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A couple of weeks ago, my student Yan Mary He presented a nice proof of Liouville’s theorem to me during our weekly meeting. The proof was the one from Benedetti-Petronio’s Lectures on Hyperbolic Geometry, which in my book gets lots of points for giving careful and complete details, and being self-contained and therefore accessible to beginning graduate students. Liouville’s Theorem is the fact that a conformal map between open subsets of Euclidean space of dimension at least 3 are Mobius transformations — i.e. they look locally like the restriction of a composition of Euclidean similarities and inversions on round spheres. This implies that the image of a piece of a plane or round sphere is a piece of a plane or round sphere, a highly rigid constraint. This sort of rigidity is in stark contrast to the case of conformal maps in dimension 2: any holomorphic (or antiholomorphic) map between open regions in the complex plane is a conformal map (and conversely). The proof given in Benedetti-Petronio is certainly clear and readable, and gives all the details; but Mary and I were a bit unsatisfied that it did not really provide any geometric insight into the meaning of the theorem. So the purpose of this blog post is to give a short sketch of a proof of Liouville’s theorem which is more geometric, and perhaps easier to remember.

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.

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:

The 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:

If $f:S \to \bf{R}^2$ 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.