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One of the nice things about living in Hyde Park is the proximity to the University of Chicago. Consequently, over the summer I came in to the department from time to time to work in my office, where I have all my math books, fast internet connection, etc. One day in early September (note: the Chicago quarter doesn’t start until October, so technically this was still “summer”) I happened to run in to Volodya Drinfeld in the hall, and he asked me what I knew about fundamental groups of (complex) projective varieties. I answered that I knew very little, but that what I did know (by hearsay) was that the most significant known restrictions on fundamental groups of projective varieties arise simply from the fact that such manifolds admit a Kähler structure, and that as far as anyone knows, the class of fundamental groups of projective varieties, and of Kähler manifolds, is the same.

Based on this brief interaction, Volodya asked me to give a talk on the subject in the Geometric Langlands seminar. On the face of it, this was a ridiculous request, in a department that contains Kevin Corlette and Madhav Nori, both of whom are world experts on the subject of fundamental groups of Kähler manifolds. But I agreed to the request, on the basis that I (at least!) would get a lot out of preparing for the talk, even if nobody else did.

Anyway, I ended up giving two talks for a total of about 5 hours in the seminar in successive weeks, and in the course of preparing for these talks I learned a lot about fundamental groups of Kähler manifolds. Most of the standard accounts of this material are aimed at people whose background is quite far from mine; so I thought it would be useful to describe, in a leisurely fashion, and in terms that I find more comfortable, some elements of this theory over the course of a few blog posts, starting with this one.

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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