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Last Friday, Henry Wilton gave a talk at Caltech about his recent joint work with Sang-hyun Kim on polygonal words in free groups. Their work is motivated by the following well-known question of Gromov:
Question(Gromov): Let be a one-ended word-hyperbolic group. Does contain a subgroup isomorphic to the fundamental group of a closed hyperbolic surface?
Let me briefly say what “one-ended” and “word-hyperbolic” mean.
A group is said to be word-hyperbolic if it acts properly and cocompactly by isometries on a proper -hyperbolic path metric space — i.e. a path metric space in which there is a constant so that geodesic triangles in the metric space have the property that each side of the triangle is contained in the -neighborhood of the union of the other two sides (colloquially, triangles are thin). This condition distills the essence of negative curvature in the large, and was shown by Gromov to be equivalent to several other conditions (eg. that the group satisfies a linear isoperimetric inequality; that every ultralimit of the group is an -tree). Free groups are hyperbolic; fundamental groups of closed manifolds with negative sectional curvature (eg surfaces with negative Euler characteristic) are word-hyperbolic; “random” groups are hyperbolic — and so on. In fact, it is an open question whether a group that admits a finite is word hyperbolic if and only if it does not contain a copy of a Baumslag-Solitar group for (note that the group is the special case ); in any case, this is a very good heuristic for identifying the word-hyperbolic groups one typically meets in examples.
If is a finitely generated group, the ends of really means the ends (as defined by Freudenthal) of the Cayley graph of with respect to some finite generating set. Given a proper topological space , the set of compact subsets of gives rise to an inverse system of inclusions, where includes into whenever is a subset of . This inverse system defines an inverse system of maps of discrete spaces , and the inverse limit of this system is a compact, totally disconnected space , called the space of ends of . A proper topological space is canonically compactified by its set of ends; in fact, the compactification is the “biggest” compactification of by a totally disconnected space, in the sense that for any other compactification where is zero dimensional, there is a continuous map which is the identity on .
For a word-hyperbolic group , the Cayley graph can be compactified by adding the ideal boundary , but this is typically not totally disconnected. In this case, the ends of can be recovered as the components of .
A group acts on its own ends . An elementary argument shows that the cardinality of is one of (if a compact set disconnects then infinitely many translates of converging to separate from infinitely many other ends accumulating on ). A group has no ends if and only if it is finite. Stallings famously showed that a (finitely generated) group has at least ends if and only if it admits a nontrivial description as an HNN extension or amalgamated free product over a finite group. One version of the argument proceeds more or less as follows, at least when is finitely presented. Let be an -dimensional Riemannian manifold with fundamental group , and let denote the universal cover. We can identify the ends of with the ends of . Let be a least (-dimensional) area hypersurface in amongst all hypersurfaces that separate some end from some other (here the hypothesis that has at least two ends is used). Then every translate of by an element of is either equal to or disjoint from it, or else one could use the Meeks-Yau “roundoff trick” to find a new with strictly lower area than . The translates of decompose into pieces, and one can build a tree whose vertices correspond to to components of , and whose edges correspond to the translates . The group acts on this tree, with finite edge stabilizers (by the compactness of ), exhibiting either as an HNN extension or an amalgamated product over the edge stabilizers. Note that the special case occurs if and only if has a finite index subgroup which is isomorphic to .
Free groups and virtually free groups do not contain closed surface subgroups; Gromov’s question more or less asks whether these are the only examples of word-hyperbolic groups with this property.
Kim and Wilton study Gromov’s question in a very, very concrete case, namely that case that is the double of a free group along a word ; i.e. (hereafter denoted ). Such groups are known to be one-ended if and only if is not contained in a proper free factor of (it is clear that this condition is necessary), and to be hyperbolic if and only if is not a proper power, by a result of Bestvina-Feighn. To see that this condition is necessary, observe that the double is isomorphic to the fundamental group of a Seifert fiber space, with base space a disk with two orbifold points of order ; such a group contains a . One might think that such groups are too simple to give an insight into Gromov’s question. However, these groups (or perhaps the slightly larger class of graphs of free groups with cyclic edge groups) are a critical case for at least two reasons:
- The “smaller” a group is, the less room there is inside it for a surface group; thus the “simplest” groups should have the best chance of being a counterexample to Gromov’s question.
- If is word-hyperbolic and one-ended, one can try to find a surface subgroup by first looking for a graph of free groups in , and then looking for a surface group in . Since a closed surface group is itself a graph of free groups, one cannot “miss” any surface groups this way.
Not too long ago, I found an interesting construction of surface groups in certain graphs of free groups with cyclic edge groups. In fact, I showed that every nontrivial element of in such a group is virtually represented by a sum of surface subgroups. Such surface subgroups are obtained by finding maps of surface groups into which minimize the Gromov norm in their (projective) homology class. I think it is useful to extend Gromov’s question by making the following
Conjecture: Let be a word-hyperbolic group, and let be nonzero. Then some multiple of is represented by a norm-minimizing surface (which is necessarily -injective).
Note that this conjecture does not generalize to wider classes of groups. There are even examples of groups with nonzero homology classes with positive, rational Gromov norm, for which there are no -injective surfaces representing a multiple of at all.
It is time to define polygonal words in free groups.
Definition: Let be free. Let be a wedge of circles whose edges are free generators for . A cyclically reduced word in these generators is polygonal if there exists a van-Kampen graph on a surface such that:
- every complementary region is a disk whose boundary is a nontrivial (possibly negative) power of ;
- the (labelled) graph immerses in in a label preserving way;
- the Euler characteristic of is strictly less than the number of disks.
The last condition rules out trivial examples; for example, the double of a single disk whose boundary is labeled by . Notice that it is very important to allow both positive and negative powers of as boundaries of complementary regions. In fact, if is not in the commutator subgroup, then the sum of the powers over all complementary regions is necessarily zero (and if is in the commutator subgroup, then has nontrivial , so one already knows that there is a surface subgroup).
Condition 2. means that at each vertex of , there is at most one oriented label corresponding to each generator of or its inverse. This is really the crucial geometric property. If is a van-Kampen graph as above, then a theorem of Marshall Hall implies that there is a finite cover of into which embeds (in fact, this observation underlies Stallings’s work on foldings of graphs). If we build a -complex with by attaching two ends of a cylinder to suitable loops in two copies of , then a tubular neighborhood of in (i.e. what is sometimes called a “fatgraph” ) embeds in a finite cover of , and its double — a surface of strictly negative Euler characteristic — embeds as a closed surface in , and is therefore -injective. Hence if is polygonal, contains a surface subgroup.
Not every word is polygonal. Kim-Wilton discuss some interesting examples in their paper, including:
- suppose is a cyclically reduced product of proper powers of the generators or their inverses (e.g a word like but not a word like ); then is polygonal;
- a word of the form is polygonal if for each ;
- the word is not polygonal.
To see 3, suppose there were a van-Kampen diagram with more disks than Euler characteristic. Then there must be some vertex of valence at least . Since is positive, the complementary regions must have boundaries which alternate between positive and negative powers of , so the degree of the vertex must be even. On the other hand, since must immerse in a wedge of two circles, the degree of every vertex must be at most , so there is consequently some vertex of degree exactly . Since each is isolated, at least edges must be labelled ; hence exactly two. Hence exactly two edges are labelled . But one of these must be incoming and one outgoing, and therefore these are adjacent, contrary to the fact that does not contain a .
1 above is quite striking to me. When is in the commutator subgroup, one can consider van-Kampen diagrams as above without the injectivity property, but with the property that every power of on the boundary of a disk is positive; call such a van-Kampen diagram monotone. It turns out that monotone van-Kampen diagrams always exist when , and in fact that norm-minimizing surfaces representing powers of the generator of are associated to certain monotone diagrams. The construction of such surfaces is an important step in the argument that stable commutator length (a kind of relative Gromov norm) is rational in free groups. In my paper scl, sails and surgery I showed that monomorphisms of free groups that send every generator to a power of that generator induce isometries of the norm; in other words, there is a natural correspondence between certain equivalence classes of monotone surfaces for an arbitrary word in and for a word of the kind that Kim-Wilton show is polygonal (Note: Henry Wilton tells me that Brady, Forester and Martinez-Pedroza have independently shown that contains a surface group for such , but I have not seen their preprint (though I would be very grateful to get a copy!)).
In any case, if not every word is polygonal, all is not lost. To show that contains a surface subgroup is suffices to show that contains a surface subgroup, where and differ by an automorphism of . Kim-Wilton conjecture that one can always find an automorphism so that is polygonal. In fact, they make the following:
Conjecture (Kim-Wilton; tiling conjecture): A word not contained in a proper free factor of shortest length (in a given generating set) in its orbit under is polygonal.
If true, this would give a positive answer to Gromov’s question for groups of the form .
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 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.
The development and scope of modern biology is often held out as a fantastic opportunity for mathematicians. The accumulation of vast amounts of biological data, and the development of new tools for the manipulation of biological organisms at microscopic levels and with unprecedented accuracy, invites the development of new mathematical tools for their analysis and exploitation. I know of several examples of mathematicians who have dipped a toe, or sometimes some more substantial organ, into the water. But it has struck me that I know (personally) few mathematicians who believe they have something substantial to learn from the biologists, despite the existence of several famous historical examples. This strikes me as odd; my instinctive feeling has always been that intellectual ruts develop so easily, so deeply, and so invisibly, that continual cross-fertilization of ideas is essential to escape ossification (if I may mix biological metaphors . . .)
It is not necessarily easy to come up with profound examples of biological ideas or principles that can be easily translated into mathematical ones, but it is sometimes possible to come up with suggestive ones. Let me try to give a tentative example.
Deoxiribonucleic acid (DNA) is a nucleic acid that contains the genetic blueprint for all known living things. This blueprint takes the form of a code — a molecule of DNA is a long polymer strand composed of simple units called nucleotides; such a molecule is typically imagined as a string in a four character alphabet , which stand for the nucleotides Adenine, Thymine, Guanine, and Cytosine. These molecular strands like to arrange themselves in tightly bound oppositely aligned pairs, matching up nucleotides in one string with complementary nucleotides in the other, so that matches with , and with .
The geometry of a strand of DNA is very complicated — strands can be tangled, knotted, linked in complicated ways, and the fundamental interactions between strands (e.g. transcription, recombination) are facilitated or obstructed by mechanical processes depending on this geometry. Topology, especially knot theory, has been used in the study of some of these processes; the value of topological methods in this context include their robustness (fault-tolerance) and the discreteness of their invariants (similar virtues motivate some efforts to build topological quantum computers). A complete mathematical description of the salient biochemistry, mechanics, and semantic content of a configuration of DNA in a single cell is an unrealistic goal for the foreseeable future, and therefore attempts to model such systems depends on ignoring, or treating statistically, certain features of the system. One such framework ignores the ambient geometry entirely, and treats the system using symbolic, or combinatorial methods which have some of the flavor of geometric group theory.
One interesting approach is to consider a mapping from the alphabet of nucleotides to a standard generating set for , the free group on two generators; for example, one can take the mapping where are free generators for , and denote their inverses. Then a pair of oppositely aligned strands of DNA translates into an edge of a van Kampen diagram — the “words” obtained by reading the letters along an edge on either side are inverse in .
Strands of DNA in a configuration are not always paired along their lengths; sometimes junctions of three or more strands can form; certain mobile four-strand junctions, so-called “Holliday junctions”, perform important functions in the process of genetic recombination, and are found in a wide variety of organisms. A configuration of several strands with junctions of varying valences corresponds in the language of van Kampen diagrams to a fatgraph — i.e. a graph together with a choice of cyclic ordering of edges at each vertex — with edges labeled by inverse pairs of words in (note that this is quite different from the fatgraph model of proteins developed by Penner-Knudsen-Wiuf-Andersen). The energy landscape for branch migration (i.e. the process by which DNA strands separate or join along some segment) is very complicated, and it is challenging to model it thermodynamically. It is therefore not easy to predict in advance what kinds of fatgraphs are more or less likely to arise spontaneously in a prepared “soup” of free DNA strands.
As a thought experiment, consider the following “toy” model, which I do not suggest is physically realistic. We make the assumption that the energy cost of forming a junction of valence is for some fixed constant . Consequently, the energy of a configuration is proportional to , i.e. the negative of Euler characteristic of the underlying graph. Let be a reduced word, representing an element of , and imagine a soup containing some large number of copies of the strand of DNA corresponding to the string . In thermodynamic equilibrium, the partition function has the form where is Boltzmann’s constant, is temperature, and is the energy of a configuration (which by hypothesis is proportional to ). At low temperature, minimal energy configurations tend to dominate; these are those that minimize per unit “volume”. Topologically, a fatgraph corresponding to such a configuration can be thickened to a surface with boundary. The words along the edges determine a homotopy class of map from such a surface to a (e.g. a once-punctured torus) whose boundary components wrap multiply around the free homotopy class corresponding to the conjugacy class of . The infimum of where is the winding degree on the boundary, taken over all configurations, is precisely the stable commutator length of ; see e.g. here for a definition.
Anyway, this example is perhaps a bit strained (and maybe it owes more to thermodynamics than to biology), but already it suggests a new mathematical object of study, namely the partition function as above, and one is already inclined to look for examples for which the partition function obeys a symmetry like that enjoyed by the Riemann zeta function, or to specialize temperature to other values, as in random matrix theory. The introduction of new methods into the study of a classical object — for example, the decision to use thermodynamic methods to organize the study of van Kampen diagrams — bends the focus of the investigation towards those examples and contexts where the methods and tools are most informative. Phenomena familiar in one context (power laws, frequency locking, phase transitions etc.) suggest new questions and modes of enquiry in another. Uninspired or predictable research programs can benefit tremendously from such infusions, whether the new methods are borrowed from other intellectual disciplines (biology, physics), or depend on new technology (computers), or new methods of indexing (google) or collaboration (polymath).
One of my intellectual heroes — Wolfgang Haken — worked for eight years in R+D for Siemens in Munich after completing his PhD. I have a conceit (unsubstantiated as far as I know by biographical facts) that his experience working for a big engineering firm colored his approach to mathematics, and made it possible for him to imagine using industrial-scale “engineering” tools (e.g. integer programming, exhaustive computer search of combinatorial possibilities) to solve two of the most significant “pure” mathematical open problems in topology at the time — the knot recognition problem, and the four-color theorem. It is an interesting exercise to try to imagine (fantastic) variations. If I sit down and decide to try to prove (for example) Cannon’s conjecture, I am liable to try minor variations on things I have tried before, appeal for my intuition to examples that I understand well, read papers by others working in similar ways on the problem, etc. If I imagine that I have been given a billion dollars to prove the conjecture, I am almost certain to prioritize the task in different ways, and to entertain (and perhaps create) much more ambitious or innovative research programs to tackle the task. This is the way in which I understand the following quote by John Dewey, which I used as the colophon of my first book:
Every great advance in science has issued from a new audacity of the imagination.
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.