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Last week I was at Oberwolfach for a meeting on geometric group theory. My friend and collaborator Koji Fujiwara gave a very nice talk about constructing actions of groups on quasi-trees (i.e. spaces quasi-isometric to trees). The construction is inspired by the famous subsurface projection construction, due to Masur-Minsky, which was a key step in their proof that the complex of curves (a natural simplicial complex on which the mapping class group acts cocompactly) is hyperbolic. Koji’s talk was very stimulating, and shook up my thinking about a few related matters; the purpose of this blog post is therefore for me to put some of my thoughts in order: to describe the Masur-Minsky construction, to point out a connection to certain geometric phenomena like winding numbers of curves on surfaces, and to note that a variation on their construction gives rise directly to certain natural chiral invariants of surface automorphisms (and their generalizations) which should be relevant to 4-manifold topologists.

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On page 10 of Besse’s famous book on Einstein manifolds one finds the following quote:

It would seem that Riemannian and Lorentzian geometry have much in common: canonical connections, geodesics, curvature tensor, etc. . . . But in fact this common part is only a common disposition at the onset: one soon enters different realms.

I will not dispute this. But it is not clear to me whether this divergence is a necessary consequence of the nature of the objects of study (in either case), or an artefact of the schism between mathematics and physics during much of the 20th century. In any case, in this blog post I have the narrow aim of describing some points of contact between Lorentzian (and more generally, causal) geometry and other geometries (hyperbolic, symplectic), which plays a significant role in some of my research.

The first point of contact is the well-known duality between geodesics in the hyperbolic plane and points in the (projectivized) “anti de-Sitter plane”. Let \mathbb{R}^{2,1} denote a 3-dimensional vector space equipped with a quadratic form

q(x,y,z) = x^2 + y^2 - z^2

If we think of the set of rays through the origin as a copy of the real projective plane \mathbb{RP}^2, the hyperbolic plane is the set of projective classes of vectors v with q(v)<0, the (projectivized) anti de-Sitter plane is the set of projective classes of vectors v with q(v)>0, and their common boundary is the set of projective classes of (nonzero) vectors v with q(v)=0. Topologically, the hyperbolic plane is an open disk, the anti de-Sitter plane is an open Möbius band, and their boundary is the “ideal circle” (note: what people usually call the anti de-Sitter plane is actually the annulus double-covering this Möbius band; this is like the distinction between spherical geometry and elliptic geometry). Geometrically, the hyperbolic plane is a complete Riemannian surface of constant curvature -1, whereas the anti de-Sitter plane is a complete Lorentzian surface of constant curvature -1.

In this projective model, a hyperbolic geodesic \gamma is an open straight line segment which is compactified by adding an unordered pair of points in the ideal circle. The straight lines in the anti de-Sitter plane tangent to the ideal circle at these two points intersect at a point p_\gamma. Moreover, the set of geodesics \gamma in the hyperbolic plane passing through a point q are dual to the set of points p_\gamma in the anti de-Sitter plane that lie on a line which does not intersect the ideal circle. In the figure, three concurrent hyperbolic geodesics are dual to three colinear anti de-Sitter points.

The anti de-Sitter geometry has a natural causal structure. There is a cone field whose extremal vectors at every point p are tangent to the straight lines through p that are also tangent to the ideal circle. A smooth curve is timelike if its tangent at every point is supported by this cone field, and spacelike if its tangent is everywhere not supported by the cone field. A timelike curve corresponds to a family of hyperbolic geodesics which locally intersect each other; a spacelike curve corresponds to a family of disjoint hyperbolic geodesics that foliate some region.

One can distinguish (locally) between future and past along a timelike trajectory, by (arbitrarily) identifying the “future” direction with a curve which winds positively around the ideal circle. The fact that one can distinguish in a consistent way between the positive and negative direction is equivalent to the existence of a nonzero section of timelike vectors. On the other hand, there does not exist a nonzero section of spacelike vectors, so one cannot distinguish in a consistent way between left and right (this is a manifestation of the non-orientability of the Möbius band).

The duality between the hyperbolic plane and the anti de-Sitter plane is a manifestation of the fact that (at least at the level of Lie algebras) they have the same (infinitesimal) symmetries. Let O(2,1) denote the group of real 3\times 3 matrices which preserve q; i.e. matrices A for which q(A(v)) = q(v) for all vectors v. This contains a subgroup SO^+(2,1) of index 4 which preserves the “positive sheet” of the hyperboloid q=-1, and acts on it in an orientation-preserving way. The hyperbolic plane is the homogeneous space for this group whose point stabilizers are a copy of SO(2) (which acts as an elliptic “rotation” of the tangent space to their common fixed point). The anti de-Sitter plane is the homogeneous space for this group whose point stabilizers are a copy of SO^+(1,1) (which acts as a hyperbolic “translation” of the geodesic in hyperbolic space dual to the given point in anti de-Sitter space). The ideal circle is the homogeneous space whose point stabilizers are a copy of the affine group of the line. The hyperbolic plane admits a natural Riemannian metric, and the anti de-Sitter plane a Lorentz metric, which are invariant under these group actions. The causal structure on the anti de-Sitter plane limits to a causal structure on the ideal circle.

Now consider the 4-dimensional vector space \mathbb{R}^{2,2} and the quadratic form q(v) = x^2 + y^2 - z^2 - w^2. The (3-dimensional) sheets q=1 and q=-1 both admit homogeneous Lorentz metrics whose point stabilizers are copies of SO^+(1,2) and SO^+(2,1) (which are isomorphic but sit in SO(2,2) in different ways). These 3-manifolds are compactified by adding the projectivization of the cone q=0. Topologically, this is a Clifford torus in \mathbb{RP}^3 dividing this space into two open solid tori which can be thought of as two Lorentz 3-manifolds. The causal structure on the pair of Lorentz manifolds limits to a pair of complementary causal structures on the Clifford torus. (edited 12/10)

Let’s go one dimension higher, to the 5-dimensional vector space \mathbb{R}^{2,3} and the quadratic form q(v) = x^2 + y^2 - u^2 - z^2 - w^2. Now only the sheet q=1 is a Lorentz manifold, whose point stabilizers are copies of SO^+(1,3), with an associated causal structure. The projectivized cone q=0 is a non-orientable twisted S^2 bundle over the circle, and it inherits a causal structure in which the sphere factors are spacelike, and the circle direction is timelike. This ideal boundary can be thought of in quite a different way, because of the exceptional isomorphism at the level of (real) Lie algebras so(2,3)= sp(4), where sp(4) denotes the Lie algebra of the symplectic group in dimension 4. In this manifestation, the ideal boundary is usually denoted \mathcal{L}_2, and can be thought of as the space of Lagrangian planes in \mathbb{R}^4 with its usual symplectic form. One way to see this is as follows. The wedge product is a symmetric bilinear form on \Lambda^2 \mathbb{R}^4 with values in \Lambda^4 \mathbb{R}^4 = \mathbb{R}. The associated quadratic form vanishes precisely on the “pure” 2-forms — i.e. those associated to planes. The condition that the wedge of a given 2-form with the symplectic form vanishes imposes a further linear condition. So the space of Lagrangian 2-planes is a quadric in \mathbb{RP}^4, and one may verify that the signature of the underlying quadratic form is (2,3). The causal structure manifests in symplectic geometry in the following way. A choice of a Lagrangian plane \pi lets us identify symplectic \mathbb{R}^4 with the cotangent bundle T^*\pi. To each symmetric homogeneous quadratic form q on \pi (thought of as a smooth function) is associated a linear Lagrangian subspace of T^*\pi, namely the (linear) section dq. Every Lagrangian subspace transverse to the fiber over 0 is of this form, so this gives a parameterization of an open, dense subset of \mathcal{L}_2 containing the point \pi. The set of positive definite quadratic forms is tangent to an open cone in T_\pi \mathcal{L}_2; the field of such cones as \pi varies defines a causal structure on \mathcal{L}_2 which agrees with the causal structure defined above.

These examples can be generalized to higher dimension, via the orthogonal groups SO(n,2) or the symplectic groups Sp(2n,\mathbb{R}). As well as two other infinite families (which I will not discuss) there is a beautiful “sporadic” example, connected to what Freudenthal called octonion symplectic geometry associated to the noncompact real form E_7(-25) of the exceptional Lie group, where the ideal boundary S^1\times E_6/F_4 has an invariant causal structure whose timelike curves wind around the S^1 factor; see e.g. Clerc-Neeb for a more thorough discussion of the theory of Shilov boundaries from the causal geometry point of view, or see here or here for a discussion of the relationship between the octonions and the exceptional Lie groups.

The causal structure on these ideal boundaries gives rise to certain natural 2-cocycles on their groups of automorphisms. Note in each case that the ideal boundary has the topological structure of a bundle over S^1 with spacelike fibers. Thus each closed timelike curve has a well-defined winding number, which is just the number of times it intersects any one of these spacelike slices. Let C be an ideal boundary as above, and let \tilde{C} denote the cyclic cover dual to a spacelike slice. If p is a point in \tilde{C}, we let p+n denote the image of p under the nth power of the generator of the deck group of the covering. If g is a homeomorphism of C preserving the causal structure, we can lift g to a homeomorphism \tilde{g} of \tilde{C}. For any such lift, define the rotation number of \tilde{g} as follows: for any point p \in \tilde{C} and any integer n, let r_n be the the smallest integer for which there is a causal curve from p to \tilde{g}(p) to p+r_n, and then define rot(\tilde{g}) = \lim_{n \to \infty} r_n/n. This function is a quasimorphism on the group of causal automorphisms of \tilde{C}, with defect equal to the least integer n such that any two points p,q in C are contained in a closed causal loop with winding number n. In the case of the symplectic group Sp(2n,\mathbb{R}) with causal boundary \mathcal{L}_n, the defect is n, and the rotation number is (sometimes) called the symplectic rotation number; it is a quasimorphism on the universal central extension of Sp(2n,\mathbb{R}), whose coboundary descends to the Maslov class (an element of 2-dimensional bounded cohomology) on the symplectic group.

Causal structures in groups of symplectomorphisms or contactomorphisms are intensely studied; see for instance this paper by Eliashberg-Polterovich.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

Mapping class groups (also called modular groups) are of central importance in many fields of geometry. If S is an oriented surface (i.e. a 2-manifold), the group \text{Homeo}^+(S) of orientation-preserving self-homeomorphisms of S is a topological group with the compact-open topology. The mapping class group of S, denoted \text{MCG}(S) (or \text{Mod}(S) by some people) is the group of path-components of \text{Homeo}^+(S), i.e. \pi_0(\text{Homeo}^+(S)), or equivalently \text{Homeo}^+(S)/\text{Homeo}_0(S) where \text{Homeo}_0(S) is the subgroup of homeomorphisms isotopic to the identity.

When S is a surface of finite type (i.e. a closed surface minus finitely many points), the group \text{MCG}(S) is finitely presented, and one knows a great deal about the algebra and geometry of this group. Less well-studied are groups of the form \text{MCG}(S) when S is of infinite type. However, such groups do arise naturally in dynamics.

Example: Let G be a group of (orientation-preserving) homeomorphisms of the plane, and suppose that G has a bounded orbit (i.e. there is some point p for which the orbit Gp is contained in a compact subset of the plane). The closure of such an orbit Gp is compact and G-invariant. Let K be the union of the closure of Gp with the set of bounded open complementary regions. Then K is compact, G-invariant, and has connected complement. Define an equivalence relation \sim on the plane whose equivalence classes are the points in the complement of K, and the connected components of K. The quotient of the plane by this equivalence relation is again homeomorphic to the plane (by a theorem of R. L. Moore), and the image of K is a totally disconnected set k. The original group G admits a natural homomorphism to the mapping class group of \mathbb{R}^2 - k. After passing to a G-invariant closed subset of k if necessary, we may assume that k is minimal (i.e. every orbit is dense). Since k is compact, it is either a finite discrete set, or it is a Cantor set.

The mapping class group of \mathbb{R}^2 - \text{finite set} contains a subgroup of finite index fixing the end of \mathbb{R}^2; this subgroup is the quotient of a braid group by its center. There are many tools that show that certain groups G cannot have a big image in such a mapping class group.

Much less studied is the case that k is a Cantor set. In the remainder of this post, we will abbreviate \text{MCG}(\mathbb{R}^2 - \text{Cantor set}) by \Gamma. Notice that any homeomorphism of \mathbb{R}^2 - \text{Cantor set} extends in a unique way to a homeomorphism of S^2, fixing the point at infinity, and permuting the points of the Cantor set (this can be seen by thinking of the “missing points” intrinsically as the space of ends of the surface). Let \Gamma' denote the mapping class group of S^2 - \text{Cantor set}. Then there is a natural surjection \Gamma \to \Gamma' whose kernel is \pi_1(S^2 - \text{Cantor set}) (this is just the familiar Birman exact sequence).

The following is proved in the first section of my paper “Circular groups, planar groups and the Euler class”. This is the first step to showing that any group G of orientation-preserving diffeomorphisms of the plane with a bounded orbit is circularly orderable:

Proposition: There is an injective homomorphism \Gamma \to \text{Homeo}^+(S^1).

Sketch of Proof: Choose a complete hyperbolic structure on S^2 - \text{Cantor set}. The Birman exact sequence exhibits \Gamma as a group of (equivalence classes) of homeomorphisms of the universal cover of this hyperbolic surface which commute with the deck group. Each such homeomorphism extends in a unique way to a homeomorphism of the circle at infinity. This extension does not depend on the choice of a representative in an equivalence class, and one can check that the extension of a nontrivial mapping class is nontrivial at infinity. qed.

This property of the mapping class group \Gamma does not distinguish it from mapping class groups of surfaces of finite type (with punctures); in fact, the argument is barely sensitive to the topology of the surface at all. By contrast, the next theorem demonstrates a significant difference between mapping class groups of surfaces of finite type, and \Gamma. Recall that for a surface S of finite type, the group \text{MCG}(S) acts simplicially on the complex of curves \mathcal{C}(S), a simplicial complex whose simplices are the sets of isotopy classes of essential simple closed curves in S that can be realized mutually disjointly. A fundamental theorem of Masur-Minsky says that \mathcal{C}(S) (with its natural simplicial path metric) is \delta-hyperbolic (though it is not locally finite). Bestvina-Fujiwara show that any reasonably big subgroup of \text{MCG}(S) contains lots of elements that act on \mathcal{C}(S) weakly properly, and therefore such groups admit many nontrivial quasimorphisms. This has many important consequences, and shows that for many interesting classes of groups, every homomorphism to a mapping class group (of finite type) factors through a finite group. In view of the potential applications to dynamics as above, one would like to be able to construct quasimorphisms on mapping class groups of infinite type.

Unfortunately, this does not seem so easy.

Proposition: The group \Gamma' is uniformly perfect.

Proof: Remember that \Gamma' denotes the mapping class group of S^2 - \text{Cantor set}. We denote the Cantor set in the sequel by C.

A closed disk D is a dividing disk if its boundary is disjoint from C, and separates C into two components (both necessarily Cantor sets). An element g \in \Gamma is said to be local if it has a representative whose support is contained in a dividing disk. Note that the closure of the complement of a dividing disk is also a dividing disk. Given any dividing disk D, there is a homeomorphism of the sphere \varphi permuting C, that takes D off itself, and so that the family of disks \varphi^n(D) are pairwise disjoint, and converge to a limiting point x \in C. Define h to be the infinite product h = \prod_i \varphi^i g \varphi^{-i}. Notice that h is a well-defined homeomorphism of the plane permuting C. Moreover, there is an identity [h^{-1},\varphi] = g, thereby exhibiting g as a commutator. The theorem will therefore be proved if we can exhibit any element of \Gamma' as a bounded product of local elements.

Now, let g be an arbitrary homeomorphism of the sphere permuting C. Pick an arbitrary p \in C. If g(p)=p then let h be a local homeomorphism taking p to a disjoint point q, and define g' = hg. So without loss of generality, we can find g' = hg where h is local (possibly trivial), and g'(p) = q \ne p. Let {}E be a sufficiently small dividing disk containing p so that g'(E) is disjoint from {}E, and their union does not contain every point of C. Join {}E to g'(E) by a path in the complement of C, and let D be a regular neighborhood, which by construction is a dividing disk. Let f be a local homeomorphism, supported in D, that interchanges {}E and g'(E), and so that f g' is the identity on D. Then fg' is itself local, because the complement of the interior of a dividing disk is also a dividing disk, and we have expressed g as a product of at most three local homeomorphisms. This shows that the commutator length of g is at most 3, and since g was arbitrary, we are done. qed.

The same argument just barely fails to work with \Gamma in place of \Gamma'. One can also define dividing disks and local homeomorphisms in \Gamma, with the following important difference. One can show by the same argument that local homeomorphisms in \Gamma are commutators, and that for an arbitrary element g \in \Gamma there are local elements h,f so that fhg is the identity on a dividing disk; i.e. this composition is anti-local. However, the complement of the interior of a dividing disk in the plane is not a dividing disk; the difference can be measured by keeping track of the point at infinity. This is a restatement of the Birman exact sequence; at the level of quasimorphisms, one has the following exact sequence: Q(\Gamma') \to Q(\Gamma) \to Q(\pi_1(S^2 - C))^{\Gamma'}.

The so-called “point-pushing” subgroup \pi_1(S^2 - C) can be understood geometrically by tracking the image of a proper ray from C to infinity. We are therefore motivated to consider the following object:

Definition: The ray graph R is the graph whose vertex set is the set of isotopy classes of proper rays r, with interior in the complement of C, from a point in C to infinity, and whose edges are the pairs of such rays that can be realized disjointly.

One can verify that the graph R is connected, and that the group \Gamma acts simplicially on R by automorphisms, and transitively on vertices.

Lemma: Let g \in \Gamma and suppose there is a vertex v \in R such that v,g(v) share an edge. Then g is a product of at most two local homeomorphisms.

Sketch of proof: After adjusting g by an isotopy, assume that r and g(r) are actually disjoint. Let E,g(E) be sufficiently small disjoint disks about the endpoint of r and g(r), and \alpha an arc from {}E to g(E) disjoint from r and g(r), so that the union r \cup E \cup \alpha \cup g(E) \cup g(r) does not separate the part of C outside E \cup g(E). Then this union can be engulfed in a punctured disk D' containing infinity, whose complement contains some of C. There is a local h supported in a neighborhood of E \cup \alpha \cup g(E) such that hg is supported (after isotopy) in the complement of D' (i.e. it is also local). qed.

It follows that if g \in\Gamma has a bounded orbit in R, then the commutator lengths of the powers of g are bounded, and therefore \text{scl}(g) vanishes. If this is true for every g \in \Gamma, then Bavard duality implies that \Gamma admits no nontrivial homogeneous quasimorphisms. This motivates the following questions:

Question: Is the diameter of R infinite? (Exercise: show \text{diam}(R)\ge 3)

Question: Does any element of \Gamma act on R with positive translation length?

Question: Can one use this action to construct nontrivial quasimorphisms on \Gamma?

The purpose of this post is to discuss my recent paper with Koji Fujiwara, which will shortly appear in Ergodic Theory and Dynamical Systems, both for its own sake, and in order to motivate some open questions that I find very intriguing. The content of the paper is a mixture of ergodic theory, geometric group theory, and computer science, and was partly inspired by a paper of Jean-Claude Picaud. To state the results of the paper, I must first introduce a few definitions and some background.

Let \Gamma be a finite directed graph (hereafter a digraph) with an initial vertex, and edges labeled by elements of a finite set S in such a way that each vertex has at most one outgoing edge with any given label. A finite directed path in \Gamma starting at the initial vertex determines a word in the alphabet S, by reading the labels on the edges traversed (in order). The set L \subset S^* of words obtained in this way is an example of what is called a regular language, and is said to be parameterized by \Gamma. Note that this is not the most general kind of regular language; in particular, any language L of this kind will necessarily be prefix-closed (i.e. if w \in L then every prefix of w is also in L). Note also that different digraphs might parameterize the same (prefix-closed) regular language L.

If S is a set of generators for a group G, there is an obvious map L \to G called the evaluation map that takes a word w to the element of G represented by that word.

Definition: Let G be a group, and S a finite generating set. A combing of G is a (prefix-closed) regular language L for which the evaluation map L \to G is a bijection, and such that every w \in L represents a geodesic in G.

The intuition behind this definition is that the set of words in L determines a directed spanning tree in the Cayley graph C_S(G) starting at \text{id}, and such that every directed path in the tree is a geodesic in C_S(G). Note that there are other definitions of combing in the literature; for example, some authors do not require the evaluation map to be a bijection, but only a coarse bijection.

Fundamental to the theory of combings is the following Theorem, which paraphrases one of the main results of this paper:

Theorem: (Cannon) Let G be a hyperbolic group, and let S be a finite generating set. Choose a total order on the elements of S. Then the language L of lexicographically first geodesics in G is a combing.

The language L described in this theorem is obviously geodesic and prefix-closed, and the evaluation map is bijective; the content of the theorem is that L is regular, and parameterized by some finite digraph \Gamma. In the sequel, we restrict attention exclusively to hyperbolic groups G.

Given a (hyperbolic) group G, a generating set S, a combing L, one makes the following definition:

Definition: A function \phi:G \to \mathbb{Z} is weakly combable (with respect to S,L) if there is a digraph \Gamma parameterizing L and a function d\phi from the vertices of \Gamma to \mathbb{Z} so that for any w \in L, corresponding to a path \gamma in \Gamma, there is an equality \phi(w) = \sum_i d\phi(\gamma(i)).

In other words, a function \phi is weakly combable if it can be obtained by “integrating” a function d\phi along the paths of a combing. One furthermore says that a function is combable if it changes by a bounded amount under right-multiplication by an element of S, and bicombable if it changes by a bounded amount under either left or right multiplication by an element of S. The property of being (bi-)combable does not depend on the choice of a generating set S or a combing L.

Example: Word length (with respect to a given generating set S) is bicombable.

Example: Let \phi:G \to \mathbb{Z} be a homomorphism. Then \phi is bicombable.

Example: The Brooks counting quasimorphisms (on a free group) and the Epstein-Fujiwara counting quasimorphisms are bicombable.

Example: The sum or difference of two (bi-)combable functions is (bi-)combable.

A particularly interesting example is the following:

Example: Let S be a finite set which generates G as a semigroup. Let \phi_S denote word length with respect to S, and \phi_{S^{-1}} denote word length with respect to S^{-1} (which also generates G as a semigroup). Then the difference \psi_S:= \phi_S - \phi_{S^{-1}} is a bicombable quasimorphism.

The main theorem proved in the paper concerns the statistical distribution of values of a bicombable function.

Theorem: Let G be a hyperbolic group, and let \phi be a bicombable function on G. Let \overline{\phi}_n be the value of \phi on a random word in G of length n (with respect to a certain measure \widehat{\nu} depending on a choice of generating set). Then there are algebraic numbers E and \sigma so that as distributions, n^{-1/2}(\overline{\phi}_n - nE) converges to a normal distribution with standard deviation \sigma.

One interesting corollary concerns the length of typical words in one generating set versus another. The first thing that every geometric group theorist learns is that if S_1, S_2 are two finite generating sets for a group G, then there is a constant K so that every word of length n in one generating set has length at most nK and at least n/K in the other generating set. If one considers an example like \mathbb{Z}^2, one sees that this is the best possible estimate, even statistically. However, if one restricts attention to a hyperbolic group G, then one can do much better for typical words:

Corollary: Let G be hyperbolic, and let S_1,S_2 be two finite generating sets. There is an algebraic number \lambda_{1,2} so that almost all words of length n with respect to the S_1 generating set have length almost equal to n\lambda_{1,2} with respect to the S_2 generating set, with error of size O(\sqrt{n}).

Let me indicate very briefly how the proof of the theorem goes.

Sketch of Proof: Let \phi be bicombable, and let d\phi be a function from the vertices of \Gamma to \mathbb{Z}, where \Gamma is a digraph parameterizing L. There is a bijection between the set of elements in G of word length n and the set of directed paths in \Gamma of length n that start at the initial vertex. So to understand the distribution of \phi, we need to understand the behaviour of a typical long path in \Gamma.

Define a component of \Gamma to be a maximal subgraph with the property that there is a directed path (in the component) from any vertex to any other vertex. One can define a new digraph C(\Gamma) without loops, with one vertex for each component of \Gamma, in an obvious way. Each component C determines an adjacency matrix M_C, with ij-entry equal to 1 if there is a directed edge from vertex i to vertex j, and equal to 0 otherwise. A component C is big if the biggest real eigenvalue \lambda of M_C is at least as big as the biggest real eigenvalue of the matrices associated to every other component. A random long walk in \Gamma will spend most of its time entirely in big components, so these are the only components we need to consider to understand the statistical distribution of \phi.

A theorem of Coornaert implies that there are no big components of C(\Gamma) in series; i.e. there are no directed paths in C(\Gamma) from one big component to another (one also says that the big components do not communicate). This means that a typical long walk in \Gamma is entirely contained in a single big component, except for a (relatively short) path at the start and the end of the walk. So the distribution of \phi gets independent contributions, one from each big component.

The contribution from an individual big component is not hard to understand: the central limit theorem for stationary Markov chains says that for elements of G corresponding to paths that spend almost all their time in a given big component C there is a central limit theorem  n^{-1/2}(\overline{\phi}_n - nE_C) \to N(0,\sigma_C) where the mean E_C and standard deviation \sigma_C depend only on C. The problem is to show that the means and standard deviations associated to different big components are the same. Everything up to this point only depends on weak combability of \phi; to finish the proof one must use bicombability.

It is not hard to show that if \gamma is a typical infinite walk in a component C, then the subpaths of \gamma of length n are distributed like random walks of length n in C. What this means is that the mean and standard deviation E_C,\sigma_C associated to a big component C can be recovered from the distribution of \phi on a single infinite “typical” path in C. Such an infinite path corresponds to an infinite geodesic in G, converging to a definite point in the Gromov boundary \partial G. Another theorem of Coornaert (from the same paper) says that the action of G on its boundary \partial G is ergodic with respect to a certain natural measure called a Patterson-Sullivan measure (see Coornaert’s paper for details). This means that there are typical infinite geodesics \gamma,\gamma' associated to components C and C' for which some g \in G takes \gamma to a geodesic g\gamma ending at the same point in \partial G as \gamma'. Bicombability implies that the values of \phi on \gamma and g\gamma differ by a bounded amount. Moreover, since g\gamma and \gamma' are asymptotic to the same point at infinity, combability implies that the values of \phi on g\gamma and \gamma' also differ by a bounded amount. This is enough to deduce that E_C = E_{C'} and \sigma_C = \sigma_{C'}, and one obtains a (global) central limit theorem for \phi on G. qed.

This obviously raises several questions, some of which seem very hard, including:

Question 1: Let \phi be an arbitrary quasimorphism on a hyperbolic group G (even the case G is free is interesting). Does \phi satisfy a central limit theorem?

Question 2: Let \phi be an arbitrary quasimorphism on a hyperbolic group G. Does \phi satisfy a central limit theorem with respect to a random walk on G? (i.e. one considers the distribution of values of \phi not on the set of elements of G of word length n, but on the set of elements obtained by a random walk on G of length n, and lets n go to infinity)

All bicombable quasimorphisms satisfy an important property which is essential to our proof of the central limit theorem: they are local, which is to say, they are defined as a sum of local contributions. In the continuous world, they are the analogue of the so-called de Rham quasimorphisms on \pi_1(M) where M is a closed negatively curved Riemannian manifold; such quasimorphisms are defined by choosing a 1-form \alpha, and defining \phi_\alpha(g) to be equal to the integral \int_{\gamma_g} \alpha, where \gamma_g is the closed oriented based geodesic in M in the homotopy class of g. De Rham quasimorphisms, being local, also satisfy a central limit theorem.

This locality manifests itself in another way, in terms of defects. Let \phi be a quasimorphism on a hyperbolic group G. Recall that the defect D(\phi) is the supremum of |\phi(gh) - \phi(g) -\phi(h)| over all pairs of elements g,h \in G. A quasimorphism is further said to be homogeneous if \phi(g^n) = n\phi(g) for all integers n. If \phi is an arbitrary quasimorphism, one may homogenize it by taking a limit \psi(g) = \lim_{n \to \infty} \phi(g^n)/n; one says that \psi is the homogenization of \phi in this case. Homogenization typically does not preserve defects; however, there is an inequality D(\psi) \le 2D(\phi). If \phi is local, one expects this inequality to be an equality. For, in a hyperbolic group, the contribution to the defect of a local quasimorphism all arises from the interaction of the suffix of (a geodesic word representing the element) g with the prefix of h (with notation as above). When one homogenizes, one picks up another contribution to the defect from the interaction of the prefix of g with the suffix of h; since these two contributions are essentially independent, one expects that homogenizing a local quasimorphism should exactly double the defect. This is the case for bicombable and de Rham quasimorphisms, and can perhaps be used to define locality for a quasimorphism on an arbitrary group.

This discussion provokes the following key question:

Question 3: Let G be a group, and let \psi be a homogeneous quasimorphism. Is there a quasimorphism \phi with homogenization \psi, satisfying D(\psi) = 2D(\phi)?

Example: The answer to question 3 is “yes” if \psi is the rotation quasimorphism associated to an action of G on S^1 by orientation-preserving homeomorphisms (this is nontrivial; see Proposition 4.70 from my monograph).

Example: Let C be any homologically trivial group 1-boundary. Then there is some extremal homogeneous quasimorphism \psi for C (i.e. a quasimorphism achieving equality \text{scl}(C) = \psi(C)/2D(\psi) under generalized Bavard duality; see this post) for which there is \phi with homogenization \psi satisfying D(\psi) = 2D(\phi). Consequently, if every point in the boundary of the unit ball in the \text{scl} norm is contained in a unique supporting hyperplane, the answer to question 3 is “yes” for any quasimorphism on G.

Any quasimorphism on G can be pulled back to a quasimorphism on a free group, but this does not seem to make anything easier. In particular, question 3 is completely open (as far as I know) when G is a free group. An interesting test case might be the homogenization of an infinite sum of Brooks functions \sum_w h_w for some infinite non-nested family of words \lbrace w \rbrace.  

If the answer to this question is false, and one can find a homogeneous quasimorphism \psi which is not the homogenization of any “local” quasimorphism, then perhaps \psi does not satisfy a central limit theorem. One can try to approach this problem from the other direction:

Question 4: Given a function f defined on the ball of radius n in a free group F, one defines the defect D(f) in the usual way, restricted to pairs of elements g,h for which g,h,gh are all of length at most n. Under what conditions can f be extended to a function on the ball of radius n+1 without increasing the defect?

If one had a good procedure for building a quasimorphism “by hand” (so to speak), one could try to build a quasimorphism that failed to satisfy a central limit theorem, or perhaps find reasons why this was impossible.

A basic reference for the background to this post is my monograph.

Let G be a group, and let [G,G] denote the commutator subgroup. Every element of [G,G] can be expressed as a product of commutators; the commutator length of an element g is the minimum number of commutators necessary, and is denoted \text{cl}(g). The stable commutator length is the growth rate of the commutator lengths of powers of an element; i.e. \text{scl}(g) = \lim_{n \to \infty} \text{cl}(g^n)/n. Recall that a group G is said to satisfy a law if there is a nontrivial word w in a free group F for which every homomorphism from F to G sends w to \text{id}.

The purpose of this post is to give a very short proof of the following proposition (modulo some background that I wanted to talk about anyway):

Proposition: Suppose G obeys a law. Then the stable commutator length vanishes identically on [G,G].

The proof depends on a duality between stable commutator length and a certain class of functions, called homogeneous quasimorphisms

Definition: A function \phi:G \to \mathbb{R} is a quasimorphism if there is some least number D(\phi)\ge 0 (called the defect) so that for any pair of elements g,h \in G there is an inequality |\phi(x) + \phi(y) - \phi(xy)| \le D(\phi). A quasimorphism is homogeneous if it satisfies \phi(g^n) = n\phi(g) for all integers n.

Note that a homogeneous quasimorphism with defect zero is a homomorphism (to \mathbb{R}). The defect satisfies the following formula:

Lemma: Let f be a homogeneous quasimorphism. Then D(\phi) = \sup_{g,h} \phi([g,h]).

A fundamental theorem, due to Bavard, is the following:

Theorem: (Bavard duality) There is an equality \text{scl}(g) = \sup_\phi \frac {\phi(g)} {2D(\phi)} where the supremum is taken over all homogeneous quasimorphisms with nonzero defect.

In particular, \text{scl} vanishes identically on [G,G] if and only if every homogeneous quasimorphism on G is a homomorphism.

One final ingredient is another geometric definition of \text{scl} in terms of Euler characteristic. Let X be a space with \pi_1(X) = G, and let \gamma:S^1 \to X be a free homotopy class representing a given conjugacy class g. If S is a compact, oriented surface without sphere or disk components, a map f:S \to X is admissible if the map on \partial S factors through \partial f:\partial S \to S^1 \to X, where the second map is \gamma. For an admissible map, define n(S) by the equality [\partial S] \to n(S) [S^1] in H_1(S^1;\mathbb{Z}) (i.e. n(S) is the degree with which \partial S wraps around \gamma). With this notation, one has the following:

Lemma: There is an equality \text{scl}(g) = \inf_S \frac {-\chi^-(S)} {2n(S)}.

Note: the function -\chi^- is the sum of -\chi over non-disk and non-sphere components of S. By hypothesis, there are none, so we could just write -\chi. However, it is worth writing -\chi^- and observing that for more general (orientable) surfaces, this function is equal to the function \rho defined in a previous post.

We now give the proof of the Proposition.

Proof. Suppose to the contrary that stable commutator length does not vanish on [G,G]. By Bavard duality, there is a homogeneous quasimorphism \phi with nonzero defect. Rescale \phi to have defect 1. Then for any \epsilon there are elements g,h with \phi([g,h]) \ge 1-\epsilon, and consequently \text{scl}([g,h]) \ge 1/2 - \epsilon/2 by Bavard duality. On the other hand, if X is a space with \pi_1(X)=G, and \gamma:S^1 \to X is a loop representing the conjugacy class of [g,h], there is a map f:S \to X from a once-punctured torus S to X whose boundary represents \gamma. The fundamental group of S is free on two generators x,y which map to the class of g,h respectively. If w is a word in x,y mapping to the identity in G, there is an essential loop \alpha in S that maps inessentially to X. There is a finite cover \widetilde{S} of S, of degree d depending on the word length of w, for which \alpha lifts to an embedded loop. This can be compressed to give a surface S' with -\chi^-(S') \le -\chi^-(\widetilde{S})-2. However, Euler characteristic is multiplicative under coverings, so -\chi^-(\widetilde{S}) = -\chi^-(S)\cdot d. On the other hand, n(S') = n(\widetilde{S})=d so \text{scl}([g,h]) \le 1/2 - 1/d. If G obeys a law, then d is fixed, but \epsilon can be made arbitrarily small. So G does not obey a law. qed.


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