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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 denote a -dimensional vector space equipped with a quadratic form
If we think of the set of rays through the origin as a copy of the real projective plane , the hyperbolic plane is the set of projective classes of vectors with , the (projectivized) anti de-Sitter plane is the set of projective classes of vectors with , and their common boundary is the set of projective classes of (nonzero) vectors with . 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 , whereas the anti de-Sitter plane is a complete Lorentzian surface of constant curvature .
In this projective model, a hyperbolic geodesic 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 . Moreover, the set of geodesics in the hyperbolic plane passing through a point are dual to the set of points 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 are tangent to the straight lines through 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 denote the group of real matrices which preserve ; i.e. matrices for which for all vectors . This contains a subgroup of index which preserves the “positive sheet” of the hyperboloid , 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 (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 (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 -dimensional vector space and the quadratic form . The (-dimensional) sheets and both admit homogeneous Lorentz metrics whose point stabilizers are copies of and (which are isomorphic but sit in in different ways). These -manifolds are compactified by adding the projectivization of the cone . Topologically, this is a Clifford torus in dividing this space into two open solid tori which can be thought of as two Lorentz -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 -dimensional vector space and the quadratic form . Now only the sheet is a Lorentz manifold, whose point stabilizers are copies of , with an associated causal structure. The projectivized cone is a non-orientable twisted 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 , where denotes the Lie algebra of the symplectic group in dimension . In this manifestation, the ideal boundary is usually denoted , and can be thought of as the space of Lagrangian planes in with its usual symplectic form. One way to see this is as follows. The wedge product is a symmetric bilinear form on with values in . The associated quadratic form vanishes precisely on the “pure” -forms — i.e. those associated to planes. The condition that the wedge of a given -form with the symplectic form vanishes imposes a further linear condition. So the space of Lagrangian -planes is a quadric in , and one may verify that the signature of the underlying quadratic form is . The causal structure manifests in symplectic geometry in the following way. A choice of a Lagrangian plane lets us identify symplectic with the cotangent bundle . To each symmetric homogeneous quadratic form on (thought of as a smooth function) is associated a linear Lagrangian subspace of , namely the (linear) section . Every Lagrangian subspace transverse to the fiber over is of this form, so this gives a parameterization of an open, dense subset of containing the point . The set of positive definite quadratic forms is tangent to an open cone in ; the field of such cones as varies defines a causal structure on which agrees with the causal structure defined above.
These examples can be generalized to higher dimension, via the orthogonal groups or the symplectic groups . 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 of the exceptional Lie group, where the ideal boundary has an invariant causal structure whose timelike curves wind around the 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 -cocycles on their groups of automorphisms. Note in each case that the ideal boundary has the topological structure of a bundle over 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 be an ideal boundary as above, and let denote the cyclic cover dual to a spacelike slice. If is a point in , we let denote the image of under the th power of the generator of the deck group of the covering. If is a homeomorphism of preserving the causal structure, we can lift to a homeomorphism of . For any such lift, define the rotation number of as follows: for any point and any integer , let be the the smallest integer for which there is a causal curve from to to , and then define . This function is a quasimorphism on the group of causal automorphisms of , with defect equal to the least integer such that any two points in are contained in a closed causal loop with winding number . In the case of the symplectic group with causal boundary , the defect is , and the rotation number is (sometimes) called the symplectic rotation number; it is a quasimorphism on the universal central extension of , whose coboundary descends to the Maslov class (an element of -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 , a quasimorphism is a function for which there is some least real number (called the defect) such that for all pairs of elements there is an inequality . Bounded functions are quasimorphisms, although in an uninteresting way, so one is usually only interested in quasimorphisms up to the equivalence relation that if the difference is bounded. It turns out that each equivalence class of quasimorphism contains a unique representative which has the extra property that for all and . Such quasimorphisms are said to be homogeneous. Any quasimorphism may be homogenized by defining (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 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 be a knot invariant. Then one can extend to an invariant of pure braids on strands by where , and the “hat” denotes plat closure. It is an interesting question to ask: under what conditions on 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 . We briefly describe this group, and a natural class of homomorphisms.
Two oriented knots in the -sphere are said to be concordant if there is a (locally flat) properly embedded annulus in with and . 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 be an arbitrary knot, and let denote the mirror image of with the opposite orientation. Arrange in space so that they are symmetric with respect to reflection in a dividing plane. There is an immersed annulus in which connects each point on to its mirror image on , and the self-intersections of this annulus are all disjoint embedded arcs, corresponding to the crossings of in the projection to the mirror. This annulus is an example of what is called a ribbon surface. Connect summing to by pushing out a finger of each into an arc in the mirror connects the ribbon annulus to a ribbon disk spanning . A ribbon surface (in particular, a ribbon disk) can be pushed into a (smoothly) embedded surface in a -ball bounding . Puncturing the -ball at some point on this smooth surface, one obtains a concordance from to the unknot, as claimed.
The resulting group is known as the concordance group 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 -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 be a knot (in for simplicity) with Seifert surface of genus . If are loops in , define to be the linking number of with , which is obtained from by pushing it to the positive side of . The function is a bilinear form on , and after choosing generators, it can be expressed in terms of a matrix (called the Seifert matrix of ). The signature of , denoted , is the signature (in the usual sense) of the symmetric matrix . Changing the orientation of a knot does not affect the signature, whereas taking mirror image multiplies it by . Moreover, if are Seifert surfaces for , one can form a Seifert surface for for which there is some sphere that intersects in a separating arc, so that the pieces on either side of the sphere are isotopic to the , and therefore the Seifert matrix of can be chosen to be block diagonal, with one block for each of the Seifert matrices of the ; it follows that . In fact it turns out that is a homomorphism from to ; equivalently (by the arguments above), it is zero on knots which are topologically slice. To see this, suppose bounds a locally flat disk in the -ball. The union is an embedded bicollared surface in the -ball, which bounds a -dimensional Seifert “surface” whose interior may be taken to be disjoint from . Now, it is a well-known fact that for any oriented -manifold , the inclusion induces a map whose kernel is Lagrangian (with respect to the usual symplectic pairing on of an oriented surface). Geometrically, this means we can find a basis for the homology of (which is equal to the homology of ) for which half of the basis elements bound -chains in . Let be obtained by pushing off in the positive direction. Then chains in and chains in are disjoint (since and are disjoint) and therefore the Seifert matrix of has a block form for which the lower right block is identically zero. It follows that also has a zero 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 ; equivalently, to the structure of the maximal metabelian quotient of . More sophisticated “twisted” and signatures can be obtained by studying further derived subgroups of 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 gives rise to a quasimorphism on braid groups if there is a constant so that , where denotes -ball genus.
The proof is roughly the following: given pure braids one forms the knots , and . It is shown that the connect sum 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 -ball, the hypothesis of the theorem says that is uniformly bounded on . Properties of 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 one can (usually) form a hyperbolic -manifold which fibers over the circle, with fiber and monodromy . The -invariant of is the signature defect where is a -manifold with with a product metric near the boundary, and is the first Pontriagin form on (expressed in terms of the curvature of the metric). Is a quasimorphism on some subgroup of (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 , when is there an immersion for which factors through ? Obviously this depends on . Consider the following examples:
The first immersed circle obviously bounds an immersed disk; in fact, an embedded disk.
The second circle does not bound such a disk. One way to see this is to use the Gauss map, i.e. the map 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 (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 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 , let be an embedded arc, transverse to , that starts in the region and ends up “far away” (ideally “at infinity”). The arc determines a homological intersection number that we denote , where each point of intersection contributes depending on orientations. In this example, there are three bounded regions, which get the numbers , , respectively:
If is any map of any oriented surface with one boundary component whose boundary factors through , then the (homological) degree with which maps over each region complementary to the image of is the number we have just defined. Hence if 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 cuts up the plane into regions , and each region gets an integer . Take “copies” of each region , 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 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 and consider an immersed -manifold . One would like to know which such -manifolds bound an immersion of a surface. One piece of subtlety is the fact that there are examples where itself does not bound, but a finite cover of (e.g. two copies of ) does bound. It is also useful to restrict the class of -manifolds that one considers. For the sake of concreteness then, let be a hyperbolic surface with geodesic boundary, and let be an oriented immersed geodesic -manifold in . An immersion is said to virtually bound if the map factors as a composition where the second map is , and where the first map is a covering map with some degree . The fundamental question, then is:
Question: Which immersed geodesic -manifolds in 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 is any group and , the commutator length of , denoted , is the smallest number of commutators in whose product is equal to , and the stable commutator length is the limit . One can geometrize this definition as follows. Let be a space with , and let be a homotopy class of loop representing the conjugacy class of . Then over all surfaces (possibly with multiple boundary components) mapping to whose boundary wraps a total of times around . One can extend this definition to -manifolds in the obvious way, and one gets a definition of stable commutator length for formal sums of elements in which represent in homology. Let denote the vector space of real finite linear combinations of elements in whose sum represents zero in (real group) homology (i.e. in the abelianization of , tensored with ). Let be the subspace spanned by chains of the form and . Then descends to a (pseudo)-norm on the quotient which we denote hereafter by ( for homogeneous).
There is a dual definition of this norm, in terms of quasimorphisms.
Definition: Let be a group. A function is a homogeneous quasimorphism if there is a least non-negative real number (called the defect) so that for all and one has
A function satisfying the second condition but not the first is an (ordinary) quasimorphism. The vector space of quasimorphisms on is denoted , and the vector subspace of homogeneous quasimorphisms is denoted . Given , one can homogenize it, by defining . Then and . A quasimorphism has defect zero if and only if it is a homomorphism (i.e. an element of ) and makes the quotient into a Banach space.
Examples of quasimorphisms include the following:
- Let be a free group on a generating set . Let be a reduced word in and for each reduced word , define to be the number of copies of in . If denotes the corresponding element of , define (note this is well-defined, since each element of a free group has a unique reduced representative). Then define . This quasimorphism is not yet homogeneous, but can be homogenized as above (this example is due to Brooks).
- Let be a closed hyperbolic manifold, and let be a -form. For each let be the geodesic representative in the free homotopy class of . Then define . By Stokes’ theorem, and some basic hyperbolic geometry, is a homogeneous quasimorphism with defect at most .
- Let be an orientation-preserving action of on a circle. The group of homeomorphisms of the circle has a natural central extension , the group of homeomorphisms of that commute with integer translation. The preimage of in this extension is an extension . Given , define ; this descends to a -valued function on , Poincare’s so-called rotation number. But on , this function is a homogeneous quasimorphism, typically with defect .
- Similarly, the group has a universal cover with deck group . The symplectic group acts on the space of Lagrangian subspaces in . This is equal to the coset space , and we can therefore define a function . After picking a basepoint, one obtains an -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 be a group, and let . Then there is an equality where the supremum is taken over all homogeneous quasimorphisms.
This duality theorem shows that with the defect norm is the dual of with the norm. (this theorem is proved for elements 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 , the set of extremal quasimorphisms for , i.e. those homogeneous quasimorphisms satisfying . By the duality theorem, the space of such extremal quasimorphisms are a nonempty closed convex cone, dual to the set of hyperplanes in that contain and support the unit ball of the norm. The fewer supporting hyperplanes, the smaller the set of extremal quasimorphisms for , and the more rigid such extremal quasimorphisms will be.
When is a free group, the unit ball in the norm in is a rational polyhedron. Every nonzero chain has a nonzero multiple contained in the boundary of this polyhedron; let denote the face of the polyhedron containing this multiple in its interior. The smaller the codimension of , the smaller the dimension of the cone of extremal quasimorphisms for , and the more rigidity we will see. The best circumstance is when has codimension one, and an extremal quasimorphism for is unique, up to scale, and elements of .
An infinite dimensional polyhedron need not necessarily have any top dimensional faces; thus it is natural to ask: does the unit ball in 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 be a free group. For every isomorphism (up to conjugacy) where is a compact oriented surface, there is a well-defined chain . This satisfies the following properties:
- The projective class of intersects the interior of a codimension one face of the norm ball
- The unique extremal quasimorphism dual to (up to scale and elements of ) is the rotation quasimorphism (to be defined below) associated to any complete hyperbolic structure on
- A homologically trivial geodesic -manifold in is virtually bounded by an immersed surface in if and only if the projective class of (thought of as an element of ) intersects . Equivalently, if and only if is extremal for . Equivalently, if and only if .
It remains to give a definition of . In fact, we give two definitions.
First, a hyperbolic structure on and the isomorphism determines a representation . This lifts to , since is free. The composition with rotation number is a homogeneous quasimorphism on , well-defined up to . Note that because the image in is discrete and torsion-free, this quasimorphism is integer valued (and has defect ). This quasimorphism is .
Second, a geodesic -manifold in cuts the surface up into regions . For each such region, let be an arc transverse to , joining to . Let denote the homological (signed) intersection number. Then define .
We now show how 3 follows. Given , we compute as above. Let be such a surface, mapping to . We adjust the map by a homotopy so that it is pleated; i.e. so that 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 . On the other hand, wraps times around (homologically) so where the sign in each case depends on whether the ideal triangle is mapped in with positive or negative orientation. Consequently with equality if and only if the sign of every triangle is . This holds if and only if the map is an immersion; on the other hand, equality holds if and only if is extremal for . This proves part 3 of the theorem above.
Incidentally, this fact gives a fast algorithm to determine whether 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 can be computed in polynomial time (see section 4.2 of my monograph for details). So one can determine whether projectively intersects , 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 .
One application to rigidity is a new proof of the following theorem:
Corollary (Goldman, Burger-Iozzi-Wienhard): Let be a closed oriented surface of positive genus, and a Zariski dense representation. Let be the Euler class associated to the action. Suppose that (note: by a theorem of Domic and Toledo, one always has ). Then is discrete.
Here is the first Chern class of the bundle associated to . The proof is as follows: cut along an essential loop into two subsurfaces . One obtains homogeneous quasimorphisms on each group (i.e. the symplectic rotation number associated to ), and the hypothesis of the theorem easily implies that they are extremal for . Consequently the symplectic rotation number is equal to , at least on the commutator subgroup. But this latter quasimorphism takes only integral values; it follows that each element in fixes a Lagrangian subspace under . But this implies that 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 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 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 is left-orderable if there is a total order on the elements satisfying if and only if for all . 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 be two compact manifolds, and two non-negative integers or infinity. Suppose the connected components of the identity of and are isomorphic as abstract groups. Then 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 is a short exact sequence, and both and are left orderable, then so is . Furthermore, it is a simple but useful theorem of Burns and Hale that a group is left orderable if and only if for every finitely generated subgroup there is a left orderable group and a surjective homomorphism . 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 to be 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 is a group of homeomorphisms of some space , with a nonempty fixed point set. If is a finitely generated subgroup of , then there is a point in the frontier of so that has a nontrivial image in the group of germs of homeomorphisms of at . If this group of germs is left-orderable for all , then so is by Burns-Hale.
Example: (Rolfsen-Wiest) Let be the group of PL homeomorphisms of the unit disk (thought of as a PL square in the plane) fixed on the boundary. If is a finitely generated subgroup, there is a point in the frontier of . Note that has a nontrivial image in the group of piecewise linear homeomorphisms of the projective space of lines through . 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 fixes some line through , and therefore has a nontrivial image in the group of homeomorphisms of an interval. By Burns-Hale, is left orderable.
Example: Let be the group of diffeomorphisms of the unit disk, fixed on the boundary. If is a finitely generated subgroup, then at a non-isolated point in the group fixes some tangent vector to (a limit of short straight lines from to nearby fixed points). Consequently the image of in 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 at is trivial, and therefore by the Thurston stability theorem, there is a nontrivial homomorphism from to the (orderable) group of translations of the plane. By Burns-Hale, we conclude that is left orderable.
The second example does not require infinite differentiability, just , 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 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 is a group of real analytic diffeomorphisms of the line, fixing the point , then every can be expanded as a power series: . The function is a multiplicative homomorphism; however, if the logarithm of is identically zero, then if is the first index for which some is nonzero, then is an additive homomorphism. The choice of coefficient is a “gauge”, adapted to , 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 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 () 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 , one can associate the lattice of (nonempty, for the sake of argument) closed subsets of , ordered by inclusion. One can reconstruct the space from this lattice, since points in correspond to minimal elements. However, any surjective map defines an embedding , so there are many structure-preserving morphisms between such lattices. The lattice is an -space in an obvious way, and one can study algebraic maps together with homomorphisms for which the algebraic maps respect the induced -structures. A weaker “localization” of this condition asks merely that for points (i.e. minimal elements) in the same -orbit, their images in are in the same -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 is a hyperbolic three-manifold fibering over the circle with fiber , then the universal cover of a fiber is properly embedded in hyperbolic -space, and its ideal boundary (a circle) maps surjectively and finitely-to-one to the sphere at infinity of hyperbolic -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 be a group defined by a presentation, where is a word in the letters and , and is a word in the letters and . In any left-orderable quotient in which both and are nontrivial, after reversing the orientation if necessary, we can assume that . If further then , contrary to the fact that . If , then , contrary to the fact that . In either case we get a contradiction. One can try to build by hand nontrivial homeomorphisms of the unit disk, fixed on the boundary, that satisfy . 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 can be arbitrarily well-approximated by smooth ; nevertheless at least one of the words evaluated on any smooth 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 when ), 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 . 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 , 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, admits a projective action on ). 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 ) do not contain such distorted elements. Some naturally occurring non-left orderable groups include some (rare) hyperbolic -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 -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 is an oriented surface (i.e. a -manifold), the group of orientation-preserving self-homeomorphisms of is a topological group with the compact-open topology. The mapping class group of , denoted (or by some people) is the group of path-components of , i.e. , or equivalently where is the subgroup of homeomorphisms isotopic to the identity.
When is a surface of finite type (i.e. a closed surface minus finitely many points), the group 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 when is of infinite type. However, such groups do arise naturally in dynamics.
Example: Let be a group of (orientation-preserving) homeomorphisms of the plane, and suppose that has a bounded orbit (i.e. there is some point for which the orbit is contained in a compact subset of the plane). The closure of such an orbit is compact and -invariant. Let be the union of the closure of with the set of bounded open complementary regions. Then is compact, -invariant, and has connected complement. Define an equivalence relation on the plane whose equivalence classes are the points in the complement of , and the connected components of . 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 is a totally disconnected set . The original group admits a natural homomorphism to the mapping class group of . After passing to a -invariant closed subset of if necessary, we may assume that is minimal (i.e. every orbit is dense). Since is compact, it is either a finite discrete set, or it is a Cantor set.
The mapping class group of contains a subgroup of finite index fixing the end of ; this subgroup is the quotient of a braid group by its center. There are many tools that show that certain groups cannot have a big image in such a mapping class group.
Much less studied is the case that is a Cantor set. In the remainder of this post, we will abbreviate by . Notice that any homeomorphism of extends in a unique way to a homeomorphism of , 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 denote the mapping class group of . Then there is a natural surjection whose kernel is (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 of orientation-preserving diffeomorphisms of the plane with a bounded orbit is circularly orderable:
Proposition: There is an injective homomorphism .
Sketch of Proof: Choose a complete hyperbolic structure on . The Birman exact sequence exhibits 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 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 . Recall that for a surface of finite type, the group acts simplicially on the complex of curves , a simplicial complex whose simplices are the sets of isotopy classes of essential simple closed curves in that can be realized mutually disjointly. A fundamental theorem of Masur-Minsky says that (with its natural simplicial path metric) is -hyperbolic (though it is not locally finite). Bestvina-Fujiwara show that any reasonably big subgroup of contains lots of elements that act on 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 is uniformly perfect.
Proof: Remember that denotes the mapping class group of . We denote the Cantor set in the sequel by .
A closed disk is a dividing disk if its boundary is disjoint from , and separates into two components (both necessarily Cantor sets). An element 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 , there is a homeomorphism of the sphere permuting , that takes off itself, and so that the family of disks are pairwise disjoint, and converge to a limiting point . Define to be the infinite product . Notice that is a well-defined homeomorphism of the plane permuting . Moreover, there is an identity , thereby exhibiting as a commutator. The theorem will therefore be proved if we can exhibit any element of as a bounded product of local elements.
Now, let be an arbitrary homeomorphism of the sphere permuting . Pick an arbitrary . If then let be a local homeomorphism taking to a disjoint point , and define . So without loss of generality, we can find where is local (possibly trivial), and . Let be a sufficiently small dividing disk containing so that is disjoint from , and their union does not contain every point of . Join to by a path in the complement of , and let be a regular neighborhood, which by construction is a dividing disk. Let be a local homeomorphism, supported in , that interchanges and , and so that is the identity on . Then is itself local, because the complement of the interior of a dividing disk is also a dividing disk, and we have expressed as a product of at most three local homeomorphisms. This shows that the commutator length of is at most , and since was arbitrary, we are done. qed.
The same argument just barely fails to work with in place of . One can also define dividing disks and local homeomorphisms in , with the following important difference. One can show by the same argument that local homeomorphisms in are commutators, and that for an arbitrary element there are local elements so that 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: .
The so-called “point-pushing” subgroup can be understood geometrically by tracking the image of a proper ray from to infinity. We are therefore motivated to consider the following object:
Definition: The ray graph is the graph whose vertex set is the set of isotopy classes of proper rays , with interior in the complement of , from a point in to infinity, and whose edges are the pairs of such rays that can be realized disjointly.
One can verify that the graph is connected, and that the group acts simplicially on by automorphisms, and transitively on vertices.
Lemma: Let and suppose there is a vertex such that share an edge. Then is a product of at most two local homeomorphisms.
Sketch of proof: After adjusting by an isotopy, assume that and are actually disjoint. Let be sufficiently small disjoint disks about the endpoint of and , and an arc from to disjoint from and , so that the union does not separate the part of outside . Then this union can be engulfed in a punctured disk containing infinity, whose complement contains some of . There is a local supported in a neighborhood of such that is supported (after isotopy) in the complement of (i.e. it is also local). qed.
It follows that if has a bounded orbit in , then the commutator lengths of the powers of are bounded, and therefore vanishes. If this is true for every , then Bavard duality implies that admits no nontrivial homogeneous quasimorphisms. This motivates the following questions:
Question: Is the diameter of infinite? (Exercise: show )
Question: Does any element of act on with positive translation length?
Question: Can one use this action to construct nontrivial quasimorphisms on ?
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 be a finite directed graph (hereafter a digraph) with an initial vertex, and edges labeled by elements of a finite set in such a way that each vertex has at most one outgoing edge with any given label. A finite directed path in starting at the initial vertex determines a word in the alphabet , by reading the labels on the edges traversed (in order). The set of words obtained in this way is an example of what is called a regular language, and is said to be parameterized by . Note that this is not the most general kind of regular language; in particular, any language of this kind will necessarily be prefix-closed (i.e. if then every prefix of is also in ). Note also that different digraphs might parameterize the same (prefix-closed) regular language .
If is a set of generators for a group , there is an obvious map called the evaluation map that takes a word to the element of represented by that word.
Definition: Let be a group, and a finite generating set. A combing of is a (prefix-closed) regular language for which the evaluation map is a bijection, and such that every represents a geodesic in .
The intuition behind this definition is that the set of words in determines a directed spanning tree in the Cayley graph starting at , and such that every directed path in the tree is a geodesic in . 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 be a hyperbolic group, and let be a finite generating set. Choose a total order on the elements of . Then the language of lexicographically first geodesics in is a combing.
The language described in this theorem is obviously geodesic and prefix-closed, and the evaluation map is bijective; the content of the theorem is that is regular, and parameterized by some finite digraph . In the sequel, we restrict attention exclusively to hyperbolic groups .
Given a (hyperbolic) group , a generating set , a combing , one makes the following definition:
Definition: A function is weakly combable (with respect to ) if there is a digraph parameterizing and a function from the vertices of to so that for any , corresponding to a path in , there is an equality .
In other words, a function is weakly combable if it can be obtained by “integrating” a function 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 , and bicombable if it changes by a bounded amount under either left or right multiplication by an element of . The property of being (bi-)combable does not depend on the choice of a generating set or a combing .
Example: Word length (with respect to a given generating set ) is bicombable.
Example: Let be a homomorphism. Then 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 be a finite set which generates as a semigroup. Let denote word length with respect to , and denote word length with respect to (which also generates as a semigroup). Then the difference is a bicombable quasimorphism.
The main theorem proved in the paper concerns the statistical distribution of values of a bicombable function.
Theorem: Let be a hyperbolic group, and let be a bicombable function on . Let be the value of on a random word in of length (with respect to a certain measure depending on a choice of generating set). Then there are algebraic numbers and so that as distributions, converges to a normal distribution with standard deviation .
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 are two finite generating sets for a group , then there is a constant so that every word of length in one generating set has length at most and at least in the other generating set. If one considers an example like , one sees that this is the best possible estimate, even statistically. However, if one restricts attention to a hyperbolic group , then one can do much better for typical words:
Corollary: Let be hyperbolic, and let be two finite generating sets. There is an algebraic number so that almost all words of length with respect to the generating set have length almost equal to with respect to the generating set, with error of size .
Let me indicate very briefly how the proof of the theorem goes.
Sketch of Proof: Let be bicombable, and let be a function from the vertices of to , where is a digraph parameterizing . There is a bijection between the set of elements in of word length and the set of directed paths in of length that start at the initial vertex. So to understand the distribution of , we need to understand the behaviour of a typical long path in .
Define a component of 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 without loops, with one vertex for each component of , in an obvious way. Each component determines an adjacency matrix , with -entry equal to if there is a directed edge from vertex to vertex , and equal to otherwise. A component is big if the biggest real eigenvalue of is at least as big as the biggest real eigenvalue of the matrices associated to every other component. A random long walk in 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 .
A theorem of Coornaert implies that there are no big components of in series; i.e. there are no directed paths in from one big component to another (one also says that the big components do not communicate). This means that a typical long walk in 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 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 corresponding to paths that spend almost all their time in a given big component there is a central limit theorem where the mean and standard deviation depend only on . 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 ; to finish the proof one must use bicombability.
It is not hard to show that if is a typical infinite walk in a component , then the subpaths of of length are distributed like random walks of length in . What this means is that the mean and standard deviation associated to a big component can be recovered from the distribution of on a single infinite “typical” path in . Such an infinite path corresponds to an infinite geodesic in , converging to a definite point in the Gromov boundary . Another theorem of Coornaert (from the same paper) says that the action of on its boundary 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 associated to components and for which some takes to a geodesic ending at the same point in as . Bicombability implies that the values of on and differ by a bounded amount. Moreover, since and are asymptotic to the same point at infinity, combability implies that the values of on and also differ by a bounded amount. This is enough to deduce that and , and one obtains a (global) central limit theorem for on . qed.
This obviously raises several questions, some of which seem very hard, including:
Question 1: Let be an arbitrary quasimorphism on a hyperbolic group (even the case is free is interesting). Does satisfy a central limit theorem?
Question 2: Let be an arbitrary quasimorphism on a hyperbolic group . Does satisfy a central limit theorem with respect to a random walk on ? (i.e. one considers the distribution of values of not on the set of elements of of word length , but on the set of elements obtained by a random walk on of length , and lets 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 where is a closed negatively curved Riemannian manifold; such quasimorphisms are defined by choosing a -form , and defining to be equal to the integral , where is the closed oriented based geodesic in in the homotopy class of . De Rham quasimorphisms, being local, also satisfy a central limit theorem.
This locality manifests itself in another way, in terms of defects. Let be a quasimorphism on a hyperbolic group . Recall that the defect is the supremum of over all pairs of elements . A quasimorphism is further said to be homogeneous if for all integers . If is an arbitrary quasimorphism, one may homogenize it by taking a limit ; one says that is the homogenization of in this case. Homogenization typically does not preserve defects; however, there is an inequality . If 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) with the prefix of (with notation as above). When one homogenizes, one picks up another contribution to the defect from the interaction of the prefix of with the suffix of ; 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 be a group, and let be a homogeneous quasimorphism. Is there a quasimorphism with homogenization , satisfying ?
Example: The answer to question 3 is “yes” if is the rotation quasimorphism associated to an action of on by orientation-preserving homeomorphisms (this is nontrivial; see Proposition 4.70 from my monograph).
Example: Let be any homologically trivial group -boundary. Then there is some extremal homogeneous quasimorphism for (i.e. a quasimorphism achieving equality under generalized Bavard duality; see this post) for which there is with homogenization satisfying . Consequently, if every point in the boundary of the unit ball in the norm is contained in a unique supporting hyperplane, the answer to question 3 is “yes” for any quasimorphism on .
Any quasimorphism on 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 is a free group. An interesting test case might be the homogenization of an infinite sum of Brooks functions for some infinite non-nested family of words .
If the answer to this question is false, and one can find a homogeneous quasimorphism which is not the homogenization of any “local” quasimorphism, then perhaps does not satisfy a central limit theorem. One can try to approach this problem from the other direction:
Question 4: Given a function defined on the ball of radius in a free group , one defines the defect in the usual way, restricted to pairs of elements for which are all of length at most . Under what conditions can be extended to a function on the ball of radius 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 be a group, and let denote the commutator subgroup. Every element of can be expressed as a product of commutators; the commutator length of an element is the minimum number of commutators necessary, and is denoted . The stable commutator length is the growth rate of the commutator lengths of powers of an element; i.e. . Recall that a group is said to satisfy a law if there is a nontrivial word in a free group for which every homomorphism from to sends to .
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 obeys a law. Then the stable commutator length vanishes identically on .
The proof depends on a duality between stable commutator length and a certain class of functions, called homogeneous quasimorphisms.
Definition: A function is a quasimorphism if there is some least number (called the defect) so that for any pair of elements there is an inequality . A quasimorphism is homogeneous if it satisfies for all integers .
Note that a homogeneous quasimorphism with defect zero is a homomorphism (to ). The defect satisfies the following formula:
Lemma: Let be a homogeneous quasimorphism. Then .
A fundamental theorem, due to Bavard, is the following:
Theorem: (Bavard duality) There is an equality where the supremum is taken over all homogeneous quasimorphisms with nonzero defect.
In particular, vanishes identically on if and only if every homogeneous quasimorphism on is a homomorphism.
One final ingredient is another geometric definition of in terms of Euler characteristic. Let be a space with , and let be a free homotopy class representing a given conjugacy class . If is a compact, oriented surface without sphere or disk components, a map is admissible if the map on factors through , where the second map is . For an admissible map, define by the equality in (i.e. is the degree with which wraps around ). With this notation, one has the following:
Lemma: There is an equality .
Note: the function is the sum of over non-disk and non-sphere components of . By hypothesis, there are none, so we could just write . However, it is worth writing and observing that for more general (orientable) surfaces, this function is equal to the function 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 . By Bavard duality, there is a homogeneous quasimorphism with nonzero defect. Rescale to have defect . Then for any there are elements with , and consequently by Bavard duality. On the other hand, if is a space with , and is a loop representing the conjugacy class of , there is a map from a once-punctured torus to whose boundary represents . The fundamental group of is free on two generators which map to the class of respectively. If is a word in mapping to the identity in , there is an essential loop in that maps inessentially to . There is a finite cover of , of degree depending on the word length of , for which lifts to an embedded loop. This can be compressed to give a surface with . However, Euler characteristic is multiplicative under coverings, so . On the other hand, so . If obeys a law, then is fixed, but can be made arbitrarily small. So does not obey a law. qed.