into regions. For each bounded region R, let \alpha:[0,1] \to \bf{R}^2 be an embedded arc, transverse to \gamma, that starts in the region R and ends up “far away” (ideally “at infinity”). The arc \alpha determines a homological intersection number that we denote \alpha \cap \gamma, where each point of intersection contributes \pm 1 depending on orientations. In this example, there are three bounded regions, which get the numbers 1, -1, 1 respectively:

immersions2If f:S \to \bf{R}^2 is any map of any oriented surface with one boundary component whose boundary factors through \gamma, then the (homological) degree with which S maps over each region complementary to the image of \gamma is the number we have just defined. Hence if \gamma bounds an immersed disk, these numbers must all be positive (or all negative, if we reverse orientation). This rules out the third example.

The complete answer of which immersed circles in the plane bound immersed disks was given by S. Blank, in his Ph.D. thesis at Brandeis in 1967 (unfortunately, this does not appear to be available online). The answer is in the form of an algorithm to decide the question. One such algorithm (not Blank’s, but related to it) is as follows. The image of \gamma cuts up the plane into regions R_i, and each region R_i gets an integer n_i. Take n_i “copies” of each region R_i, and think of these as pieces of a jigsaw puzzle. Try to glue them together along their edges so that they fit together nicely along \gamma and make a disk with smooth boundary. If you are successful, you have constructed an immersion. If you are not successful (after trying all possible ways of gluing the puzzle pieces together), no such immersion exists. This answer is a bit unsatisfying, since in the first place it does not give any insight into which loops bound and which don’t, and in the second place the algorithm is quite slow and impractial.

As usual, more insight can be gained by generalizing the question. Fix a compact oriented surface \Sigma and consider an immersed 1-manifold \Gamma: \coprod_i S^1 \to \Sigma. One would like to know which such 1-manifolds bound an immersion of a surface. One piece of subtlety is the fact that there are examples where \Gamma itself does not bound, but a finite cover of \Gamma (e.g. two copies of \Gamma) does bound. It is also useful to restrict the class of 1-manifolds that one considers. For the sake of concreteness then, let \Sigma be a hyperbolic surface with geodesic boundary, and let \Gamma be an oriented immersed geodesic 1-manifold in \Sigma. An immersion f:S \to \Sigma is said to virtually bound \Gamma if the map \partial f factors as a composition \partial S \to \coprod_i S^1 \to \Sigma where the second map is \Gamma, and where the first map is a covering map with some degree n(S). The fundamental question, then is:

Question: Which immersed geodesic 1-manifolds \Gamma in \Sigma are virtually bounded by an immersed surface?

It turns out that this question is unexpectedly connected to stable commutator length, symplectic rigidity, and several other geometric issues; I hope to explain how in the remainder of this post.

First, recall that if G is any group and g \in [G,G], the commutator length of g, denoted \text{cl}(g), is the smallest number of commutators in G whose product is equal to g, and the stable commutator length \text{scl}(g) is the limit \text{scl}(g) = \lim_{n \to \infty} \text{cl}(g^n)/n. One can geometrize this definition as follows. Let X be a space with \pi_1(X) = G, and let \gamma:S^1 \to X be a homotopy class of loop representing the conjugacy class of g. Then \text{scl}(g) = \inf_S -\chi^-(S)/2n(S) over all surfaces S (possibly with multiple boundary components) mapping to X whose boundary wraps a total of n(S) times around \gamma. One can extend this definition to 1-manifolds \Gamma:\coprod_i S^1 \to X in the obvious way, and one gets a definition of stable commutator length for formal sums of elements in G which represent 0 in homology. Let B_1(G) denote the vector space of real finite linear combinations of elements in G whose sum represents zero in (real group) homology (i.e. in the abelianization of G, tensored with \bf{R}). Let H be the subspace spanned by chains of the form g^n - ng and g - hgh^{-1}. Then \text{scl} descends to a (pseudo)-norm on the quotient B_1(G)/H which we denote hereafter by B_1^H(G) (H for homogeneous).

There is a dual definition of this norm, in terms of quasimorphisms.

Definition: Let G be a group. A function \phi:G \to \bf{R} is a homogeneous quasimorphism if there is a least non-negative real number D(\phi) (called the defect) so that for all g,h \in G and n \in \bf{Z} one has

  1. \phi(g^n) = n\phi(g) (homogeneity)
  2. |\phi(gh) - \phi(g) - \phi(h)| \le D(\phi) (quasimorphism)

A function satisfying the second condition but not the first is an (ordinary) quasimorphism. The vector space of quasimorphisms on G is denoted \widehat{Q}(G), and the vector subspace of homogeneous quasimorphisms is denoted Q(G). Given \phi \in \widehat{Q}(G), one can homogenize it, by defining \overline{\phi}(g) = \lim_{n \to \infty} \phi(g^n)/n. Then \overline{\phi} \in Q(G) and D(\overline{\phi}) \le 2D(\phi). A quasimorphism has defect zero if and only if it is a homomorphism (i.e. an element of H^1(G)) and D(\cdot) makes the quotient Q/H^1 into a Banach space.

Examples of quasimorphisms include the following:

  1. Let F be a free group on a generating set S. Let \sigma be a reduced word in S^* and for each reduced word w \in S^*, define C_\sigma(w) to be the number of copies of \sigma in w. If \overline{w} denotes the corresponding element of F, define C_\sigma(\overline{w}) = C_\sigma(w) (note this is well-defined, since each element of a free group has a unique reduced representative). Then define H_\sigma = C_\sigma - C_{\sigma^{-1}}. This quasimorphism is not yet homogeneous, but can be homogenized as above (this example is due to Brooks).
  2. Let M be a closed hyperbolic manifold, and let \alpha be a 1-form. For each g \in \pi_1(M) let \gamma_g be the geodesic representative in the free homotopy class of g. Then define \phi_\alpha(g) = \int_{\gamma_g} \alpha. By Stokes’ theorem, and some basic hyperbolic geometry, \phi_\alpha is a homogeneous quasimorphism with defect at most 2\pi \|d\alpha\|.
  3. Let \rho: G \to \text{Homeo}^+(S^1) be an orientation-preserving action of G on a circle. The group of homeomorphisms of the circle has a natural central extension \text{Homeo}^+(\bf{R})^{\bf{Z}}, the group of homeomorphisms of \bf{R} that commute with integer translation. The preimage of G in this extension is an extension \widehat{G}. Given g \in \text{Homeo}^+(\bf{R})^{\bf{Z}}, define \text{rot}(g) = \lim_{n \to \infty} (g^n(0) - 0)/n; this descends to a \bf{R}/\bf{Z}-valued function on \text{Homeo}^+(S^1), Poincare’s so-called rotation number. But on \widehat{G}, this function is a homogeneous quasimorphism, typically with defect 1.
  4. Similarly, the group \text{Sp}(2n,\bf{R}) has a universal cover \widetilde{\text{Sp}}(2n,\bf{R}) with deck group \bf{Z}. The symplectic group acts on the space \Lambda_n of Lagrangian subspaces in \bf{R}^{2n}. This is equal to the coset space \Lambda_n = U(n)/O(n), and we can therefore define a function \text{det}^2:\Lambda_n \to S^1. After picking a basepoint, one obtains an S^1-valued function on the symplectic group, which lifts to a real-valued function on its universal cover. This function is a quasimorphism on the covering group, whose homogenization is sometimes called the symplectic rotation number; see e.g. Barge-Ghys.

Quasimorphisms and stable commutator length are related by Bavard Duality:

Theorem (Bavard duality): Let G be a group, and let \sum t_i g_i \in B_1^H(G). Then there is an equality \text{scl}(\sum t_i g_i) = \sup_\phi \sum t_i \phi(g_i)/2D(\phi) where the supremum is taken over all homogeneous quasimorphisms.

This duality theorem shows that Q/H^1 with the defect norm is the dual of B_1^H with the \text{scl} norm. (this theorem is proved for elements g \in [G,G] by Bavard, and in generality in my monograph, which is a reference for the content of this post.)

What does this have to do with rigidity (or, for that matter, immersions)? Well, one sees from the examples (and many others) that homogeneous quasimorphisms arise from geometry — specifically, from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures). One expects to find rigidity in extremal circumstances, and therefore one wants to understand, for a given chain C \in B_1^H(G), the set of extremal quasimorphisms for C, i.e. those homogeneous quasimorphisms \phi satisfying \text{scl}(C) = \phi(C)/2D(\phi). By the duality theorem, the space of such extremal quasimorphisms are a nonempty closed convex cone, dual to the set of hyperplanes in B_1^H that contain C/|C| and support the unit ball of the \text{scl} norm. The fewer supporting hyperplanes, the smaller the set of extremal quasimorphisms for C, and the more rigid such extremal quasimorphisms will be.

When F is a free group, the unit ball in the \text{scl} norm in B_1^H(F) is a rational polyhedron. Every nonzero chain C \in B_1^H(F) has a nonzero multiple C/|C| contained in the boundary of this polyhedron; let \pi_C denote the face of the polyhedron containing this multiple in its interior. The smaller the codimension of \pi_C, the smaller the dimension of the cone of extremal quasimorphisms for C, and the more rigidity we will see. The best circumstance is when \pi_C has codimension one, and an extremal quasimorphism for C is unique, up to scale, and elements of H^1.

An infinite dimensional polyhedron need not necessarily have any top dimensional faces; thus it is natural to ask: does the unit ball in B_1^H(F) have any top dimensional faces? and can one say anything about their geometric meaning? We have now done enough to motivate the following, which is the main theorem from my paper “Faces of the scl norm ball”:

Theorem: Let F be a free group. For every isomorphism F \to \pi_1(\Sigma) (up to conjugacy) where \Sigma is a compact oriented surface, there is a well-defined chain \partial \Sigma \in B_1^H(F). This satisfies the following properties:

  1. The projective class of \partial \Sigma intersects the interior of a codimension one face \pi_\Sigma of the \text{scl} norm ball
  2. The unique extremal quasimorphism dual to \pi_\Sigma (up to scale and elements of H^1) is the rotation quasimorphism \text{rot}_\Sigma (to be defined below) associated to any complete hyperbolic structure on \Sigma
  3. A homologically trivial geodesic 1-manifold \Gamma in \Sigma is virtually bounded by an immersed surface S in \Sigma if and only if the projective class of \Gamma (thought of as an element of B_1^H(F)) intersects \pi_\Sigma. Equivalently, if and only if \text{rot}_\Sigma is extremal for \Gamma. Equivalently, if and only if \text{scl}(\Gamma) = \text{rot}_\Sigma(\Gamma)/2.

It remains to give a definition of \text{rot}_\Sigma. In fact, we give two definitions.

First, a hyperbolic structure on \Sigma and the isomorphism F\to \pi_1(\Sigma) determines a representation F \to \text{PSL}(2,\bf{R}). This lifts to \widetilde{\text{SL}}(2,\bf{R}), since F is free. The composition with rotation number is a homogeneous quasimorphism on F, well-defined up to H^1. Note that because the image in \text{PSL}(2,\bf{R}) is discrete and torsion-free, this quasimorphism is integer valued (and has defect 1). This quasimorphism is \text{rot}_\Sigma.

Second, a geodesic 1-manifold \Gamma in \Sigma cuts the surface up into regions R_i. For each such region, let \alpha_i be an arc transverse to \Gamma, joining R_i to \partial \Sigma. Let (\alpha_i \cap \Gamma) denote the homological (signed) intersection number. Then define \text{rot}_\Sigma(\Gamma) = 1/2\pi \sum_i (\alpha_i \cap \Gamma) \text{area}(R_i).

We now show how 3 follows. Given \Gamma, we compute \text{scl}(\Gamma) = \inf_S -\chi^-(S)/2n(S) as above. Let S be such a surface, mapping to \Sigma. We adjust the map by a homotopy so that it is pleated; i.e. so that S is itself a hyperbolic surface, decomposed into ideal triangles, in such a way that the map is a (possibly orientation-reversing) isometry on each ideal triangle. By Gauss-Bonnet, we can calculate \text{area}(S) = -2\pi \chi^-(S) = \pi \sum_\Delta 1. On the other hand, \partial S wraps n(S) times around \Gamma (homologically) so \text{rot}_\Sigma(\Gamma) = \pi/2\pi n(S) \sum_\Delta \pm 1 where the sign in each case depends on whether the ideal triangle \Delta is mapped in with positive or negative orientation. Consequently \text{rot}_\Sigma(\Gamma)/2 \le -\chi^-(S)/2n(S) with equality if and only if the sign of every triangle is 1. This holds if and only if the map S \to \Sigma is an immersion; on the other hand, equality holds if and only if \text{rot}_\Sigma is extremal for \Gamma. This proves part 3 of the theorem above.

Incidentally, this fact gives a fast algorithm to determine whether \Gamma is the virtual boundary of an immersed surface. Stable commutator length in free groups can be computed in polynomial time in word length; likewise, the value of \text{rot}_\Sigma can be computed in polynomial time (see section 4.2 of my monograph for details). So one can determine whether \Gamma projectively intersects \pi_\Sigma, and therefore whether it is the virtual boundary of an immersed surface. In fact, these algorithms are quite practical, and run quickly (in a matter of seconds) on words of length 60 and longer in F_2.

One application to rigidity is a new proof of the following theorem:

Corollary (Goldman, Burger-Iozzi-Wienhard): Let \Sigma be a closed oriented surface of positive genus, and \rho:\pi_1(\Sigma) \to \text{Sp}(2n,\bf{R}) a Zariski dense representation. Let e_\rho \in H^2(\Sigma;\mathbb{Z}) be the Euler class associated to the action. Suppose that |e_\rho([\Sigma])| = -n\chi(\Sigma) (note: by a theorem of Domic and Toledo, one always has |e_\rho([\Sigma])| \le -n\chi(\Sigma)). Then \rho is discrete.

Here e_\rho is the first Chern class of the bundle associated to \rho. The proof is as follows: cut \Sigma along an essential loop \gamma into two subsurfaces \Sigma_i. One obtains homogeneous quasimorphisms on each group \pi_1(\Sigma_i) (i.e. the symplectic rotation number associated to \rho), and the hypothesis of the theorem easily implies that they are extremal for \partial \Sigma_i. Consequently the symplectic rotation number is equal to \text{rot}_{\Sigma_i}, at least on the commutator subgroup. But this latter quasimorphism takes only integral values; it follows that each element in \pi_1(\Sigma_i) fixes a Lagrangian subspace under \rho. But this implies that \rho is not dense, and since it is Zariski dense, it is discrete. (Notes: there are a couple of details under the rug here, but not many; furthermore, the hypothesis that \rho is Zariski dense is not necessary (but can be derived as a conclusion with more work), and one can just as easily treat representations of compact surface groups as closed ones; finally, Burger-Iozzi-Wienhard prove more than just this statement; for instance, they show that the space of maximal representations is always real semialgebraic, and describe it in some detail).

More abstractly, we have shown that extremal quasimorphisms on \partial \Sigma are unique. In other words, by prescribing the value of a quasimorphism on a single group element, one determines its values on the entire commutator subgroup. If such a quasimorphism arises from some geometric or dynamical context, this can be interpreted as a kind of rigidity theorem, of which the Corollary above is an example.

The development and scope of modern biology is often held out as a fantastic opportunity for mathematicians. The accumulation of vast amounts of biological data, and the development of new tools for the manipulation of biological organisms at microscopic levels and with unprecedented accuracy, invites the development of new mathematical tools for their analysis and exploitation. I know of several examples of mathematicians who have dipped a toe, or sometimes some more substantial organ, into the water. But it has struck me that I know (personally) few mathematicians who believe they have something substantial to learn from the biologists, despite the existence of several famous historical examples.  This strikes me as odd; my instinctive feeling has always been that intellectual ruts develop so easily, so deeply, and so invisibly, that continual cross-fertilization of ideas is essential to escape ossification (if I may mix biological metaphors . . .)

It is not necessarily easy to come up with profound examples of biological ideas or principles that can be easily translated into mathematical ones, but it is sometimes possible to come up with suggestive ones. Let me try to give a tentative example.

Deoxiribonucleic acid (DNA) is a nucleic acid that contains the genetic blueprint for all known living things. This blueprint takes the form of a code — a molecule of DNA is a long polymer strand composed of simple units called nucleotides; such a molecule is typically imagined as a string in a four character alphabet \lbrace A,T,G,C \rbrace, which stand for the nucleotides Adenine, Thymine, Guanine, and Cytosine. These molecular strands like to arrange themselves in tightly bound oppositely aligned pairs, matching up nucleotides in one string with complementary nucleotides in the other, so that A matches with T, and C with G.

The geometry of a strand of DNA is very complicated — strands can be tangled, knotted, linked in complicated ways, and the fundamental interactions between strands (e.g. transcription, recombination) are facilitated or obstructed by mechanical processes depending on this geometry. Topology, especially knot theory, has been used in the study of some of these processes; the value of topological methods in this context include their robustness (fault-tolerance) and the discreteness of their invariants (similar virtues motivate some efforts to build topological quantum computers). A complete mathematical description of the salient biochemistry, mechanics, and semantic content of a configuration of DNA in a single cell is an unrealistic goal for the foreseeable future, and therefore attempts to model such systems depends on ignoring, or treating statistically, certain features of the system. One such framework ignores the ambient geometry entirely, and treats the system using symbolic, or combinatorial methods which have some of the flavor of geometric group theory.

One interesting approach is to consider a mapping from the alphabet of nucleotides to a standard generating set for F_2, the free group on two generators; for example, one can take the mapping T \to a, A \to A, C \to b, G \to B where a,b are free generators for F_2, and {}A,B denote their inverses. Then a pair of oppositely aligned strands of DNA translates into an edge of a van Kampen diagram — the “words” obtained by reading the letters along an edge on either side are inverse in F_2.

Strands of DNA in a configuration are not always paired along their lengths; sometimes junctions of three or more strands can form; certain mobile four-strand junctions, so-called “Holliday junctions”, perform important functions in the process of genetic recombination, and are found in a wide variety of organisms. A configuration of several strands with junctions of varying valences corresponds in the language of van Kampen diagrams to a fatgraph — i.e. a graph together with a choice of cyclic ordering of edges at each vertex — with edges labeled by inverse pairs of words in F_2 (note that this is quite different from the fatgraph model of proteins developed by Penner-Knudsen-Wiuf-Andersen). The energy landscape for branch migration (i.e. the process by which DNA strands separate or join along some segment) is very complicated, and it is challenging to model it thermodynamically. It is therefore not easy to predict in advance what kinds of fatgraphs are more or less likely to arise spontaneously in a prepared “soup” of free DNA strands.

As a thought experiment, consider the following “toy” model, which I do not suggest is physically realistic. We make the assumption that the energy cost of forming a junction of valence v is c(v-2) for some fixed constant c. Consequently, the energy of a configuration is proportional to -\chi, i.e. the negative of Euler characteristic of the underlying graph. Let w be a reduced word, representing an element of F_2, and imagine a soup containing some large number of copies of the strand of DNA corresponding to the string \dot{w}:=\cdots www \cdots. In thermodynamic equilibrium, the partition function has the form Z = \sum_i e^{-E_i/k_BT} where k_B is Boltzmann’s constant, T is temperature, and E_i is the energy of a configuration (which by hypothesis is proportional to -\chi). At low temperature, minimal energy configurations tend to dominate; these are those that minimize -\chi per unit “volume”. Topologically, a fatgraph corresponding to such a configuration can be thickened to a surface with boundary. The words along the edges determine a homotopy class of map from such a surface to a K(F_2,1) (e.g. a once-punctured torus) whose boundary components wrap multiply around the free homotopy class corresponding to the conjugacy class of w. The infimum of -\chi/2d where d is the winding degree on the boundary, taken over all configurations, is precisely the stable commutator length of w; see e.g. here for a definition.

Anyway, this example is perhaps a bit strained (and maybe it owes more to thermodynamics than to biology), but already it suggests a new mathematical object of study, namely the partition function Z as above, and one is already inclined to look for examples for which the partition function obeys a symmetry like that enjoyed by the Riemann zeta function, or to specialize temperature to other values, as in random matrix theory. The introduction of new methods into the study of a classical object — for example, the decision to use thermodynamic methods to organize the study of van Kampen diagrams — bends the focus of the investigation towards those examples and contexts where the methods and tools are most informative. Phenomena familiar in one context (power laws, frequency locking, phase transitions etc.) suggest new questions and modes of enquiry in another. Uninspired or predictable research programs can benefit tremendously from such infusions, whether the new methods are borrowed from other intellectual disciplines (biology, physics), or depend on new technology (computers), or new methods of indexing (google) or collaboration (polymath).

One of my intellectual heroes — Wolfgang Haken — worked for eight years in R+D for Siemens in Munich after completing his PhD. I have a conceit (unsubstantiated as far as I know by biographical facts) that his experience working for a big engineering firm colored his approach to mathematics, and made it possible for him to imagine using industrial-scale “engineering” tools (e.g. integer programming, exhaustive computer search of combinatorial possibilities) to solve two of the most significant “pure” mathematical open problems in topology at the time — the knot recognition problem, and the four-color theorem. It is an interesting exercise to try to imagine (fantastic) variations. If I sit down and decide to try to prove (for example) Cannon’s conjecture, I am liable to try minor variations on things I have tried before, appeal for my intuition to examples that I understand well, read papers by others working in similar ways on the problem, etc. If I imagine that I have been given a billion dollars to prove the conjecture, I am almost certain to prioritize the task in different ways, and to entertain (and perhaps create) much more ambitious or innovative research programs to tackle the task. This is the way in which I understand the following quote by John Dewey, which I used as the colophon of my first book:

Every great advance in science has issued from a new audacity of the imagination.

I have struggled for a long time (and I continue to struggle) with the following question:

Question: Is the group of self-homeomorphisms of the unit disk (in the plane) that fix the boundary pointwise a left-orderable group?

Recall that a group G is left-orderable if there is a total order < on the elements satisfying g<h if and only if fg < fh for all f,g,h \in G. For a countable group, the property of being left orderable is equivalent to the property that the group admits a faithful action on the interval by orientation-preserving homeomorphisms; however, this equivalence is not “natural” in the sense that there is no natural way to extract an ordering from an action, or vice-versa. This formulation of the question suggests that one is trying to embed the group of homeomorphisms of the disk into the group of homeomorphisms of the interval, an unlikely proposition, made even more unlikely by the following famous theorem of Filipkiewicz:

Theorem: (Filipkiewicz) Let M_1,M_2 be two compact manifolds, and r_1,r_2 two non-negative integers or infinity. Suppose the connected components of the identity of \text{Diff}^{r_1}(M_1) and \text{Diff}^{r_2}(M_2) are isomorphic as abstract groups. Then r_1=r_2 and the isomorphism is induced by some diffeomorphism.

The hard(est?) part of the argument is to identify a subgroup stabilizing a point in purely algebraic terms. It is a fundamental and well-studied problem, in some ways a natural outgrowth of Klein’s Erlanger programme, to perceive the geometric structure on a space in terms of algebraic properties of its automorphism group. The book by Banyaga is the best reference I know for this material, in the context of “flexible” geometric structures, with big transformation groups (it is furthermore the only math book I know with a pink cover).

Left orderability is inherited under extensions. I.e. if K \to G \to H is a short exact sequence, and both K and H are left orderable, then so is G. Furthermore, it is a simple but useful theorem of Burns and Hale that a group G is left orderable if and only if for every finitely generated subgroup H there is a left orderable group H' and a surjective homomorphism H \to H'. The necessity of this condition is obvious: a subgroup of a left orderable group is left orderable (by restricting the order), so one can take H' to be H and the surjection to be the identity. One can exploit this strategy to show that certain transformation groups are left orderable, as follows:

Example: Suppose G is a group of homeomorphisms of some space X, with a nonempty fixed point set. If H is a finitely generated subgroup of G, then there is a point y in the frontier of \text{fix}(H) so that H has a nontrivial image in the group of germs of homeomorphisms of X at y. If this group of germs is left-orderable for all y, then so is G by Burns-Hale.

Example: (Rolfsen-Wiest) Let G be the group of PL homeomorphisms of the unit disk (thought of as a PL square in the plane) fixed on the boundary. If H is a finitely generated subgroup, there is a point p in the frontier of \text{fix}(H). Note that H has a nontrivial image in the group of piecewise linear homeomorphisms of the projective space of lines through p. Since the fixed point set of a finitely generated subgroup is equal to the intersection of the fixed point sets of a finite generating set, it is itself a polyhedron. Hence H fixes some line through p, and therefore has a nontrivial image in the group of homeomorphisms of an interval. By Burns-Hale, G is left orderable.

Example: Let G be the group of diffeomorphisms of the unit disk, fixed on the boundary. If H is a finitely generated subgroup, then at a non-isolated point p in \text{fix}(H) the group H fixes some tangent vector to p (a limit of short straight lines from p to nearby fixed points). Consequently the image of H in \text{GL}(T_p) is reducible, and is conjugate into an affine subgroup, which is left orderable. If the image is nontrivial, we are done by Burns-Hale. If it is trivial, then the linear part of H at p is trivial, and therefore by the Thurston stability theorem, there is a nontrivial homomorphism from H to the (orderable) group of translations of the plane. By Burns-Hale, we conclude that G is left orderable.

The second example does not require infinite differentiability, just C^1, the necessary hypothesis to apply the Thurston stability theorem. This is such a beautiful and powerful theorem that it is worth making an aside to discuss it. Thurston’s theorem says that if H is a finitely generated group of germs of diffeomorphisms of a manifold fixing a common point, then a suitable limit of rescaled actions of the group near the fixed point converge to a nontrivial action by translations. One way to think of this is in terms of power series: if H is a group of real analytic diffeomorphisms of the line, fixing the point 0, then every h \in H can be expanded as a power series: h(x) = c_1(h)x + c_2(h)x^2 + \cdots. The function h \to c_1(h) is a multiplicative homomorphism; however, if the logarithm of c_1 is identically zero, then if i is the first index for which some c_i(h) is nonzero, then h \to c_i(h) is an additive homomorphism. The choice of coefficient i is a “gauge”, adapted to H, that sees the most significant nontrivial part; this leading term is a character (i.e. a homomorphism to an abelian group), since the nonabelian corrections have strictly higher degree. Thurston’s insight was to realize that for a finitely generated group of germs of C^1 diffeomorphisms with trivial linear part, one can find some gauge that sees the most significant nontrivial part of the action of the group, and at this gauge, the action looks abelian. There is a subtlety, that one must restrict attention to finitely generated groups of homeomorphisms: on each scale of a sequence of finer and finer scales, one of a finite generating set differs the most from the identity; one must pass to a subsequence of scales for which this one element is constant (this is where the finite generation is used). The necessity of this condition is demonstrated by a theorem of Sergeraert: the group of germs of (C^\infty) diffeomorphisms of the unit interval, infinitely tangent to the identity at both endpoints (i.e. with trivial power series at each endpoint) is perfect, and therefore admits no nontrivial homomorphism to an abelian group.

Let us now return to the original question. The examples above suggest that it might be possible to find a left ordering on the group of homeomorphisms of the disk, fixed on the boundary. However, I think this is misleading. The construction of a left ordering in either category (PL or smooth) was ad hoc, and depended on locality in two different ways: the locality of the property of left orderability (i.e. Burns-Hale) and the tameness of groups of PL or smooth homeomorphisms blown up near a common fixed point. Rescaling an arbitrary homeomorphism about a fixed point does not make things any less complicated. Burns-Hale and Filipkiewicz together suggest that one should look for a structural dissimilarity between the group of homeomorphisms of the disk and of the interval that persists in finitely generated subgroups. The simplest way to distinguish between the two spaces algebraically is in terms of their lattices of closed (or equivalently, open) subsets. To a topological space X, one can associate the lattice \Lambda(X) of (nonempty, for the sake of argument) closed subsets of X, ordered by inclusion. One can reconstruct the space X from this lattice, since points in X correspond to minimal elements. However, any surjective map X \to Y defines an embedding \Lambda(Y) \to \Lambda(X), so there are many structure-preserving morphisms between such lattices. The lattice \Lambda(X) is an \text{Aut}(X)-space in an obvious way, and one can study algebraic maps \Lambda(Y) \to \Lambda(X) together with homomorphisms \rho:\text{Aut}(Y) \to \text{Aut}(X) for which the algebraic maps respect the induced \text{Aut}(Y)-structures. A weaker “localization” of this condition asks merely that for points (i.e. minimal elements) p,p' \in \Lambda(Y) in the same \text{Aut}(Y)-orbit, their images in \Lambda(X) are in the same \text{Aut}(X)-orbit. This motivates the following:

Proposition: There is a surjective map from the unit interval to the unit disk so that the preimages of any two points are homeomorphic.

Sketch of Proof: This proposition follows from two simpler propositions. The first is that there is a surjective map from the unit interval to itself so that every point preimage is a Cantor set. The second is that there is a surjective map from the unit interval to the unit disk so that the preimage of any point is finite. A composition of these two maps gives the desired map, since a finite union of Cantor sets is itself a Cantor set.

There are many surjective maps from the unit interval to the unit disk so that the preimage of any point is finite. For example, if M is a hyperbolic three-manifold fibering over the circle with fiber S, then the universal cover of a fiber \widetilde{S} is properly embedded in hyperbolic 3-space, and its ideal boundary (a circle) maps surjectively and finitely-to-one to the sphere at infinity of hyperbolic 3-space. Restricting to a suitable subinterval gives the desired map.

To obtain the first proposition, one builds a surjective map from the interval to itself inductively; there are many possible ways to do this, and details are left to the reader. qed.

It is not clear how much insight such a construction gives.

Another approach to the original question involves trying to construct an explicit (finitely generated) subgroup of the group of homeomorphisms of the disk that is not left orderable. There is a “cheap” method to produce finitely presented groups with no left-orderable quotients. Let G = \langle x,y \; | \; w_1, w_2 \rangle be a group defined by a presentation, where w_1 is a word in the letters x and y, and w_2 is a word in the letters x and y^{-1}. In any left-orderable quotient in which both x and y are nontrivial, after reversing the orientation if necessary, we can assume that x > \text{id}. If further y>\text{id} then w_1 >\text{id}, contrary to the fact that w_1 = \text{id}. If y^{-1} >\text{id}, then w_2 >\text{id}, contrary to the fact that w_2=\text{id}. In either case we get a contradiction. One can try to build by hand nontrivial homeomorphisms x,y of the unit disk, fixed on the boundary, that satisfy w_1,w_2 =\text{id}. Some evidence that this will be hard to do comes from the fact that the group of smooth and PL homeomorphisms of the disk are in fact left-orderable: any such x,y can be arbitrarily well-approximated by smooth x',y'; nevertheless at least one of the words w_1,w_2 evaluated on any smooth x',y' will be nontrivial. Other examples of finitely presented groups that are not left orderable include higher Q-rank lattices (e.g. subgroups of finite index in \text{SL}(n,\mathbb{Z}) when n\ge 3), by a result of Dave Witte-Morris. Suppose such a group admits a faithful action by homeomorphisms on some closed surface of genus at least 1. Since such groups do not admit homogeneous quasimorphisms, their image in the mapping class group of the surface is finite, so after passing to a subgroup of finite index, one obtains a (lifted) action on the universal cover. If the genus of the surface is at least 2, this action can be compactified to an action by homeomorphisms on the unit disk (thought of as the universal cover of a hyperbolic surface) fixed pointwise on the boundary. Fortunately or unfortunately, it is already known by Franks-Handel (see also Polterovich) that such groups admit no area-preserving actions on closed oriented surfaces (other than those factoring through a finite group), and it is consistent with the so-called “Zimmer program” that they should admit no actions even without the area-preserving hypothesis when the genus is positive (of course, \text{SL}(3,\mathbb{R}) admits a projective action on S^2). Actually, higher rank lattices are very fragile, because of Margulis’ normal subgroup theorem. Every normal subgroup of such a lattice is either finite or finite index, so to prove the results of Franks-Handel and Polterovich, it suffices to find a single element in the group of infinite order that acts trivially. Unipotent elements are exponentially distorted in the word metric (i.e. the cyclic subgroups they generate are not quasi-isometrically embedded), so one “just” needs to show that groups of area-preserving diffeomorphisms of closed surfaces (of genus at least 1) do not contain such distorted elements. Some naturally occurring non-left orderable groups include some (rare) hyperbolic 3-manifold groups, amenable but not locally indicable groups, and a few others. It is hard to construct actions of such groups on a disk, although certain flows on 3-manifolds give rise to actions of the fundamental group on a plane.

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