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I have just uploaded a paper to the arXiv, entitled “Scl, sails and surgery”. The paper discusses a connection between stable commutator length in free groups and the geometry of sails. This is an interesting example of what sometimes happens in geometry, where a complicated topological problem in low dimensions can be translated into a “simple” geometric problem in high dimensions. Other examples include the Veronese embedding in Algebraic geometry (i.e. the embedding of one projective space into another taking a point with homogeneous co-ordinates to the point whose homogeneous co-ordinates are the monomials of some fixed degree in the ), which lets one exhibit any projective variety as an intersection of a Veronese variety (whose geometry is understood very well) with a linear subspace.
In my paper, the fundamental problem is to compute stable commutator length in free groups, and more generally in free products of Abelian groups. Let’s focus on the case of a group where are free abelian of finite rank. A is just a wedge of tori of dimension equal to the ranks of . Let be a free homotopy class of -manifold in , which is homologically trivial. Formally, we can think of as a chain in , the vector space of group -boundaries, modulo homogenization; i.e. quotiented by the subspace spanned by chains of the form and . One wants to find the simplest surface mapping to that rationally bounds . I.e. we want to find a map such that factors through , and so that the boundary wraps homologically times around each loop of , in such a way as to infimize . This infimum, over all maps of all surfaces of all possible genus, is the stable commutator length of the chain . Computing this quantity for all such finite chains is tantamount to understanding the bounded cohomology of a free group in dimension .
Given such a surface , one can cut it up into simpler pieces, along the preimage of the basepoint . Since is a surface with boundary, these simpler pieces are surfaces with corners. In general, understanding how a surface can be assembled from an abstract collection of surfaces with corners is a hopeless task. When one tries to glue the pieces back together, one runs into trouble at the corners — how does one decide when a collection of surfaces “closes up” around a corner? The wrong decision leads to branch points; moreover, a decision made at one corner will propogate along an edge and lead to constraints on the choices one can make at other corners. This problem arises again and again in low-dimensional topology, and has several different (and not always equivalent) formulations and guises, including -
- Given an abstract branched surface and a weight on that surface, when is there an unbranched surface carried by the abstract branched surface and realizing the weight?
- Given a triangulation of a -manifold and a collection of normal surface types in each simplex satisfying the gluing constraints but *not* necessarily satisfying the quadrilateral condition (i.e. there might be more than one quadrilateral type per simplex), when is there an immersed unbranched normal surface in the manifold realizing the weight?
- Given an immersed curve in the plane, when is there an immersion from the disk to the plane whose boundary is the given curve?
- Given a polyhedral surface (arising e.g. in computer graphics), how can one choose smooth approximations of the polygonal faces that mesh smoothly at the vertices?
I think of all these problems as examples of what I like to call the holonomy problem, since all of them can be reduced, in one way or another, to studying representations of fundamental groups of punctured surfaces into finite groups. The fortunate “accident” in this case is that every corner arises by intersecting a cut with a boundary edge of . Consequently, one never wants to glue more than two pieces up at any corner, and the holonomy problem does not arise. Hence in principle, to understand the surface one just needs to understand the pieces of that can arise by cutting, and the ways in which they can be reassembled.
This is still not a complete solution of the problem, since infinitely many kinds of pieces can arise by cutting complicated surfaces . The -manifold decomposes into a collection of arcs in the tori and which we denote respectively, and the surface (hereafter abbreviated to ) has edges that alternate between elements of , and edges mapping to . Since is a torus, handles of mapping to can be compressed, reducing the complexity of , and thereby , so one need only consider planar surfaces .
Let denote the real vector space with basis the set of ordered pairs of elements of (not necessarily distinct), and the real vector space with basis the elements of . A surface determines a non-negative integral vector , by counting the number of times a given pair of edges appear in succession on one of the (oriented) boundary components of . The vector satisfies two linear constraints. First, there is a map defined on a basis vector by . The vector satisfies . Second, each element is a based loop in , and therefore corresponds to an element in the free abelian group . Define on a basis vector by (warning: the notation obscures the fact that and map to quite different vector spaces). Then ; moreover, a non-negative rational vector satisfying has a multiple of the form for some as above. Denote the subspace of consisting of non-negative vectors in the kernel of and by . This is a rational polyhedral cone — i.e. a cone with finitely many extremal rays, each spanned by a rational vector.
Although every integral is equal to for some , many different correspond to a given . Moreover, if we are allowed to consider formal weighted sums of surfaces, then even more possibilities. In order to compute stable commutator length, we must determine, for a given vector , an expression where the are positive real numbers, which minimizes . Here denotes orbifold Euler characteristic of a surface with corners; each corner contributes to . The reason one counts complexity using this modified definition is that the result is additive: . The contribution to from corners is a linear function on . Moreover, a component with can be covered by a surface of high genus and compressed (increasing ); so such a term can always be replaced by a formal sum for which . Thus the only nonlinear contribution to comes from the surfaces whose underlying topological surface is a disk.
Call a vector a disk vector if where is topologically a disk (with corners). It turns out that the set of disk vectors has the following simple form: it is equal to the union of the integer lattice points contained in certain of the open faces of (those satisfying a combinatorial criterion). Define the sail of to be equal to the boundary of the convex hull of the polyhedron (where here denotes Minkowski sum). The Klein function is the unique continuous function on , linear on rays, that is equal to exactly on the sail. Then over expressions satisfies where denotes norm. To calculate stable commutator length, one minimizes over contained in a certain rational polyhedron in .
Sails are considered elsewhere by several authors; usually, people take to be the set of all integer vectors except the vertex of the cone, and the sail is therefore the boundary of the convex hull of this (simpler) set. Klein introduced sails as a higher-dimensional generalization of continued fractions: if is a polyhedral cone in two dimensions (i.e. a sector in the plane, normalized so that one edge is the horizontal axis, say), the vertices of the sail are the continued fraction approximations of the boundary slope. Arnold has revived the study of such objects in recent years. They arise in many different interesting contexts, such as numerical analysis (especially diophantine approximation) and algebraic number theory. For example, let be a matrix with irreducible characteristic equation, and all eigenvalues real and positive. There is a basis for consisting of eigenvalues, spanning a convex cone . The cone — and therefore its sail — is invariant under ; moreover, there is a subgroup of consisting of matrices with the same set of eigenvectors; this observation follows from Dirichlet’s theorem on the units in a number field, and is due to Tsuchihashi. This abelian group acts freely on the sail with quotient a (topological) torus of dimension , together with a “canonical” cell decomposition. This connection between number theory and combinatorics is quite mysterious; for example, Arnold asks: which cell decompositions can arise? This is unknown even in the case .
The most interesting aspect of this correspondence, between stable commutator length and sails, is that it allows one to introduce parameters. An element in a free group can be expressed as a word in letters , e.g. , which is usually abbreviated with exponential notation, e.g. . Having introduced this notation, one can think of the exponents as parameters, and study stable commutator length in families of words, e.g. . Under the correspondence above, the parameters only affect the coefficients of the linear map , and therefore one obtains families of polyhedral cones whose extremal rays depend linearly on the exponent parameters. This lets one prove many facts about the stable commutator length spectrum in a free group, including:
Theorem: The image of a nonabelian free group of rank at least under scl in is precisely .
Theorem: For each , the image of the free group under scl contains a well-ordered sequence of values with ordinal type . The image of contains a well-ordered sequence of values with ordinal type .
One can also say things about the precise dependence of scl on parameters in particular families. More conjecturally, one would like to use this correspondence to say something about the statistical distribution of scl in free groups. Experimentally, this distribution appears to obey power laws, in the sense that a given (reduced) fraction appears in certain infinite families of elements with frequency proportional to for some power (which unfortunately depends in a rather opaque way on the family). Such power laws are reminiscent of Arnold tongues in dynamics, one of the best-known examples of phase locking of coupled nonlinear oscillators. Heuristically one expects such power laws to appear in the geometry of “random” sails — this is explained by the fact that the (affine) geometry of a sail depends only on its orbit, and the existence of invariant measures on a natural moduli space; see e.g. Kontsevich and Suhov. The simplest example concerns the (-dimensional) cone spanned by a random integral vector in . The orbit of such a vector depends only on the gcd of the two co-ordinates. As is easy to see, the probability distribution of the gcd of a random pair of integers obeys a power law: with probability . The rigorous justification of the power laws observed in the scl spectrum of free groups remains the focus of current research by myself and my students.
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 ?
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.
As an experiment, I plan to spend the next five weeks documenting my current research on this blog. This research comprises several related projects, but most are concerned in one way or another with the general program of studying the geometry of a space by probing it with surfaces. Since I am nominally a topologist, these surfaces are real -manifolds, and I am usually interested in working in the homotopy category (or some rational “quotient” of it). I am especially concerned with surfaces with boundary, and even (occasionally) with corners.
Since it is good to have a “big question” lurking somewhere in the background (for the purposes of motivation and advertising, if nothing else), I should admit from the start that I am interested in Gromov’s well-known question about surface subgroups, which asks:
Question (Gromov): Does every one-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?
I don’t have strong feelings about whether the answer to this question is “yes” or “no”, but I do think the question can be sharpened usefully in many ways, and it is my intention to do so. Gromov’s question is certainly inspired by questions such as Waldhausen’s conjecture and the virtual fibration conjecture in -manifold topology, but it is hard to imagine that a proof of one of these conjectures would shed much light on Gromov’s question in general. At least one essential tool in -manifold topology — namely Dehn’s lemma — has no meaningful analogue in geometric group theory, and I think it is important to try to imagine different methods of constructing surface groups from “first principles”.
Another long-term project that informs much of my current research is the problem of understanding stable commutator length in free groups. The interested reader can learn something about this from my monograph (which can be downloaded from this page). I hope to explain why this is a fundamental and interesting problem, with rich structure and many potential applications.