Riemannian manifolds are not primitive mathematical objects, like numbers, or functions, or graphs. They represent a compromise between local Euclidean geometry and global smooth topology, and another sort of compromise between precognitive geometric intuition and precise mathematical formalism.

Don’t ask me precisely what I meant by that; rather observe the repeated use of the key word *compromise*. The study of Riemannian geometry is — at least to me — fraught with compromise, a compromise which begins with language and notation. On the one hand, one would like a language and a formalism which treats Riemannian manifolds on their own terms, without introducing superfluous extra structure, and in which the fundamental objects and their properties are highlighted; on the other hand, in order to actually compute or to use the all-important tools of vector calculus and analysis one must introduce coordinates, indices, and cryptic notation which trips up beginners and experts alike.

Actually, my complicated relationship began the first time I was introduced to vectors. It was 1986, I was at a training camp for Australian mathematics olympiad hopefuls, and Ben Robinson gave me a 2 minute introduction to the subject over lunch. I found the notation overwhelming, and there was no connection in my mind between the letters and subscripts on one side of the page and the squiggly arrows and parallelograms on the other side. By the time the subject came up again a few years later in high school, somehow the mystery had faded, and the vocabulary and meaning of vectors, inner products, determinants etc. was crystal clear. I think that the difference this time around was that I concentrated first on learning what vectors *were*, and only when I had gotten the point did I engage with the question of how to represent them or calculate with them. In a similar way, my introduction to div, grad and curl was equally painless, since we learned the subject in physics class (in the last couple of years of high school) in the context of classical electrodynamics. I might have been challenged to grasp the abstract idea of a “vector field” as it is introduced in some textbooks, but those little pictures of lines of force running from positive to negative charges made immediate and intuitive sense. In fact, the whole idea of describing a vector field as a partial differential operator such as obscures an enormous complexity; it’s easy enough to *compute* with an expression like this, but as a mathematical object itself it is quite sophisticated, since even to define it we need not just one coordinate but an entire system of coordinates on some nearby smooth patch. Contrast this with the intuitive idea of a particle moving along a line of force, and being subjected to some influence which varies along the trajectory. I’m grateful to whoever designed the Melbourne high school science curriculum in the late 1980′s for integrating the maths and physics curricula so successfully.

A few years later, as an undergraduate at the University of Melbourne, I was attending Marty Ross’ reading group as we attempted to go through Cheeger and Ebin’s *Comparison theorems in Riemannian geometry*, and the confusion was back. Noel Hicks’ MathSciNet review calls this book a “tight, elegant, and delightful addition to the literature on global Riemannian geometry”, although he remarks that the “tightness of the exposition and a few misprints leave the reader with some challenging work”. Today I love this book, and recommend it to anyone; but at the time it was a terrible book to learn Riemannian geometry from for the first time (actually, since I was not a maths major, my confusion was amplified by many gaps in my intermediate education). Some aspects of the book I could appreciate — at least we were not drowning in indices, and the formulae were almost readable. But I was simply at a loss to understand the rules of the game — what sort of manipulations of formulae were allowed? how do you contract a vector field with a form? why am I allowed to choose coordinates at this point so that everything magically simplifies? how would anyone ever stumble on the formula for the Ricci curvature and see that it was invariant and had such nice properties? and so on.

And yet again, the duration of a couple of years made a world of difference. As a graduate student at Berkeley taking classes from Shoshichi Kobayashi and Sasha Givental, suddenly everything made sense (well, not everything, but at least the rudiments of Riemannian geometry). The difference again was that the notation and the calculations *followed* a discussion of what the objects *were*, and what information they contained and why you might want to use them or talk about them. And, crucially, this initial discussion was carried out first informally in *words* rather than by beginning with a formal definition or a formula.

So with this backstory in mind, I hope it might be useful to the graduate student out there who is struggling with the elements of the tensor calculus to go through a brief informal discussion of the meaning of some of the basic differential operators, which are the ingredients out of which much of the beauty of the subject can be synthesized.

Let’s get down to brass tacks. We start with a smooth manifold M and a vector field X. What is a vector field? For me I always think of it dynamically as a *flow: *the manifold is something like a fluid, and an object in M will be swept along by this flow and moved along the flowlines, or *integral curves* of the vector field. On a smooth manifold without a metric it doesn’t make sense to talk about whether the flow is moving “fast” or “slow”, but it *does* make sense to look at the places where it is stationary (the zeros of the vector field) and see whether the zeros are isolated or not, stable or unstable, or come in families. If f is a smooth function on M, the value of f varies along the integral curves of the vector field, and we can look at the rate at which the value changes; this is the derivative of f in the direction X, and denoted Xf. It is a smooth function on M; we can iterate this procedure and compute X(Xf), X(X(Xf)) and so on. The level sets of a smooth function f are (generically) smooth manifolds, and the whole idea of calculus is to approximate smooth things locally by linear things; thus generically through most points we can look at the level set of f through that point, and the tangent space to that level set. This is a hyperplane, and is spanned locally by the vector fields for which Xf is zero at the given point. More precisely, we can define a 1-form df just by setting df(X) = Xf; where df is nonzero, the kernel of df is the tangent space to the level set to f as described above.

**Grad. **Now we introduce a Riemannian metric, which is a smooth choice of inner product on the tangent space at each point. It does two things for us: first, it lets us talk about the *speed* of a flow generated by a vector field X (or equivalently, the *size* of the vectors); and second, it lets us measure the *angle* between two vectors at each point, in particular it lets us say what it means for vectors to be *perpendicular*. If f is a smooth function on a Riemannian manifold, we can do more than just construct the level sets of f; we can ask in which direction the value of f increases the fastest (and we can further ask how fast it increases in that direction). The answer to this question is the *gradient*; the gradient of f is a vector field which points always in the direction in which f increases the fastest, and with a magnitude proportional to the rate at which it increases there. In terms of the level sets of the function f, any vector field can be decomposed into a part which is *tangent* to the level sets (this is the part of the vector field whose flow keeps f unchanged) and a part which is *perpendicular* to it; the gradient is thus everywhere perpendicular to the level sets of f.

The inner product lets us give isomorphisms between vector fields and 1-forms called the *sharp* and *flat* isomorphisms. If is a 1-form, and X is a vector field, we define the vector field and the 1-form by the formulae

and

Sharp and flat are inverse operations. In words, a vector field and a 1-form are related by these operations if at each point they have the same magnitude, and the direction of the vector field is perpendicular to the kernel of the 1-form (i.e. the tangent space on which the 1-form vanishes). Using these isomorphisms, the gradient of a function f is just the vector field obtained by applying the sharp isomorphism to the 1-form df. In other words, it is the unique vector field such that for any other vector field X there is an identity

The zeros of the gradient are the critical points of f; for instance, the gradient vanishes at the minimum and the maximum of f.

**Div.** In Euclidean space of some dimension n, a collection of n linearly independent vectors form the edges of a parallelepiped. The volume of the parallelepiped is the determinant of the matrix whose columns are the given vectors. Actually there is a subtlety here — we need to choose an ordering of the vectors to take the determinant. A permutation might change the determinant by a factor of -1 if the sign of the permutation is odd. On an oriented Riemannian n-manifold if we have n vectors at a point, we can convert them to 1-forms and wedge them together — the result is an n-form. On an n-dimensional vector space, any two n-forms are proportional. Wedging together the 1-forms associated to a basis of perpendicular vectors of length 1 (an *orthonormal* collection) gives an n-form at each point which we call the *volume form*, and denote it . For any other n-tuple of vectors the volume of the parallelepiped is equal to the ratio of the n-form they determine (by taking sharp and wedging) and the volume form.

Now, there is an operator called *Hodge star* which acts on differential forms as follows. A k-form can be wedged with an (n-k) form to make an n-form, and this n-form can be compared in size to the volume form. We define the (n-k) form to be the *smallest* form such that

In other words, is perpendicular to the subspace of forms with . With this notation is the constant function equal to 1 everywhere; conversely for any smooth function f we have .

If X is a vector field, the flow generated by X carries along not just points, but tensor fields of all kinds. Covariant tensor fields are pushed forward by the flow, contravariant ones are pulled back. Thus a stationary observer at a point in M sees a one-parameter family of tensors of some fixed kind flowing through their point, and they may differentiate this family. The result is the *Lie derivative* of the tensor field, and is denote . The *divergence* of a vector field X measures the extent to which the flow generated by X does or does not preserve volume. It is a function which vanishes where the field infinitesimally preserves volume, and is biggest where the flow expands volume the most and smallest where the flow compresses volume the most.

The Lie derivative of the volume form is an n-form; taking Hodge star gives a function, and this function is the divergence. Thus:

In terms of the operators we have described above, applying flat to a vector field X gives a 1-form . Applying Hodge star to this one form gives rise to an (n-1)-form, then applying d gives an n-form, and this n-form (finally) is precisely . Thus,

Gradient and divergence are “almost” dual to each other under Hodge star, in the following sense. Let’s suppose we have some function f and some vector field X. We can take the gradient and form , and then we can look at the inner product of the gradient with X to obtain a function, and then integrate this function over the manifold. I.e.

But

If M is closed, the integral of an exact form over M is zero, so we deduce that

so that -div is a formal adjoint to grad.

**Laplacian.** If f is a function, we can first apply the gradient and then the divergence to obtain another function; this composition (or rather its negative) is the *Laplacian*, and is denoted . In other words,

Note that there are competing conventions here: it is common to denote the negative of this quantity (i.e. the composition div grad itself) as the Laplacian. But this convention is also common, and has the advantage that the Laplacian is a *non-negative self-adjoint operator*. The Laplacian governs the flow of *heat* in the manifold; if we imagine our manifold is filled with some collection of microscopic particles buzzing around randomly at great speed and carrying kinetic energy around, then the *temperature* is a measure of the amount of energy per unit of volume. If the temperature is constant, then although the particles can move from point to point, on average for each particle that moves out of a small box, there will be another particle that moves in from the outside; thus the ensemble of particles is in “thermal equilibrium”. However, if there is a local hot spot — i.e. a concentration of high energy particles — then these particles will have a tendency to spread out, in the sense that the average number of particles that leave the small hot box will exceed the number of particles that enter from neighboring cooler boxes. Thus, heat will tend to spread out by the vector field which is its negative gradient, and where this vector field diverges, the heat will dissipate and the temperature will cool. In other words, if f is the temperature, then the derivative of temperature over time satisfies the *heat equation *. Actually, since heat can come in or out from any direction, what is important is how the heat at a point deviates from the *average* of the heat at nearby points. The stationary heat distributions — i.e. the functions f with — are therefore the functions which satisfy an (infinitesimal) mean value property. These functions are called *harmonic*.

The erratic motion of the infinitesimal particles as they bump into each other and drift around is called *Brownian motion*, after the botanist Robert Brown, who is known to Australians for being the naturalist on the scientific voyage of the Investigator which sailed to Western Australia in 1801. Later, in 1827, he observed the jittery motion of minute particles ejected from pollen grains, and the phenomenon came to be named after him. Thus, a function on a Riemannian manifold is *harmonic* if its *expected* value stays constant under random Brownian motion, and the Laplacian describes the way that the expected value of the function changes under such motion.

**Curl.** After converting a vector field to a 1-form with the flat operator, one can apply the operator d to obtain a closed 2-form. On an arbitrary Riemannian manifold, this is more or less the end of the story, but on a 3-manifold, applying Hodge star to a 2-form gives back a 1-form, which can then be converted back to a vector field with the sharp operator. This composition is the *curl* of a vector field; i.e.

Notice that this satisfies the identities

and

Thus one of the functions of the curl operator is to give a necessary condition on a vector field to arise as the gradient of some function; such a function, if it exists, is called a *potential* for the vector field. Since a gradient flows from places where the function is small to where it is large, it does not recur or circulate; hence in a sense the curl measures the tendency of the vector field to circulate, or to form closed orbits. Actually there is a subtlety here which is that the curl will vanish precisely on vector fields which are *locally* the gradient of a smooth function. The topology of M — in particular its first homology group with real coefficients — parameterizes curl-free vector fields modulo those which are gradients of smooth functions.

As mentioned above, the curl measures the tendency of the vector field to spiral around an axis (locally); the direction of this axis of spiraling is the direction of the vector field , and the magnitude is the rate of twisting. Another way to say this is that the magnitude of the curl measures the tendency of flowlines of the vector field to wind positively around each other. A vector field and its curl can be proportional; such vector fields are called *Beltrami fields* and they arise (up to rescaling) as the Reeb flows associated to contact structures.

On an arbitrary Riemannian n-manifold it is still possible to interpret the curl in terms of rotation or twisting. Using the sharp and flat isomorphisms, a 2-form determines at each point a skew-symmetric endomorphism of the tangent space. The endomorphism applies to a vector by first contracting it with the 2-form to produce a 1-form, then using the sharp operator to transform it back to a vector. The skew-symmetry of this endomorphism is equivalent to the alternating property of forms. Now, a skew-symmetric endomorphism of a vector space can be thought of as an infinitesimal rotation, since the Lie algebra of the orthogonal group consists precisely of skew-symmetric matrices. Thus a vector field X on a Riemannian manifold determines a field of infinitesimal rotations, and this field is one way of thinking of . On a 3-manifold, a rotation has a unique axis, and this axis points in the direction of the vector field . On a Kähler manifold, the Kähler form determines a field of infinitesimal rotations which rotate the complex directions at constant speed.

**Strain.** Actually, the curl, the divergence, and a third operator called the *strain* can all be put on a uniform footing, as follows. We continue to think of a vector field X as a flow on a smooth manifold M. Tensor fields are pushed or pulled around by X, and an observer at a fixed point sees a 1-parameter family of tensors (of a fixed kind) evolving over time. But we would like to be able to study the effect of X on an object which is carried about and distorted by the flow; for example, we might have a curve or a submanifold in M, and we might want to understand how the geometry of this submanifold is preserved or distorted as it is carried along by the flow. Calculus takes place in a fixed vector space, and the flow is moving our object along the flowlines. We need some way to bring the object back along the flowline to a fixed reference frame so that we can understand how it is being transformed by the flow. On a Riemannian manifold there is a canonical way to move tensor fields along flowlines: we move them by *parallel transport*. There is a unique connection on the manifold called the *Levi-Civita connection* which preserves the metric, and is torsion-free. The first condition just means that parallel transport is an isometry from one tangent space to the other. The second condition is more subtle, and it means (roughly) that there is no “unnecessary twisting” of the tangent space as it is transported around (no *yaw*, in aviation terms). Think of a car moving down a straight freeway; the geometry of the car is (hopefully!) not distorted by its motion, and the occupants of the car are not unnecessarily rotated or twisted. When the car hits some ice, it begins to skid and twist; the occupants are still moved in roughly the same overall direction, and the geometry is still not distorted (until a collision, anyway), but there is unnecessary twisting — the “torsion” of the connection.

So on a Riemannian manifold, we can flow objects away by a vector field X, and then parallel transport them back along the flowlines with the Levi-Civita connection. Now “the same” tensor experiences the effect of the vector field X while staying in “the same” vector space, so that we can compute the derivative to determine the infinitesimal effect of the flow. This derivative is the operator denoted by Kobayashi-Nomizu, and it is easy to check that it is itself a tensor field for any fixed X, and therefore determines a section of the bundle of endomorphisms of the tangent bundle.

On a Riemannian manifold, the space of endomorphisms of the tangent space at each point is a module for the Lie algebra of the orthogonal group, and it makes sense to decompose an endomorphism into components which correspond to the irreducible factors. Said more prosaically, an endomorphism is expressed (in terms of an orthonormal basis) as a *matrix*, and we can decompose this matrix into an antisymmetric and a symmetric part. Further, the symmetric part can be decomposed into its trace (a diagonal matrix, up to scale) and a trace-free part.

In this language,

- the divergence of X is the negative of the trace of ;
- the curl of X is the skew-symmetric part of ; and
- the
*strain*of X is the trace-free symmetric part of .

The strain measures the infinitesimal failure of flow by X to be conformal. Under a conformal transformation, lengths might change but angles are preserved. The strain measures the extent to which some directions are pushed and pulled by the flow of X more than others; in general relativity, this is expressed by talking about the *tidal force* of the gravitational field. An extreme example of tidal forces is the *spaghettification* experienced (briefly) by an observer falling in to a black hole. In the theory of quasiconformal analysis, a *Beltrami field* prescribes the strain of a smooth mapping between domains.

**and so on.** This is a far from exhaustive survey of some of the key players in Riemannian geometry, and yet strangely I am temporarily exhausted. It is hard work to unpack the telegraphic beauty of Levi-Civita’s calculus into a collection of stories. And this is the undeniable advantage of the notational formalism — its concision. A geometric formula can (and often does) contain an enormous amount of information — much of it explicit, but some of it implicit, and depending on the reader to be familiar with a host of conventions, simplifications, abbreviations, and even *ad hoc* identifications which might depend on context. Maybe the trick is to learn to read more slowly. Or if you have a couple of years to spare, you can always do what I did, and go away and come back later when the material is ready for you. For the curious, I have a few notes on my webpage, including notes from a class on Riemannian geometry I taught in Spring 2013, and notes from a class on minimal surfaces that I’m teaching right now (much of this blog post is adapted from the introduction to the latter). Bear in mind that these notes are not very polished in places, and the minimal surface notes are very rudimentary and only cover a couple of topics as of this writing.

Tagged: curl, div, exposition, grad, Riemannian geometry, vector field ]]>

I have tried to include at least one problem in each homework assignment which builds a connection between classical geometry and some other part of mathematics, frequently elementary number theory. For last week’s assignment I thought I would include a problem on the well-known connection between Pythagorean triples and the modular group, perhaps touching on the Euclidean algorithm, continued fractions, etc. But I have introduced the hyperbolic plane in my class mainly in the hyperboloid model, in order to stress an analogy with spherical geometry, and in order to make it easy to derive the identities for hyperbolic triangles (i.e. hyperbolic laws of sines and cosines) from linear algebra, so it made sense to try to set up the problem in the language of the orthogonal group , and the subgroup preserving the integral lattice in .

First, let’s recall the definition of the* hyperboloid model* of the hyperbolic plane. In we consider the quadratic form , and let denote the group of real matrices preserving this form. The vectors with are those lying on a 2-sheeted hyperboloid; the positive sheet H is the one consisting of vectors whose z coefficient is positive, and is the subgroup preserving this sheet. For each vector v in H, the tangent space is naturally isomorphic to the set of vectors with ; i.e. the subspace of vectors “perpendicular” to v with respect to the form. The restriction of the quadratic form to the tangent space is positive definite, so it makes H into a Riemannian manifold, in such a way that acts by isometries. This group acts transitively, and the stabilizer of a point is conjugate to ; thus H with this metric is homogeneous and isotropic, and is a model for the hyperbolic plane.

Another model is the *upper half-space model* of the hyperbolic plane. In this model, we define H to be the subspace of complex numbers with positive imaginary part, and let denote the group of real matrices, which acts on H by fractional linear transformations:

This action is not faithful; the subgroup acts trivially, so the action descends to the quotient . The group acts transitively, and the stabilizer of a point is conjugate to ; thus (again) H is homogeneous and isotropic, and is a model for the hyperbolic plane. This reflects the exceptional isomorphism of groups .

The subgroup acts discretely with finite covolume (i.e. it is a *lattice* in the Lie group ); the quotient is the *modular surface* — an orbifold with underlying surface a sphere with one puncture, and two cone points with order 2 and 3 respectively; one sometimes calls this the -triangle orbifold, since it is made from two semi-ideal hyperbolic triangles with angles at the vertices (the third “ideal” vertex is at infinity, and corresponds to the puncture). There is an associated tessellation of the hyperbolic plane by such triangles whose symmetry group is in which the ideal vertices lie exactly at the rational numbers (plus infinity) on the boundary of hyperbolic space. Thus acts in a natural way on the set of rational numbers union infinity, which can be thought of as the projective line over . As an abstract group, is the free product of two cyclic groups of order 2 and 3 respectively, corresponding to the matrices

and

and all torsion elements in are conjugate to these elements or their inverse (note that these matrices have orders 4 and 6 respectively in ; it is only in that they have orders 2 and 3).

The group is an example of what is known as an *arithmetic lattice*; roughly speaking, the arithmetic lattices in semisimple Lie groups G are those with “integer entries”, in a suitable sense. Arithmetic lattices are characterized by the existence of many *hidden symmetries *— i.e. their finite index subgroups have surprisingly large normalizers in G. More formally, for a subgroup in G, we define the *commensurator* of to be the subgroup of G consisting of elements g such that the conjugate of by g intersects in a finite index subgroup. With this definition, Margulis famously proved that the arithmetic lattices are precisely those whose commensurators are dense, and that all other lattices (i.e. the non-arithmetic ones) have a commensurator which is discrete (and hence contains the lattice itself with finite index). In , all the arithmetic lattices are derived from quaternion algebras over totally real number fields. Roughly speaking, if is a totally real number field — i.e. a finite extension of obtained by adjoining some root of an integer polynomial with all real roots — and if is a quaternion algebra over , then we can find a group consisting of “integer” elements of of norm 1. Each real embedding of embeds in a quaternion algebra over ; this is either the Hamiltonian quaternions (which is a division algebra), or the algebra of real matrices (which has zero divisors). Then embeds as a lattice in a product of copies of and , one for each real embedding in the Hamiltonian quaternions and in respectively. The factors are compact, so if there is exactly one factor, embeds as a lattice in it, and projects to a lattice in ; these are exactly the arithmetic lattices.

It is a theorem of Borel that the only way to get an arithmetic lattice in which is not cocompact is to take — in other words, .

OK, now — how to reproduce this picture in the hyperboloid model? The most natural guess is to look at — the group of matrices with integer entries preserving the quadratic form and the positive sheet of the hyperboloid. So, what exactly is this group? Let’s let A be a matrix in this group, and denote its column vectors by u,v,w. One obvious matrix to take is the identity matrix; for that matrix, the vector w is which lies on the hyperboloid H, whereas the vectors u and v are orthonormal vectors in . But this property of a triple of vectors is preserved by the action of any element of , and therefore in general there is a bijection between such matrices and triples u,v,w where w lies on H, and u,v are orthonormal vectors in .

Now consider the condition that the entries of the matrix be integers. Let’s abstract the discussion slightly. Suppose V is a real vector space of dimension n, with a symmetric nondegenerate quadratic form Q. Let L be a lattice in V; this is a slightly different use of the word “lattice” than above (at least in flavor) — it means a discrete cocompact additive subgroup, isomorphic as a group to . We suppose that the lattice L is *integral* and *unimodular*; the first condition means that is an integer for all in L, and the second means that the matrix with entries has determinant 1 or -1 for any basis of L. Now, for any nonzero vector the linear function has image of finite index (because Q is nondegenerate and L has full rank) and therefore the kernel has rank (n-1). If has norm 1 or -1, then is itself an integral unimodular lattice in the vector space with respect to the quadratic form which is the restriction of Q.

In with the quadratic form Q as above, suppose we can find an integer vector w on the hyperboloid H. Then the intersection of with the lattice of integer vectors has rank 2, and since the form Q is positive definite there, we can find an orthonormal basis u,v of integer vectors for . Hence there is a matrix A in taking to w, and acts transitively on such vectors, with stabilizer isomorphic to , the group of symmetries of the square. If we want to restrict attention to orientation-preserving symmetries, then is cyclic of order 4, generated by

Let’s find another matrix. An integral vector w on the hyperbolic H is a triple of integers x,y,z so that ; one simple example is , and then it is straightforward to find vectors and for u and v. This gives the matrix

Actually, it is pretty easy to see that no other integral vector on H is closer to than , since is not a sum of two squares. Let’s let be the group generated by R and T. Some experimentation with fundamental domains confirms that this group is a lattice, and that the quotient is a sphere with one puncture and two orbifold points of orders 2 and 4; in particular, this is the entire group , and its quotient is the triangle orbifold.

So, this group is certainly not . In fact, a rotation of order 4 realized as an element of necessarily has a trace of , so it can’t even have rational entries. But wait — this is surely an arithmetic lattice (for any conceivable definition of arithmetic), and therefore corresponds to some lattice derived from a quaternion algebra over a totally real number field. Since it is not cocompact, the only possibility is that the number field is , so that this lattice is commensurable with . At this point I vaguely recall something from a course on arithmetic lattices I took from Walter Neumann over 20 years ago in Melbourne, in which he stressed that the trace field of an arithmetic lattice (i.e. the field generated by the traces of the elements, thought of as a subgroup of ) is not by itself a commensurability invariant — rather the trace field generated by the *squares* of the elements is invariant; and the squares of the elements in this group all have integer trace after conjugating into . So mathematics is consistent after all, and I learn the surprising (to me) fact that the and triangle orbifolds are commensurable. Hard to believe I have been working with Kleinian groups for 20 years without noticing that before . . .

Here’s a picture of the tiling of the hyperbolic plane whose symmetry group is :

The center is the projection of and the adjacent 8-valent vertices are the projection of .

**(Update May 20, 2014):** As galoisrepresentations points out, the fact that the field generated by traces of squares of elements is a commensurability invariant is a theorem of Alan Reid.

Tagged: arithmetic lattice, Hyperbolic geometry, orthogonal group ]]>

This impression was dramatically shaken by Agol’s proof of the virtual Haken conjecture and virtual fibration conjectures in 3-manifold topology by an argument which depends for one of its key ingredients on the theory of non-positively curved cube complexes — a subject in geometric and combinatorial group theory which, while inspired by key examples in low-dimensions (especially surfaces in the hands of Scott, and graphs in the hands of Stallings), is definitely a high-dimensional theory with no obvious relations to manifolds at all. Even so, the transfer of information in this case is still from the “broad” world of group theory to the “special” world of 3-manifolds. It shows that 3-manifold topology is even richer than hitherto suspected, but it does not contradict the idea that the beautiful edifice of 3-manifold topology is an exceptional corner in the vast unstructured world of geometry.

I have just posted a paper to the arXiv, coauthored with Henry Wilton, and building on prior work I did with Alden Walker, that aims to challenge this idea. Let me quote the first couple of paragraphs of the introduction:

Geometric group theory was born in low-dimensional topology, in the collective visions of Klein, Poincaré and Dehn. Stallings used key ideas from 3-manifold topology (Dehn’s lemma, the sphere theorem) to prove theorems about free groups, and as a model for how to think about groups geometrically in general. The pillars of modern geometric group theory — (relatively) hyperbolic groups and hyperbolic Dehn filling, NPC cube complexes and their relations to LERF, the theory of JSJ splittings of groups and the structure of limit groups — all have their origins in the geometric and topological theory of 2- and 3-manifolds.

Despite these substantial and deep connections, the role of 3-manifolds in the larger world of group theory has been mainly to serve as a source of examples — of specific groups, and of rich and important phenomena and structure. Surfaces (especially Riemann surfaces) arise naturally throughout all of mathematics (and throughout science more generally), and are as ubiquitous as the complex numbers. But the conventional view is surely that 3-manifolds per se do not spontaneously arise in other areas of geometry (or mathematics more broadly) amongst the generic objects of study. We challenge this conventional view: 3-manifolds are everywhere.

The *generic objects* that we discuss in the paper are *random groups*, in the sense of Gromov. In fact, there are two models of random groups that one usually encounters in geometric group theory. First, fix a finite number k (at least 2) of generators , and a length n; and then throw in random relations all reduced words of length n in the generators and their inverses, chosen randomly and independently from amongst all possible words of that length. The two models are distinguished by how the number of relators (i.e. ) depends on the length n. In the *few relators* model, one takes to be a fixed (positive!) constant. In the *density* model, one fixes a constant D between 0 and 1, and lets . The point is that there are approximately possible reduced words of length n to add as relators (each successive letter of a random word could be any generator or its inverse *except* for the inverse of the previous letter) and we are choosing to throw in a fixed multiplicative density of these words.

Suppose we are interested in some property of a group; for instance, that it should be infinite, or torsion-free, or abelian, or whatever. For each fixed n, we get a probability law on groups, and we can ask what the probability is that our random group (with relators of length n) has the desired property. Then one takes n to infinity and looks at the way in which the probability behaves; usually we are interested in properties for which the probability goes to 1 as n goes to infinity. We say then that a random group has the desired property *with overwhelming probability*.

Gromov showed that there is a natural phase transition in the behavior of random groups; at any fixed density D bigger than 1/2, a random group is either trivial or isomorphic to , with overwhelming probability. Conversely, at any fixed density less than 1/2, a random group is infinite, torsion-free, hyperbolic, and 2-dimensional. Since the group is 2-dimensional and hyperbolic, the boundary is 1-dimensional. Dahmani-Guirardel-Przytycki show that the boundary is a Menger sponge with overwhelming probability — i.e. the universal compact 1-dimensional topological space that every other 1-dimensional compact topological space embeds into it (one should say “metrizable” to be really rigorous here).

So in one sense, we know what the “generic” objects look like amongst finitely generated groups. But in another sense, the answer is unsatisfying — these groups are unfamiliar, and not obviously related to the sorts of groups that we understand well, like free groups, surface groups, matrix groups, and so on. So it becomes important to try to understand the structure of *subgroups* of random groups; do they contain subgroups that are familiar, which we can use as key structural elements to understand the big group? and is this subgroup structure rich enough that we can hope to find similar structure in all hyperbolic groups?

In order to make progress, we must first be clear about what sorts of subgroups we are looking for. We are interested in our groups not only as algebraic objects, but as geometric objects (with respect to some choice of word metric), and it is important to look for subgroups whose intrinsic and extrinsic geometry are uniformly comparable, so that the geometry of the subgroup (which we understand) tells us something about the geometry of the ambient group (which we want to understand). Since the random group G is hyperbolic, this means looking for subgroups H which are quasiconvex. Such groups are themselves necessarily hyperbolic, and the boundary of a quasiconvex subgroup H embeds in the boundary of G. Since the boundary of G is (topologically) 1-dimensional, the same is true of H, so we are led to the natural question: what hyperbolic groups have 1-dimensional boundary?

The answer to this question is essentially known, by work of Kapovich-Kleiner. First of all, a hyperbolic group with disconnected boundary splits over a finite group, by Stallings theorem on ends. Second of all, a hyperbolic group with connected boundary with local cut points is either virtually a surface group or splits over a cyclic group, by Bowditch. So we are led to essentially four cases:

- a Cantor set; in this case, H is (virtually) free. All nonelementary hyperbolic groups contain free subgroups, by Klein’s ping-pong argument; so random groups certainly contain such subgroups;
- a circle; in this case, H is (virtually) a surface group. It is a famous open problem of Gromov whether all one-ended hyperbolic groups contain surface subgroups. A positive answer is known in a few cases: Kahn-Markovic showed that closed hyperbolic 3-manifold groups contain surface subgroups (a key ingredient in Agol’s theorem). About a year ago, Alden Walker and I showed that random groups contain surface subgroups, and these subgroups are quasiconvex;
- a Sierpinski carpet; in this case,
*conjecturally*H is (virtually) the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. This conjecture more-or-less reduces (by a doubling argument) to the Cannon conjecture — that a hyperbolic group has boundary homeomorphic to the 2-sphere if and only if it is virtually the fundamental group of a closed hyperbolic 3-manifold; or - a Menger sponge; this is the boundary of the random group itself!

In view of this classification, François Dahmani asked me (after hearing the proof of my theorem with Alden) whether random groups could contain subgroups isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. This is precisely the main theorem that Henry and I prove in the paper; explicitly:

**3-Manifolds Everywhere Theorem: **A random group, either in the few relators model or in the density model at any density less than 1/2, contains many quasiconvex subgroups isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary.

The proof is direct — we basically show that one can directly construct a map from such a 3-manifold group into a random group (given by a random presentation) in such a way that it is very likely to be quasiconvex and injective. The argument borrows very heavily from many parts of my earlier paper with Alden, although the construction step is much more complicated.

It is possible to say something in general terms about the combinatorial construction. Our random presentation can be realized in geometric terms by building a 2-dimensional complex K, whose 1-skeleton X is a wedge of k circles (one for each generator), to which we attach disks along loops corresponding to the relators. Let r be one such relator; it is a long (cyclic) reduced word in the generators and their inverses. We can think of this word as being written along the edges of a circle L subdivided into intervals, with one letter in each interval. Imagine taking this circle and gluing it up to itself, matching sets of edges with the same label, so that the result is a labeled graph Z. If we then attach a disk along the boundary of the circle, we get a 2-complex M(Z), and this 2-complex immerses in K. If we are careful, we can arrange for M(Z) to have the homotopy type of a 3-manifold with boundary; and if the manifold is acylindrical and freely indecomposable with infinite fundamental group, it is the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary.

Gluing up L to produce the “spine” Z so that M(Z) is homotopic to a 3-manifold is thus the bulk of the work. The spine Z will be a 4-valent graph, and the circle L will map to Z with degree 3 (i.e. every edge of Z has 3 preimages). At each vertex of Z, 6 edges in L run over the vertex in all possible ways from one incident edge of Z to another. The figure below shows three local models; the correct local model is the third one:

The key to the construction is to glue up collections of segments in L in triples, leaving a gap of three unglued segments of some fixed length which are the three edges of a theta graph (we call them “football bubbles”). Almost all the mass of L can be glued up this way, so we produce a *reservoir *of bubbles in a predictable distribution, and a *remainder *with relatively small mass. There are some operations that can then be performed on the remainder, gluing it up into the desired form, at the cost of adjusting the reservoir somewhat. Then the great mass of the reservoir is glued up into small disjoint collections whose local combinatorics can be completely specified; one particularly pretty move glues up four football bubbles (with suitably labeled edges) by draping them along the edges of a cube, each bubble aligned with one of the diagonal axes:

This idea of first performing a random matching which is “almost” right, which can then be adjusted at the cost of perturbing the distribution of an almost equidistributed “sea” of predictable pieces of bounded size, so that the rest of the matching decouples into a massive number of matching problems of uniformly bounded size that can be solved once and for all — is one that has come up in several places recently, including in the papers of Kahn-Markovic and my paper with Alden mentioned above, but also in Peter Keevash’s construction of General Steiner Systems and Designs (a paper I learned of from Gil Kalai’s blog). This is an idea with remarkable power and potential, beyond the already impressive (but well-known) power of “random constructions”. And it shows that highly constrained and beautiful combinatorial and geometric objects — designs as well as 3-manifolds — can be built out of generic pieces.

I am not sure what the moral of the story is; perhaps in every corner of the geometric desert, beautiful flowers bloom.

Tagged: 3-manifolds, acylindrical, quasiconvex group, Random groups, Sierpinski carpet ]]>

So the purpose of this blog post is to advertise that I wrote a little piece of software called *kleinian* which uses the GLUT tools to visualize Kleinian groups (or, more accurately, interesting hyperbolic polyhedra invariant under such groups). The software can be downloaded from my github repository at

https://github.com/dannycalegari/kleinian

and then compiled from the command line with “make”. It should work out of the box on OS X; Alden Walker tells me he has successfully gotten it to compile on (Ubuntu) Linux, which required tinkering with the makefile a bit, and installing freeglut3-dev. There is a manual on the github page with a detailed description of file formats and so on.

One nice feature of the program is that the user just has to give semigroup generators for their (semi)-group, and a finite list of (hyperbolic) triangle orbits; the program then computes the Cayley graph out to some (user-specified) depth, applies the resulting set of transformations to the triangles, and renders the result. The code is available, and is licensed under the GPL, and I actively encourage anyone who wants to fork it and develop it into a more powerful tool to do so.

A few examples of output are:

universal cover of a genus 3 handlebody

universal cover of the fiber of the fibration of the figure 8 knot complement

space with Sierpinski carpet limit set invariant by super-ideal simplex reflection group

I wrote this program mainly just to produce some nice figures for a recent talk I gave at U Chicago to first-year graduate students; the talk itself can be downloaded from my webpage here. If you download this program, and enjoy using it, I would be very grateful to get feedback, or just to hear about your experience.

Tagged: software, visualization ]]>

If X is a connected CW complex, by successively attaching cells of dimension 3 and higher to X we may obtain a CW complex Y for which the inclusion of X into Y induces an isomorphism on fundamental groups, while the universal cover of Y is contractible (i.e. Y is a with the fundamental group of X). The (co)-homology of Y is (by definition) the group (co)-homology of the fundamental group of X. Since Y is obtained from X by attaching cells of dimension at least 3, the map induced by inclusion is an isomorphism in dimension 0 and 1, and an injection in dimension 2 (dually, the map is a surjection, whose kernel is the image of under the Hurewicz map; so the cokernel of measures the pairing of the 2-dimensional cohomology of X with essential 2-spheres).

A surjective map f from a space X to a space S with connected fibers is surjective on fundamental groups. This basically follows from the long exact sequence in homotopy groups for a fibration; more prosaically, first note that 1-manifolds in S can be lifted locally to 1-manifolds in X, then distinct lifts of endpoints of small segments can be connected in their fibers in X. A surjection on fundamental groups induces an injection on in the other direction, and by naturality of cup product, if is a subspace of on which the cup product vanishes identically — i.e. if it is *isotropic* — then is also isotropic. If S is a closed oriented surface of genus g then cup product makes into a symplectic vector space of (real) dimension 2g, and any Lagrangian subspace V is isotropic of dimension g. Thus: a surjective map with connected fibers from a space X to a closed Riemann surface S of genus at least 2 gives rise to an isotropic subspace of of dimension at least 2.

So in a nutshell: the purpose of this blog post is to explain how the existence of isotropic subspaces in 1-dimensional cohomology of Kähler manifolds imposes very strong geometric constraints. This is true for “ordinary” cohomology on compact manifolds, and also for more exotic (i.e. ) cohomology on noncompact covers.

**1. Fibered Kähler groups**

For a compact Kähler manifold Hodge theory gives

(recall that the notation means the holomorphic p-forms). In other words, every (complex) 1-dimensional cohomology class has a unique representative 1-form which is a linear combination of holomorphic and anti-holomorphic 1-forms. Since the wedge product of holomorphic 1-forms is holomorphic (the first miracle mentioned in the previous post!), for holomorphic 1-forms we have

if and only if as *forms. *

This has the following classical application:

**Theorem (Castelnuovo-de Franchis):** Let M be a compact Kähler manifold, and let V be a subspace of the space of holomorphic 1-forms on M which is isotropic with respect to the pairing (on cohomology; but equivalently, on forms). Suppose that the dimension of V is at least 2. Then there exists a surjective holomorphic map f with connected fibers from M to a compact Riemann surface C of genus g such that V is pulled back by f from C.

Proof: Let where be a basis of V. Where two forms don’t vanish, the condition that says that they are proportional, and therefore the ratio is a holomorphic *function*. If we let U denote the open (and dense) subset of M where none of the vanish, then the ratios define the coordinates of a holomorphic map to . Since is closed, its kernel is tangent to a (complex) codimension 1 foliation on U. Since the are closed, the ratio is constant on the leaves of , so the image of U in is 1-dimensional, and the map factors through a map to a compact Riemann surface D.

A priori a holomorphic map to a Riemann surface defined on an open set U does not extend to M; the simplest example to think of is the holomorphic function

where x and y are the two coordinate functions. This map is well defined away from the origin, where it is indeterminate. On the other hand, as we approach the origin radially along a (complex) line, the ratio is constant; so the map, defined on , extends over a copy of obtained by blowing up the origin. In general therefore a map extends to where M’ is obtained from M by blowing up along the indeterminacy of the map f, and the fibers of the blow-up map from M’ to M are all copies of .

Now, the map does not necessarily have connected fibers, but it is proper. So there is a (so-called) *Stein factorization* for some intermediate compact Riemann surface C, where has connected fibers, and is finite-to-one. As a set, the points of C are just the connected components of the point preimages of . As a complex manifold, the charts on are modeled on the transverse holomorphic structure on the foliation . Notice that since (as remarked above) the 1-forms are all locally constant on the leaves of , they descend to well-defined 1-forms on (which pull back to the under the map). In particular, we deduce that has genus at least . But now we see that there was no indeterminacy at all, since the fibers of the blow up admit no non-constant holomorphic map to a surface of positive genus, and therefore the map factors through after all. qed

Now suppose M is a compact Kähler manifold, and let V be a subspace of which is isotropic with respect to cup product, and of dimension at least 2. We can choose real harmonic 1-forms which are a basis for V, and take their holomorphic (1,0)-part . Then is holomorphic, and is equal to the (2,0)-part of . Since the holomorphic 2-forms inject into cohomology, it follows that as *forms*. It is straightforward to check that the are linearly independent if the are, so we obtain an isotropic subspace of holomorphic 1-forms of the same dimension as V. Applying Castelnuovo-de Franchis, we see that M fibers over D as above (this observation is due to Catanese).

From this we easily deduce the following theorem of Siu-Beauville, proved originally by hard analytic methods (i.e. the theory of harmonic maps):

**Corollary (Siu, Beauville):** Let M be a compact Kähler manifold, and let . Then there is a holomorphic map with connected fibers from M to a compact Riemann surface C of genus at least g if and only if there is a surjective homomorphism .

Proof: A surjective map with connected fibers is surjective on fundamental groups. Conversely, a surjective map on fundamental groups pulls back injectively, and pulls back a maximal isotropic subspace of (which has dimension ) to an isotropic subspace of . qed

**Definition:** A Kähler group is *fibered* if it surjects onto the fundamental group of a compact Riemann surface of genus at least 2; equivalently, if some (equivalently: every) compact Kähler manifold with that fundamental group holomorphically fibers over a compact Riemann surface of genus at least 2 with connected fibers.

Note that the condition of being fibered implies .

**2. L2 cohomology**

Perhaps the fundamental method in geometric group theory is to study a group via its cocompact isometric action on some (typically noncompact) space. If G is the fundamental group of a manifold M, then G acts as a deck group on the universal cover of M. The aim of geometric group theory is to perceive algebraic properties of the group G in the “global” geometry of this universal cover.

The most important tool for the study of differential forms on compact Riemannian manifolds is Hodge theory. To use this tool on noncompact manifolds one must impose additional (global) restrictions on the forms that one studies. Thus Hodge theory on noncompact manifolds is related directly not to ordinary cohomology, but to more refined, quantitative versions, of which one of the most important is -cohomology.

If M is a smooth Riemannian manifold (not assumed to be compact), the pointwise inner product on forms gives rise to a global inner product which is well-defined on compactly supported forms. We say that a smooth form is in if

Now, the -forms do not usually form a chain complex, but we can pass to a subcomplex consisting of forms for which both and are -forms. Since this is a complex, and we can define cohomology:

In general, the image of d is not a closed subspace (in the topology), so we define the *reduced* cohomology to be:

The advantage of working with reduced cohomology is that there is an -analogue of the Hodge theorem. The operators and still make sense on a noncompact Riemannian manifold, and so does . We can define the harmonic forms to be those for which , and we denote by the space of harmonic p-forms which are .

Let’s impose some reasonable global conditions on our manifold M. We say that a (complete) Riemannian manifold has *bounded geometry* if it satisfies the following two conditions:

- The curvature and its derivatives satisfy uniform 2-sided bounds: for each k; and
- The injectivity radius satisfies a uniform lower bound: everywhere.

Bounded geometry is the natural condition to impose to ensure that the manifold is “precompact” in Gromov-Hausdorff space; i.e. that for any sequence of points in the sequence of pointed metric spaces contain a subsequence which converge on compact subsets to a pointed Riemannian manifold . An equivalent way to think about it is that this is the condition which ensures that the Riemannian manifold can appear as a leaf in a compact lamination. The condition of bounded geometry is automatically satisfied for any cover (infinite or not) of a compact Riemannian manifold. Since this is essentially the only class of noncompact Riemannian manifolds we will consider, we hereafter assume that all our noncompact Riemannian manifolds have bounded geometry.

**Theorem (L2 Hodge theorem):** Let M be a complete Riemannian manifold with bounded geometry. Then every cohomology class in has a unique representative minimizing . Such a form is harmonic; i.e. it is in . Moreover, there is an orthogonal decomposition

One subtlety is that it is no longer true that is a formal adjoint to d, since integration by parts gives rise to a potentially nontrivial boundary term “at infinity”. But for an form , this boundary term vanishes, and one has

(since *a priori* the forms and are not , one first interprets this by using cutoff functions, and passing to a limit). In other words, a harmonic form *which is also * is closed and coclosed; conversely, any form which is closed and coclosed is harmonic (with no analytic conditions).

On a Kähler manifold the identity still holds pointwise (since this is a consequence purely of the local properties of the metric), and so there is a further decomposition of into components which are individually harmonic. There is furthermore a Hodge decomposition

and an form satisfies if and only if and . Thus consists precisely of *holomorphic ** p-forms*.

**Example:** A harmonic form which is not does *not* have to be in the kernel of d. For instance, a function is closed if and only if it is (locally) constant, but any nonconstant holomorphic function on a domain in has harmonic real and imaginary parts. On the other hand, suppose that is harmonic and , and exact as a form, so that for some smooth function f. Then we claim that f is actually harmonic (but not closed unless ). For, and commute, so is a constant c, and by the Gaffney cutoff trick, it can be shown that c=0.

**3. Kähler hyperbolicity**

Gromov showed that under certain geometric conditions, the reduced cohomology of a Kähler manifold vanishes outside the middle dimension. To define this condition, one first introduces the notion of a *bounded* form; this is a form for which is finite, where denotes the (operator) norm of at the point p.

**Definition:** A compact Kähler manifold M is *Kähler hyperbolic* if the pullback of the symplectic form to the universal cover satisfies for some bounded 1-form .

Suppose M is Kähler hyperbolic, and let be any harmonic form on . Then is closed, and

Since is bounded, the form is . On the other hand, is bounded (because it is pulled back from a form on a compact manifold), so is . Now, (recalling the notation L for the operation of wedging with the Kähler form), the Kähler identity is a purely local calculation, and therefore on any Kähler manifold (compact or not), wedge product with the Kähler form takes harmonic forms to harmonic forms. It follows that is harmonic, , and equal to the image of an form under d; thus it vanishes identically.

But if V is a real vector space of dimension 2n, and is a nondegenerate 2-form on V, then wedging with is injective on below the middle dimension (this is the linear algebra fact which underpins the Hard Lefschetz Theorem for compact Kähler manifolds). Thus the operator L is injective on harmonic -forms below the middle dimension. Dualizing, the operator is injective above the middle dimension, and we deduce the following:

**Theorem (Gromov): **If M is compact and Kähler hyperbolic, the reduced cohomology of the universal cover vanishes outside the middle dimension.

**Example:** If M is any compact manifold with then for any closed form on M the pullback of to the universal cover is d of a bounded form. This is proved by the Poincaré Lemma, since for a complete simply-connected manifold with , coning a submanifold along geodesics to a point gives a cone whose volume is bounded by the volume of the submanifold times a constant. So every Kähler manifold with a metric of strict negative curvature is Kähler hyperbolic. More generally, if M is merely nonpositively curved, and the flat planes are isotropic for the Kähler form, then the manifold is still Kähler hyperbolic. This applies (for example) to Kähler manifolds which are compact and locally symmetric of noncompact type. Generalizing in another direction, if M is Kähler with and word-hyperbolic, then M is Kähler hyperbolic.

**4. Calibrations**

The previous section shows that vanishes whenever M is Kähler hyperbolic of complex dimension at least 2, where denotes the universal cover of M. In fact, it turns out that one can completely understand the fundamental groups of Kähler manifolds for which is nonzero: it turns out that such groups are always virtually equal to the fundamental group of a closed Riemann surface of genus at least 2.

So let’s suppose M is a compact Kähler manifold, that is its universal cover, and let’s suppose that is nonzero. Since is simply-connected, every harmonic form (which is necessarily closed) is actually *exact*. Let be a nonzero harmonic form, and let denote its (1,0)-part, which is an holomorphic 1-form. Since is also exact, we can write for some holomorphic function on . By the coarea formula we compute

or in other words, most of the level sets have finite volume. On the other hand, these level sets are complete holomorphic submanifolds, and holomorphic submanifolds of Kähler manifolds turn out to enjoy a very strong geometric property, which we now explain.

On a Kähler manifold, the symplectic form is a *calibrating* form. This means that it satisfies the following two properties:

- it is closed; and
- it satisfies a pointwise estimate for all real 2k-planes A, with equality if and only if A is a complex subspace.

It follows that if S is a holomorphic submanifold of complex dimension k, and S’ is a real 2k dimensional submanifold obtained from S by a compactly supported variation so that S and S’ are in the same (relative) homology class, there is an inequality

In other words, holomorphic submanifolds of Kähler manifolds are absolute volume minimizers in their homology classes (amongst compactly supported variations). From this one deduces the following:

**Lemma:** Let M be a Kähler manifold with bounded geometry. Then for each k there is a constant C so that if S is a complete holomorphic submanifold of complex dimension k, there is an estimate

Proof: It suffices to show that for some fixed (taken to be the injectivity radius, say), there is a constant so that the volume of is at least for any point p in S. A Kähler manifold with bounded geometry is uniformly holomorphically bilipschitz to flat in balls of size smaller than the injectivity radius, so we need only prove this estimate for holomorphic submanifolds of .

But actually, the estimate follows just from the fact that S is a minimal surface. If S is a complete minimal surface of real dimension N in a Euclidean space, passing through the origin (say), then the *Monotonicity Formula* says that for any there is an inequality

This can be proved directly by using the vanishing of the mean curvature, but there is a softer proof that where C is the volume of the unit ball in Euclidean N dimensional space, which is enough for our purposes. To see this, observe that C is the limit of as R goes to zero. Suppose on some interval that somewhere, WLOG achieving its minimum at . The value of on gives a lower bound for the volume of , by the coarea formula. But the cone on evidently has less volume than this, in violation of the fact that S is calibrated. The estimate, and the proof follow. qed

It follows from this estimate that some of the fibers of are compact. The components of these fibers are the leaves of a foliation, and since the foliation is defined locally by a closed 1-form, the set of compact leaves is open; but these leaves are all locally homologous and thus have locally constant volume and therefore uniformly bounded diameter, so the set of compact leaves is closed, and therefore every leaf is compact. The space of leaves is 1 (complex) dimensional, and we thereby obtain a proper holomorphic map with connected fibers to a Riemann surface S. Note that the group of holomorphic automorphisms of (which includes the deck group ) must permute leaves of the foliation; for, since the leaves are compact, if their image were not contained in a leaf, the map to would be nonconstant, in contradiction of the fact that a holomorphic map from a compact holomorphic manifold to a noncompact one must be constant.

In summary, the deck group acts on permuting the fibers of the map h, and thus descends to an action on S. Because the fibers have uniformly bounded diameter, and the action of the deck group on is cocompact and proper, the action on S is also cocompact and proper. Since the map h is surjective with connected fibers, S is simply-connected; since the reduced -cohomology class is pulled back from S, it follows that S is the unit disk, and therefore contains a finite index subgroup which acts freely, and is isomorphic to the fundamental group of a closed Riemann surface of genus at least 2.

Now, it turns out that for a compact manifold M, the 1-dimensional -cohomology of the universal cover depends only on the fundamental group G of M, and is equal to , where the (reduced) cohomology groups may be defined directly from the bar complex. We have therefore proved the following theorem of Gromov:

**Theorem (Gromov):** Let G be a Kähler group with . Then G is commensurable with the fundamental group of a closed Riemann surface of genus at least 2.

**5. Ends**

To apply Gromov’s theorem (and its generalizations) it is important to have some interesting examples of groups with . Let X be a locally compact topological space. Then for every compact set K we have the set of components of X-K, and an inclusion induces . The *space of ends* of X (introduced by Freudenthal) is the inverse limit:

taken with respect to the directed system of complements of compact subsets. If each is finite, the space of ends is compact.

Now, let G be a finitely generated group. For each finite generating set we can build a Cayley graph C, which has one vertex for each element of G, and one edge for each pair of elements which differ by (right) multiplication by a generator. The graph C is locally finite and connected, and we define the *space of ends of* G, denoted , to be just . It turns out that this does not depend on the choice of a finite generating set, but is really an invariant of the group.

The theory of ends of groups is completely understood, thanks to the work of Stallings:

**Theorem (Stallings, ends of groups):** Let G be a finitely generated group. Then has cardinality 0,1,2 or . Moreover,

- if and only if G is finite;
- if and only if G is virtually equal to ; and
- if and only if G splits as a nontrivial amalgam or HNN extension or where B is finite, and G is not virtually cyclic.

Actually, the only hard part of this theorem is the third bullet; the rest is elementary, and was known to Freudenthal. The third case is equivalent to the existence of a nontrivial action of G on a tree T (which is not a line) with finite edge stabilizers. It follows that groups with infinitely many ends are non-amenable.

Now, let M be a compact Riemannian manifold, and suppose that the fundamental group G has infinitely many ends. This implies that the universal cover also has infinitely many ends, and we may find a compact subset of whose complement has at least two unbounded regions. Define a function f on which is equal to 0 on some (but not all) of the unbounded regions of and 1 on the rest. Then has compact support (contained in K) and is therefore . On the other hand, if is any function with then is a constant, so is constant and nonzero on some end of , and is therefore not . It follows that is nonzero in *unreduced* .

Now, on functions f we have an equality . The Laplacian is self-adjoint, with non-negative real spectrum. So to prove that is equal to it suffices to establish a *spectral gap* for ; i.e. to prove an estimate of the form

for all functions f of compact support (which are dense in ). In exactly this context one has the following famous theorem of Brooks:

**Theorem (Brooks):** with notation as above, one has if and only if is an amenable group.

One can think of the size of as governing the rate of dissipation of the norm of a function f as it evolves by the heat equation . Geometrically it is plausible that heat dissipates at a definite rate when it is concentrated in a region whose boundary is big compared to its volume (since then a definite amount of heat can escape out the boundary). So heat should dissipate at a definite rate *unless* there are a sequence of compact regions in , exhausting , for which . To each such region one can assign a finite subset of G, by looking at which translates of a basepoint are contained in ; this sequence of subsets is known as a Følner sequence, and the existence of a Følner sequence for a countable group G is one of the definitions of amenability (the equivalence to the other standard definitions is due to Følner). The hard details of Brooks’ argument are to show that one can take subsets whose boundary is regular enough that the comparison between volumes of subsets and their boundaries in the continuous and the discrete world is uniform.

So in conclusion, if G is a group with infinitely many ends, then reduced and ordinary cohomology agree in dimension 1, and we can construct a nontrivial class as above. Putting this together we deduce the following:

**Corollary (Gromov):** A Kähler group is either finite, or has 1 end.

Proof: A group with two ends is virtually equal to , which is not Kähler because it has odd. A group with infinitely many ends has nontrivial reduced -cohomology in dimension one. But for a Kähler group, this implies the group is commensurable with the fundamental group of a closed surface of genus at least 2; such groups have only 1 end after all. qed

**6. Ends and extensions**

The arguments of Gromov can be generalized considerably. It should be remarked from the outset that at very few points in the proof of Gromov’s theorem did we use the fact that the manifold was the universal cover of M.

The following is proved by Arapura-Bressler-Ramachandran:

**Theorem (Arapura-Bressler-Ramachandran): ** Let M be a complete Kähler manifold with bounded geometry, and suppose that has dimension at least 2. Then there is a hyperbolic Riemann surface S and a proper holomorphic map with connected fibers. Moreover, the fibers of the map are permuted by the holomorphic automorphisms of M, and the map induces an isomorphism from to .

Here the subscript “ex” means the harmonic 1-forms which are exact (as ordinary forms). Given an exact harmonic form we can take the holomorphic (1,0) part which is and closed. But we *cannot* assume it is exact if is nontrivial. If we only have one , then we are more or less stuck. But if we have at least *two* such forms, then the following remarkable Lemma (due originally to Gromov) applies:

**Lemma (cup product):** Let M be a complete Kähler manifold with bounded geometry, and let be real, harmonic exact 1-forms. Let be their (1,0)-components. Then pointwise.

Proof: The first remark to make is that on a complete Kähler manifold with bounded geometry, any harmonic form is actually bounded. Equivalently, since harmonic forms are smooth, there is no sequence of points going off to infinity such that the operator norms diverge. Since the manifold has bounded geometry, we can integrate the square of on disjoint balls of definite radius centered at such points, and the claim will therefore follow if we show the integral of the square of a harmonic form on a ball of definite radius is controlled by below by its value at the center. Assume we are in flat space; then this claim is obviously true for a linear form. But a harmonic form satisfies an elliptic 2nd order equation, which shows that the higher derivatives can be controlled in terms of the first derivative; the claim follows.

Now let be an exact harmonic form, and write . Suppose is a closed form. Then is in because is bounded (as above). If we define to be equal to f where and locally constant elsewhere, then is equal to where and vanishes elsewhere. But now is bounded, so is in , whereas in . We deduce that is zero in reduced cohomology.

Finally, if we let be the decomposition of the (1,0) forms into real and imaginary parts, then we compute

Now, the imaginary part of this is harmonic and ; on the other hand, we have just shown it is trivial in reduced cohomology. Thus it must vanish identically. But then must vanish identically too, proving the lemma. qed

It follows that the space determines (by taking holomorphic parts) an *isotropic* subspace of , which by hypothesis has dimension at least 2. Each form in this space determines a complex codimension 1 foliation whose leaves are tangent to the kernel, and because the forms are closed and the space is isotropic, this foliation is independent of the choice of form. Furthermore, on any open subset where two such holomorphic 1-forms do not both vanish, the ratio defines a holomorphic map to .

At this point the following fact is extremely handy:

**Proposition:** Let M be a connected complex manifold (not assumed to be compact!) and and linearly independent closed holomorphic 1-forms with . Then has no indeterminacy; i.e. it defines a holomorphic map from M to .

This Proposition is Lemma 2.2 in a paper of Napier-Ramachandran, where they seem to suggest that the fact is standard, but give an elementary proof. Since the argument is local, one can write and and then one observes that the functions are locally constant on the fiber over each point ; the argument then follows essentially from a (co)dimension count.

Anyway, once this proposition is proved, it follows that the components of the level sets of this function agree with the leaves of , which can be taken to be the points of a Riemann surface S. An argument similar to the one above (using a pair of real harmonic functions instead of a single holomorphic function in the coarea formula) shows that some, and therefore every, leaf is compact of bounded volume. Pulling back an form on S gives something by uniform boundedness of the volume of the fibers; conversely, exact harmonic forms on M descend to S because they are constant on the leaves of the foliation. This proves the theorem.

**Corollary:** Let G be a Kähler group, and suppose there is an exact sequence

where and . Then H is commensurable to the fundamental group of a compact Riemann surface of genus at least 2.

Proof: Let M be a compact Kähler manifold with fundamental group G, and let N be the cover with fundamental group K. Then H acts on N cocompactly, and it follows that . An unbounded sequence of deck transformations must push most of the mass of an harmonic form off to infinity, so necessarily the space is infinite dimensional; since is finite dimensional, there is an infinite dimensional space of exact forms. Thus N fibers over S as above. Since the map from N to S is surjective on fundamental groups, it follows that S is of finite type (because is finite dimensional). But the deck group H acts on S discretely and cocompactly by holomorphic automorphisms (which are isometries in the hyperbolic metric), so actually S is the disk. qed

**Example (Arapura):** The pure braid group surjects onto a (virtually) free group, with finitely generated kernel, and therefore it is never Kähler. Note that so these groups can’t always be ruled out as Kähler groups on the oddness of alone. On the other hand, pure braid groups are fundamental groups of hyperplane complements: the group is the fundamental group of the space of ordered distinct n-tuples of points in , which is the complement of a hyperplane arrangement in . So it follows that this quasiprojective variety can’t be compactified in such a way as that the compactifying locus has big codimension (or one could apply the Lefschetz hyperplane theorem).

(Updated November 26: added references)

Tagged: 1-forms, amenable groups, Castelnuovo-de Franchis, ends, Gromov, Kähler groups, Kähler manifolds, L_2 cohomology, spectral gap ]]>

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

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

This post is a gentle introduction to the (mostly local) geometry of Kähler manifolds themselves. Everything I say here is completely standard, and can be found in all the standard references (e.g. Griffiths and Harris; another very nice reference is Lectures on Kähler geometry by Moroianu). The main reason to go through this material so explicitly is to make transparent what parts of the theory still hold, and what need to be modified, when one considers the geometry of *noncompact* Kähler manifolds, especially those arising as (infinite) covering spaces of compact ones; but this point will need to wait to a subsequent post to be validated. The definition of a Kähler manifold has two parts: a linear algebra condition, and an integrability condition. We discuss these in turn.

**1. Linear algebra**

A *Euclidean structure* on V is just a positive definite symmetric inner product. After a change of basis, we can identify V with with its “standard” inner product (i.e. dot product). Thus the group of linear transformations of V preserving a positive definite symmetric inner product is isomorphic to the *orthogonal group* .

A *complex structure* on V is just a linear endomorphism J which squares to -1. Since V is real, the eigenvalues of J are i and -i, each occurring with multiplicity equal to half the dimension of V (so the dimension of V had better be even). The endomorphism J extends by linearity to a *complex*-linear endomorphism of the complexification , where it becomes diagonalizable, and there is a canonical decomposition where V’ is the i-eigenspace, and V” is the -i-eigenspace of J. For any vector v in V there is a canonical decomposition

which we write as v = v’ + v”, where v’ is in V’ and v” in V”. The map from V to V’ taking v to v’ takes the operator J to multiplication by i, and identifies V with the complex vector space V’. Thus the group of (real) linear transformations of V preserving J is isomorphic to the *complex linear group* .

A *symplectic structure* on V is a non-degenerate antisymmetric inner product. This means a bilinear map satisfying , and such that for any nonzero v there is a nonzero w with . After a change of basis, we can identify V with with its “standard” symplectic product; i.e. if we choose basis vectors then

, and

Thus the group of linear transformations of V preserving a symplectic form is isomorphic to the *symplectic group* .

Thus, a real vector space V of even dimension can admit a Euclidean structure, a complex structure, and a symplectic structure. These three structures are said to be *compatible* if they satisfy

for any two vectors v and w. Note that any two of these conditions implies the third. At the level of Lie groups, compatibility can be expressed in terms of the intersection of the stabilizers of the three structures:

- ,
- , and

Thus any two of the three structures (Euclidean, complex, symplectic) are compatible if the intersections of their stabilizers are isomorphic to a copy of the *unitary group*. The unitary group is the group of complex linear automorphisms of a complex vector space preserving a Hermitian form. This arises in the following way: a symmetric definite inner product on V induces a symmetric complex bilinear pairing on , and thereby a sesquilinear pairing H defined by

The restriction of H defines a Hermitian pairing on V’; identifying V’ with V gives a complex valued (real!) linear pairing on V whose real part is the given inner product, and whose imaginary part is the given symplectic form.

**2. Integrability, and Kähler manifolds**

Now let M be a real 2n-dimensional manifold. A *Riemannian metric* on M is a smoothly varying choice of positive definite inner product on the tangent spaces to M at each point. An *almost complex structure* is a smoothly varying choice of complex structure on the tangent spaces to M at each point. An *almost symplectic structure* is a smoothly varying choice of symplectic structure on the tangent spaces to M at each point. Expressed in terms of tensors, the Riemannian metric is a symmetric 2-form g, the almost complex structure is a section J of squaring to -1 pointwise, and the almost symplectic structure is an alternating 2-form .

The field of endomorphisms J determines a splitting of the complexification of T M into T’M and T”M pointwise. An almost complex structure is *integrable* if the bundle T’M is integrable; i.e. if the Lie bracket of two sections of this bundle is also a section of this bundle. Such a structure gives M the structure of a *complex manifold*, and is equivalent to the existence of an atlas of charts modeled on for which the transition functions between charts are holomorphic. An almost symplectic structure is *integrable* if the 2-form is *closed*; i.e. if as a form. Such a structure gives M the structure of a *symplectic manifold*, and is equivalent to the existence of an atlas of charts modeled on for which the transition functions between charts are symplectomorphisms (i.e. the derivative of the transition function at every point is a symplectic matrix).

**Definition: **A real 2n-manifold is *Kähler* if it admits a Riemannian metric, a complex structure, and a symplectic structure which are compatible at every point.

Every smooth manifold admits a Riemannian metric, and a manifold admits an almost complex structure if and only if it admits an almost symplectic structure (and either condition can be expressed in terms of properties of the characteristic classes of the tangent bundle). But the condition of integrability is much more subtle (at least for closed manifolds; any almost symplectic structure on an open manifold is homotopic to an integrable one).

**Definition:** A finitely presented group G is a *Kähler group* if it is equal to the fundamental group of a closed (i.e. compact without boundary) Kähler manifold.

Note that since the Kähler condition is preserved under taking covers and products, the class of Kähler groups is closed under passing to finite index subgroups, and taking (finite) products.

On any complex manifold we can choose coordinates locally so that the vector fields

are sections of T’M. The dual 1-forms and are a local basis for the smooth complex-valued 1-forms , and any complex 2-form can be expressed locally in the form

A Hermitian metric H determines such an h by ; the Hermitian condition is equivalent to the symmetry of h (i.e. that ) and positivity (i.e. that is real and positive for all nonzero v). Any Riemannian metric on a complex manifold can be averaged under the action of J pointwise and then complexified and restricted to T’M to produce a Hermitian metric. Taking imaginary parts gives rise to an alternating 2-form

which is nondegenerate pointwise (i.e. is nowhere zero). The metric is Kähler if and only if .

Now, on a Riemannian manifold, one may always locally choose *geodesic normal coordinates*, centered at any given point, and in which the metric tensor g osculates the Euclidean metric (in these coordinates) to first order; i.e.

where O(2) denotes terms vanishing to at least 2nd order at the center. One way to find such coordinates is to take Euclidean coordinates on the tangent space at the center point, and push them forward by the exponential map. For a Hermitian metric on a complex manifold, one can choose *holomorphic* local coordinates with this property *if and only if the metric is Kähler*; that is,

**Proposition: **A Hemitian metric h on a complex manifold M is Kähler if and only if there are local *holomorphic* coordinates at any point for which

One direction of this proposition is easy: for such a choice of coordinates, the form is constant up to first order, and therefore at the given point. But the definition of exterior d is coordinate free, and therefore holds everywhere.

**3. Dolbeault Cohomology**

On any almost complex manifold M, the decomposition of the complexified tangent space into T’ and T” gives rise to a decomposition of its dual space, and we can decompose the space of complex-valued n-forms into components One coordinate-free way to see this decomposition is to extend the action of J on the (complexified) tangent space to an action of the circle (by complex linearity); this gives rise to an action of the circle on the complexified cotangent spaces, and to all its tensor powers. Thus the space of complex-valued n-forms decomposes into invariant subspaces for this circle action; the fiber of over each point is the subspace where acts as multiplication by .

If the almost complex structure is integrable, we can choose holomorphic coordinates locally, and then is spanned by forms

Thus (by differentiating in the usual way) we see that (this fact is *equivalent* to the integrability of the complex structure) and we can decompose d into and respectively, where and . These operators satisfy

So, for example, on a Kähler manifold, the symplectic form is both *real* (i.e. contained in ordinary ) and of type in .

Since , the various form a complex, whose homology groups are the *Dolbeault cohomology*, denoted . By analogy with the Poincaré lemma (which proves vanishing of ordinary de Rham cohomology of smooth manifolds locally) there is the Dolbeault Lemma, which says that any form with can be *locally* written as . This lets us take resolutions and compute cohomology; if we write for the sheaf of *holomorphic *p-forms (i.e. those forms which are in the kernel of ) then we obtain the

**Dolbeault Theorem:** for any complex manifold M, there is an isomorphism .

In particular, can be identified with the *global* holomorphic p-forms, which we denote (by abuse of notation) also by .

From the Dolbeault Lemma one can also deduce the following:

**Local **** Lemma: **if is a real 2-form of type , then if and only if we can write locally in the form for some real function .

If is exact, such a function u can be found *globally*. When M is Kähler, the symplectic form can be expressed locally in the form ; such a function u is called a (local) *Kähler potential*. Conversely, every local potential u on a complex manifold for which the form is nondegenerate (i.e. satisfies in its domain of definition) gives the manifold locally the structure of a Kähler manifold. Note that a Kähler potential cannot exist *globally* on a compact Kähler manifold.

**4. Hodge theory**

A Riemannian metric on a manifold induces inner products on the fibers of all natural bundles over the manifold, including the cotangent bundle and its tensor and exterior powers. On a Riemannian manifold of dimension n there is a Hodge star defined pointwise by

and we get an inner product on forms by .

The Hodge star operator satisfies the identity on k-forms. Define an operator from to for each k, and define the *Laplacian* to be the operator .

A form is *harmonic* if ; the harmonic p-forms are denoted . On any compact manifold there is a *Hodge decomposition*

where the summands are orthogonal. One deduces that there is an isomorphism , and that every (de Rham) cohomology class contains a unique harmonic representative, which is also the unique representative of smallest norm.

Again on a compact manifold, it turns out that is the formal adjoint of d with respect to the pairing on p-forms (for any p), and therefore that

One proves this by integration by parts, since the difference between the two sides differs by the integral of an exact form. Thus, a form is harmonic if and only if it is closed and coclosed (i.e. in the kernel of ).

On a complex manifold we extend Hodge star to complex-valued forms so that is the local Hermitian pairing. Thus . We can define formal adjoints

and Laplace operators

On a Kähler manifold, a surprisingly difficult local calculation gives the crucial identity

and therefore the (p,q) components of a harmonic p+q form are themselves harmonic!

Explicitly, we have a Hodge decomposition for (p,q)-forms using :

where are the (p,q)-forms in the kernel of , from which one deduces the Dolbeault isomorphism ; but from one also gets the decomposition

One immediate miracle is the fact that on a Kähler manifold, *holomorphic forms are harmonic.* Explicitly, a (p,q)-form on a compact manifold is harmonic if and only if and . This follows from the identity

proved as before by integrating by parts. But for a (p,0) form, the operator is identically zero (since its image is in ), and a (p,0) form is in the kernel of if and only if it is holomorphic.

One reason to be impressed by this miracle is that the condition of being harmonic depends very delicately on the choice of a Riemannian metric, whereas the condition of being holomorphic depends only on the complex structure. Usually, the harmonic forms are only as regular as the metric; a Kähler metric is typically only smooth (one sees this by starting with one Kähler form and perturbing it by adding something of the form for u a small bump function) whereas a complex structure is *analytic*. Anyway, this miracle has another miraculous consequence: since the wedge product of two holomorphic forms is holomorphic, it follows that *the wedge product of two harmonic forms of type (p,0) and (q,0) is also harmonic, of type (p+q,0)*. As a rule of thumb, wedge products of harmonic forms (even on a Kähler manifold) is almost *never* harmonic, so this is an extraordinary fact.

**Example:** Let S be a closed Riemann surface of genus at least 2. There is a natural complex structure on S, and any Riemannian metric can be averaged under J to define a Hermitian metric, whose associated 2-form is automatically closed because S is 2-dimensional (as a real manifold). So S is Kähler. Let and be two real harmonic 1-forms which are not proportional; for instance, we could take to be the generator of . A real 1-form is dual to a vector field, and on a closed manifold, the number of singularities of a vector field (counted properly) is the Euler characteristic. Since , the forms and must be singular somewhere. This implies that must vanish somewhere; but the only (real) harmonic 2-form is the area form and its multiples, which does not vanish. Thus is never harmonic.

There are further symmetries of the various operators under consideration. Complex conjugation commutes with , so is isomorphic to . Similarly, the composition of Hodge star with complex conjugation commutes with , so is isomorphic to . If we denote the (complex) dimension of by , and the ordinary betti numbers of M by , we have identities

The last fact follows because the symplectic form and all its powers are real of type (p,p), and nontrivial in cohomology. In particular, notice that is even for k odd, and is positive for k even between 0 and 2n.

**Example:** finitely generated free groups are not Kähler, since they all have finite index subgroups with odd. The fundamental group of a Klein bottle is not Kähler, since it has ; on the other hand, this group has an index 2 subgroup which *is* Kähler (namely ).

**5. Hard Lefschetz Theorem**

One consequence of Hodge theory is so special it deserves to be singled out. Define an operator by (i.e. by wedging with the symplectic form). It has a formal adjoint ; in terms of an orthonormal basis it is defined by the formula (where denotes contraction — i.e. interior product). Define “twisted” operators

Then with these definitions one has the *Kähler identities*:

From this one can deduce another miracle: — in other words, *the operators and descend to operators on *. Notice as a special case that this implies the symplectic form is harmonic (it is *not* real analytic in general); actually this already follows from the fact that is closed, and so it is coclosed. More generally, the wedge product of the (harmonic) symplectic form with *any* harmonic form is harmonic.

The commutator acts on as multiplication by ; furthermore, it is elementary that and . Thus, the operators generate a copy of the Lie algebra , in a way which makes into a module over this Lie algebra. From the classification of finite dimensional modules, we deduce the:

**Hard Lefschetz Theorem:** The map is an isomorphism, and if we denote the kernel of by then . Furthermore, if we write the intersection of with by then .

Ordinary Poincare duality on a closed oriented 2n-manifold says that the pairing

is nondegenerate. Combining this with the Hard Lefschetz Theorem we deduce the Corollary:

**Corollary:** For all the pairing defined by

is nondegenerate.

The special case is particularly important; its nondegeneracy implies that the ordinary cup product cannot be too degenerate.

**Example:** if is the fundamental group of a closed surface of genus g, the universal central extension is not Kähler, since cup product on vanishes identically.

**6. Holonomy**

On any Riemannian manifold there is a unique connection on the tangent bundle called the *Levi-Civita connection *which is torsion-free, and which preserves the metric. This connection determines connections on the cotangent bundle and its tensor and exterior powers. If M is a complex manifold, and E is a holomorphic bundle on M with a Hermitian metric, any metric connection on M gives rise to connections ; decomposing the form part into types, there is a unique metric connection on E called the *Chern connection* whose (1,0) part is , when expressed in any local (holomorphic) coordinates.

The Kähler condition for a Riemannian metric on a complex manifold is equivalent to equality for the Levi-Civita connection and the Chern connection on the tangent bundle. This is equivalent to the condition that the tensors J and are parallel under (the Levi-Civita connection). Equivalently, the holonomy group of the metric is isomorphic to a subgroup of .

The coincidence of the Levi-Civita and Chern connections simplify the expression for the curvature of many natural bundles on a Kähler manifold. The most important example is the following. Let K be the canonical bundle on M (i.e.\/ the holomorphic line bundle whose holomorphic local sections are holomorphic n-forms where n is the dimension of M). Let denote the Ricci form on M; i.e. the real (1,1)-form defined by . Then the curvature of K (with its Hermitian metric arising from the Kähler metric on M) is equal to .

Some further remarks are in order:

- The Kähler condition already implies that is a real alternating form of type (1,1), and since it is the curvature of a line bundle, it is automatically closed. So the local lemma says that it can be expressed locally in the form for some real u. In fact, if the coefficients of the Hermitian metric are given by (expressed in local coordinates), then .
- Since the canonical bundle (as a holomorphic bundle, but ignoring its Hermitian metric) only depends on the complex structure, the form represents the first Chern class . Conversely, it is a famous theorem of Yau that on a Kähler manifold, for
*every*2-form representing the class there is a*unique*Kähler metric for which . As a corollary, M admits a Ricci-flat Kähler metric if and only if . - A Kähler metric is Ricci-flat if and only if the holonomy is a subgroup of . Such a manifold is the product (locally) of a flat manifold and compact pieces of complex dimension and with irreducible holonomy exactly equal to . These irreducible factors are called
*Calabi-Yau*manifolds. A Calabi-Yau has a compact universal cover, and therefore its fundamental group is finite.

**7. Weitzenböck formulae**

Suppose is a “natural” second order elliptic operator on sections of a metric bundle E over a Riemannian manifold M. Naturality should mean that its symbol is invariant under the action of whatever orthogonal group acts in a structure preserving way on whatever bundle the symbol lies in. In many cases it is possible to take the square root of the symbol, and identify the square root as the symbol of some first-order operator D, so that and have the same (second-order) symbol. *A priori* one might expect the difference to be first order; but in many cases, the condition of naturality forces the first order term to vanish (because of the lack of an orthogonal group-invariant bundle map between and ). Thus the difference is a 0th order operator — i.e. a tensor. The only natural tensor fields on Riemannian manifolds are curvature fields, so we obtain a formula of the form

for some and some . If is in the kernel of , then by integrating we get

The integral of the first term is non-negative, and strictly positive unless vanishes. So if is a *positive* operator, the kernel of must be trivial. Such formulae are called (in this generality) *Weitzenböck formulae*, and the use of such formulae to prove triviality of the kernel of a natural elliptic operator under a curvature inequality is called the *Böchner technique*. There is a beautiful survey article on such formulae and their uses by Bourguignon.

Depending on the context, the operators might be more or less complicated. The simpler is, the more useful the formula.

**Definition:** a real (1,1)-form on a complex manifold is *positive* (resp. *negative*) if is positive definite (resp. negative definite). A cohomology class in is positive (resp. negative) if it can be represented by a positive (resp. negative) form. A holomorphic line bundle L is positive (resp. negative) if there is a Hermitian structure on L for which is positive (resp. negative) where is the curvature of the Chern connection

A line bundle is positive if and only if its first Chern class is positive (this can be proved by adjusting the curvature of the bundle by adjusting the metric, using the global form of the -Lemma).

**Example:** The Kähler form of a Kähler manifold is positive. The Ricci form of a Kähler manifold with positive Ricci curvature (in the usual sense) is positive. The canonical bundle of a Kähler manifold has curvature , so if the manifold has positive Ricci curvature, the canonical bundle is *negative*. For example, is Kähler with positive Ricci curvature (for the Fubini-Study metric), so its canonical bundle is negative. The dual of a positive line bundle is negative and vice versa, so every projective variety admits a positive line bundle (by restriction).

Kodaira applied a Weitzenböck formula to positive and negative holomorphic line bundles on compact complex manifolds, and proved the following vanishing result:

**Proposition (Kodaira):** Let L be a positive holomorphic line bundle on a compact Kähler manifold M. Then there is a positive integer so that for all and all .

From this one deduces the famous

**Theorem (Kodaira embedding):** If L is positive, then is arbitrarily large for all sufficiently large positive k. Consequently, a Kähler manifold is projective if and only if it admits a positive line bundle.

Proof: For any holomorphic bundle E, the *holomorphic Euler characteristic*

can be computed from the Atiyah-Singer index theorem by the formula

where Td is the Todd class, and ch is the Chern character, both formal power series in the Chern classes of the tangent bundle and of E respectively. All we need to know about the Todd class is that it starts with 1 in dimension 0. For a line bundle L we have

Since L is positive, is positive, and integrates over M to give a positive number. If k is big, this term dominates, and therefore is positive for all sufficiently big k. On the other hand, for all and all sufficiently big k, so we deduce that has arbitrarily many linearly independent holomorphic sections, when is big; in other words, L is *ample*. We obtain a projective embedding from ratios of these sections in the usual way. qed

(Appealing to the Atiyah-Singer index theorem is a cheap way to get nonvanishing of from vanishing of for ; Kodaira constructed his sections more directly, by building them locally, and then showing that the obstructions to patching the local sections together globally — which are parameterized by the higher — vanish.)

**8. Lefschetz hyperplane theorem**

If M is a (complex) n dimensional smooth projective variety in , its intersection V with a generic hyperplane H is smooth. The inclusion of V into M induces a map , and the classical statement of the Lefschetz hyperplane theorem says that this map is an isomorphism in dimensions and an injection in dimension .

In fact this statement about homology has a refinement at the level of *homotopy*, which can be proved by Morse theory, as observed by Bott.

**Theorem (Lefschetz hyperplane):** Let M be a complex n dimensional smooth projective variety, and let V be its intersection with a generic hyperplane. Then is an isomorphism for and is surjective for .

Bott showed how to build a Morse function on (converging to on ) such that at every critical point, the Hessian has at least n negative eigenvalues. In particular, M is obtained from V by attaching handles of dimension at least n, from which the theorem follows.

In particular, it follows that any group which can arise as the fundamental group of a smooth projective variety, can arise as the fundamental group of a smooth projective variety of complex dimension at most 2.

**9. Examples of Kähler manifolds**

**Example ():** the group acts projectively, holomorphically and transitively on , and the point stabilizers are conjugate to . Since point stabilizers are compact, it leaves invariant a Riemannian metric (unique up to scale), which is evidently compatible with the complex structure. The associated almost symplectic form is invariant under the group action, and easily seen to be parallel, and therefore the metric is Kähler. This is called the *Fubini-Study* metric. The Kähler “potential” on gives rise to a closed 2-form which is degenerate in radial directions, and descends to the Kähler form on . The curvature of the metric is pinched between 1 (in totally real directions) and 4 (in totally complex directions)

**Example (nonsingular projective varieties):** the Fubini-Study metric defines compatible complex and symplectic structures on every complex subspace of the tangent space at each point of , so it defines an almost Kähler structure on every holomorphic submanifold. The restriction of a closed form to a subspace is closed, so this structure is integrable. In the same vein, any holomorphic submanifold of a Kähler manifold is Kähler.

**Example (bounded domains and their quotients):** A bounded domain U in carries a canonical Hermitian metric, called the *Bergman metric*, which is invariant under all biholomorphic self-mappings of U. This is a Kähler metric, and descends to a canonical Kähler metric on any quotient . In fact, with respect to the Bergman metric, the canonical bundle is negative, and therefore (when is cocompact and acts without fixed points) the quotient is projective (though not obviously so from the construction). Examples of bounded domains with a lot of symmetry are Hermitian symmetric spaces, so torsion-free cocompact lattices in groups like , , are Kähler groups.

**Example (Riemann surfaces):** Riemann surfaces are Kähler manifolds, and so are their products. Atiyah–Kodaira found examples of nontrivial algebraic surface bundles over surfaces, which can be obtained as branched covers of products over certain sections.

**Example ():** If M is any Kähler manifold with then M is actually projective. For, by symmetry, so . The Kähler form can be approximated by real harmonic 2-forms with *rational* periods, and by hypothesis, these nearby forms are of type (1,1). On the other hand, nearby forms are still positive, and because the periods are rational, after multiplying to clear denominators, the form is realized as the curvature of a (positive) line bundle.

**Example (Voisin):** Voisin found examples, in every complex dimension , of Kähler manifolds which are not *homotopic* to smooth projective varieties. However, these examples have free abelian fundamental groups, which are also fundamental groups of projective varieties.

(Updated November 21: added several references)

Tagged: complex geometry, Hodge theory, Kähler groups, Kähler manifold, Symplectic geometry ]]>

Remember that a *conformal map* is one which infinitesimally takes round spheres to round spheres. That is, it is *angle preserving*, at least infinitesimally. In particular, it is smooth. So let’s think about a conformal map between open regions in Euclidean 3-space (for concreteness). The image of a flat plane P is a smooth surface f(P). Pick a point p in P and look at its image f(p). Infinitesimal round circles around p in P get taken to infinitesimal round circles around f(p) in f(P). And straight lines perpendicular to P get taken to smooth curves perpendicular to f(P). If you take a smooth surface S in Euclidean 3-space, and a small round circle in S, and push the circle off S in the perpendicular direction, some directions will be distorted more than others (typically): the infinitesimal circle gets distorted to an infinitesimal ellipse, whose major and minor axes are the directions of principal curvature on the surface S. But these ellipses are the conformal image of small round circles in the domain, and therefore should also be (almost) round. In other words: the principal curvatures at each point of f(P) should be *equal*. A point on a surface where the principal curvatures are equal is called an *umbilical point*, and a surface on which every point is umbilical is called *totally umbilical*.

It is a classical fact, proved by Meusnier in 1785, that an umbilical surface in Euclidean space is locally a piece of a plane or sphere. One way to see this is as follows. Let G denote the Gauss map, so that the condition of being umbilical at a point says exactly that dG is a multiple of the identity at that point (note: we are using here in the usual way the canonical identification between the tangent space to the surface and the tangent space to the round sphere at the image of the Gauss map at each point to think of dG as a map from the tangent space to itself). So if a surface is totally umbilical, there is some function f so that dG is equal to f times the identity at each point. Let’s denote by X a local chart on the surface giving rise to local coordinates u and v. So the definition of f says in this notation that and . But then

Since u and v are local coordinates, their tangent vectors and are independent, and therefore . This means that is (locally) constant. But this means that the surface osculates a sphere (or plane) of *fixed* curvature to first order at every point, and therefore (by developing this sphere along a path in the surface) the center of this osculating sphere is fixed and the surface agrees (locally) with the sphere (or plane). Incidentally, Gauss was only 8 years old in 1785, so whatever Meusnier’s proof was, he could not have mentioned the Gauss map by name. Does any reader know Meusnier’s argument?

Once we know that a conformal map takes subsets of planes and round spheres to subsets of planes and round spheres, we can intersect these planes and spheres with perpendicular planes and spheres to see that it takes straight segments and arcs of round circles to straight segments and arcs of round circles. From this it is easy to deduce Liouville’s theorem.

By the way, I strongly suspect that the connection between totally umbilical surfaces and conformal maps is classical and well-known, and for all I know this was how Liouville thought of his theorem in the first place.

Tagged: conformal map, Liouville's theorem, Rigidity, umbilical surface ]]>

The story starts with the following classical theorem, usually called the Jordan curve theorem, or Jordan-Schoenflies theorem:

**Theorem (Jordan-Schoenflies):** Let P be a simple closed curve in the plane. Then its complement has a unique bounded component, whose closure is homeomorphic to the disk in such a way that P becomes the boundary of the disk.

In order to make the relationship between the two complementary components more symmetric, one could express this theorem by saying that a simple closed curve P in the 2-sphere separates the 2-sphere into two components X and Y, each of which has closure homeomorphic to a disk with P as the boundary.

Based on this simple but powerful fact in dimension 2, Schoenflies asked: is it true for every n that every n-sphere P in the (n+1)-sphere splits the (n+1)-sphere into two standard (n+1)-balls?

For n=2 (i.e. 2-spheres in the 3-sphere) Alexander showed in 1924 that the answer is *no*: there is an embedding of the 2-sphere in the 3-sphere for which a complementary region is not homeomorphic to a ball (in fact, it it not even simply-connected). This counterexample is the well-known *Alexander’s horned sphere*, illustrated in the figure below:

For the example indicated in the figure, the “outside” region is not homeomorphic to a ball, and in fact its fundamental group is infinite. Interestingly enough, Alexander duality implies that the complementary regions have the *homology* of a ball, and the fundamental group, though infinite, is therefore perfect (i.e. every element can be expressed as a product of commutators).

Alexander’s sphere has a Cantor set of “wild” points where the sphere is not *locally flat*; i.e. where there is no neighborhood U in which the 2-sphere sits in the 3-sphere locally like a flat plane in 3-space. So Schoenflies question was modified to ask about locally flat n-spheres in the (n+1)-sphere. Perhaps surprisingly, the answer to this modified question turns out to be *yes:*

**Theorem (M. Brown 1960):** Every locally flat n-sphere in the (n+1)-sphere bounds a standard (n+1)-ball.

Brown’s argument depends on a certain remarkably simple infinite construction, introduced by Barry Mazur, and called the *Mazur swindle*. Morally, the argument is as follows. If some locally flat sphere were not standard, it would exhibit the (n+1)-sphere S as the connect sum of two manifolds X and Y, neither of which were themselves (n+1)-spheres; i.e. X#Y=S. But then we can form an infinite connect sum X#Y#X#Y#X#Y# . . . which is still homeomorphic to S. On the other hand, since Y#X=S we can bracket this infinite sum as X#(Y#X#Y#X# . . .)=X#S=X, so X=S contrary to hypothesis.

Because of the infinite nature of this construction, the resulting manifolds are only shown to be *topologically* standard, and not *smoothly* standard, even if P is smooth. So it is natural to wonder whether every smooth n-sphere in the (n+1)-sphere bounds a smooth (n+1)-ball. This is a question where the dimension is very important. For n=2, this is a classical theorem of Alexander:

**Theorem (Alexander 1924):** Every smooth 2-sphere in the 3-sphere bounds a smoothly standard 3-ball.

This is proved by a kind of Morse theory argument. We let P be the 2-sphere in question, and we look at its intersection with a foliation of the 3-sphere (minus the north/south poles, and assume by general position that all but finitely many planes are transverse to P, and at the exceptional level sets we have a standard Morse critical point – a local minimum, a local maximum, or a saddle. At a non-critical level, the intersection of the plane with P is a compact smooth 1-manifold, and hence a collection of circles. By the Jordan-Schoenflies theorem, some innermost circle bounds a disk, and one can cut along this disk to produce two simpler spheres which, by induction, bound balls. Thinking about how these balls are glued together along the disk we cut along proves the theorem. The base step of the induction involves looking at pieces with two critical points, which are analyzed directly. qed

In high enough dimensions too, the question is known to have a positive answer:

**Theorem (S. Smale 1960):** For n at least 4, every smooth n-sphere in the (n+1)-sphere bounds a smoothly standard (n+1)-ball.

This follows (at least for n>4) from Smale’s **h-cobordism Theorem**, which says that if W is a smooth cobordism between two simply-connected manifolds U and V which are both deformation retracts of W, then W is a smooth product UxI, and therefore U and V are diffeomorphic. A smooth n-sphere in the (n+1)-sphere is cobordant to a tiny standard sphere around a point, and therefore the region between them is a smooth product, and when capped off with a tiny ball around a point, is a smooth ball.

The last remaining case is n=3; this is the

**Schoenflies Conjecture:** Every smooth 3-sphere P in the 4-sphere bounds a smoothly standard 4-ball.

As a technical point: of course, we want P to bound a smoothly standard 4-ball on *both* sides. But it turns out that if one side is smoothly standard, the other side is too, since (for example), we could shrink one side down (by a smooth isotopy) to a very small, round ball in a small coordinate patch where a Riemannian metric looks almost flat, and recognize its complement as a standard smooth ball once it is small enough.

OK, let’s get started! It is natural to try to reproduce Alexander’s argument one dimension lower, and consider the intersections of P with a foliation of the 4-sphere minus the north/south poles by 3-spheres of constant “latitude”. We can put P into general position, so that the height function defining these level sets is Morse, and put the critical points on distinct levels in increasing index; a technical improvement due to Kearton-Lickorish says that we can arrange for all handles to be horizontal (ie contained in a level sphere), and for all collars (between handles) to be vertical.

By Alexander duality, P divides the 4-sphere into two submanifolds X and Y (Marty had the clever mnemonic that one should think of these as the Xterior and Ynterior), each with the homology of a 4-ball (actually, by Brown, we even know that they are homeomorphic to 4-balls, but perhaps not diffeomorphic). As we build up P by a handle decomposition, we can also imagine that we are building up X and Y at the same time. The effect on X and Y of attaching a handle to P depends on which “side” of P the handle is added (in its level 3-sphere); one has the

**Rising Water Principle:** adding a 3-dimensional i-handle to P on the Y side has no effect on Y, but adds a 4-dimensional i-handle to X (and vice-versa).

This is perhaps a bit counter-intuitive, unless one thinks of a “4-d printer”, building up X and Y as we go. During the collar regions between critical levels, the printer adds layer after layer to the “top” of X and the “top” of Y, building them higher, but not changing their diffeomorphism type. Adding an i-handle to the Y side has the effect of putting a “cap” on the top of some subset of Y; above this level, the printer lays down material on Y only above the part in the complement of this “cap”. From this point of view it is clear that the topology of Y is not changing – we are just adding a product collar on the top of some subset of the top face. But on the X side we are adding a new “bridge” running over the i-handle, which is unsupported on lower levels.

This is illustrated schematically (and one dimension lower) in the figure above. The Xterior is in red, and the Ynterior in blue. At some level, a (2-dimensional) 1-handle is added on the Ynterior side (the green square in the second figure). Above this level, the effect on the Xterior is to add a (3-dimensional) 1-handle, while the effect on the Ynterior is nothing.

There are also two kinds of “duality” to think about: the core of an “ascending” i handle in P can be “turned upside down” to be the cocore of a “descending” 3-i handle in P. But an i handle in P corresponds to an i handle in X or Y (depending on whether it is on the Y or the X side), so when it is turned upside-down in corresponds to a descending 4-i handle in X or Y.

Marty gave a nice example to illustrate these ideas. Suppose P can be built in such a way that all the (3-dimensional) 0 and 1 handles are attached on the X side. If we turn this picture upside down, a 0 handle on the X side becomes a 3 handle on the Y side, and a 1 handle on the X side becomes a 2 handle on the Y. side. So turning the picture upside down, Y is built without any (4-dimensional) 2 or 3 handles; i.e. it is made just from 0 and 1 handles. But this means Y is diffeomorphic to a thickened neighborhood of a graph, and since it is homeomorphic to a 4-ball (by Brown’s theorem), it is diffeomorphic to a thickened neighborhood of a *tree*, and hence is standard.

One of the first observations to make is that if we cut P along a surface H above all the 0 and 1 handles, and below all the 2 and 3 handles, then the two sides of H in P are actually handlebodies, and H is a Heegaard surface. Every Heegaard splitting of the 3-sphere is standard (by an old theorem of Haken), so this is quite reassuring. The genus of H is called the *genus** of the embedding*. An embedding P is said to be a *Heegaard embedding* if *every* (nonsingular) level set is a Heegaard surface (not just the ones between the 1 and the 2 handles). A recent preprint of Agol-Freedman shows that every embedding can be isotoped to a Heegaard embedding, possibly at the cost of raising the genus dramatically.

It is natural to try to get some insight into the Schoenflies conjecture by restricting attention to a specific (low) genus. Marty Scharlemann famously proved the conjecture for genus at most 2; his paper appeared in the journal *Topology* in 1984. Something that Marty emphasized is the (*a priori* unexpected) fact that (hard) 3-manifold topology can be used to get insight in the Schoenflies conjecture, at least in the low genus case. For example, suppose P is a smooth 3-sphere in the 4-sphere and (with increasing height function) all 0 and 1 handles are attached on the X side. It follow that X can be built using only 2 and 3 handles. Turning the handle decomposition of X upside down, we see that X can be built using only 1 and 2 handles. If there is *only one* 1 handle and canceling 2 handle, then after attaching the 1 handle X is a circle times 3-ball, with boundary a circle times 2-sphere, and then the result of attaching a 2-handle is to do 0-framed surgery on a knot in the boundary circle times 2-sphere in such a way as to obtain the 3-sphere. Turning the handle decomposition of this 3-sphere upside down, we can say conversely that a circle times 2-sphere is obtained by 0 frame surgery a knot K (the co-core of the 2 handle in X) in the 3-sphere. Now, the famous Property R conjecture, proved in 1987 by Gabai, says that if 0-framed surgery on a knot K in the 3-sphere gives rise to a circle times 2-sphere, then K was the unknot. This shows that X is standard, and therefore Y too, and therefore P.

In general, knowing that X is built only from 1 and 2 handles is *not* known to be sufficient to show that X is standard. In the particular context of this example, one can get around this by studying the handle decomposition of Y: if we turn the original Morse function upside down, all 2 and 3 handles of P are attached on the Y side, so Y is built only from 0 and 1 handles. Any 4-manifold built from 0 and 1 handles is a smooth thickening of a graph; if it is contractible, the graph is a tree, and the 4-manifold (i.e. Y) is the smooth 4-ball. So in this particular case, we find a shortcut to the proof, bypassing the need for property R in this case.

But the idea of using 3-manifold topology to tackle Schoenflies is too good to pass up, and in fact, a certain purely 3-dimensional generalization of Property R would imply the Schoenflies conjecture. We explain how.

We have a smooth 3-sphere P in the 4-sphere, and to show it is standard it suffices to show that the two sides X and Y are standard 4-balls. In fact, just showing that *one* of them is standard implies that the other is, and that P is standard. Suppose that we somehow have some completely different smooth 3-sphere P’ in the 4-sphere, with sides X’ and Y’, and suppose we know that X and X’ are diffeomorphic (but *a priori* we don’t know anything about the relationship of Y and Y’). If we could show that X’ was standard, then of course X would be standard, and therefore also Y, and P. How might we find such a 3-sphere P’? Remember, the handles attached on the X side do not affect the topology of X. So if we build up P’ with the same abstract handle decomposition as P, attaching the handles on the Y side in the “same” way, but the handles on the X side in a possibly different way, we will construct X’ and Y’ for which we know that X and X’ are diffeomorphic, without immediately knowing anything about Y and Y’.

This new 3-sphere P’, with the same exterior as P, is called a *reimbedding*. Marty showed that for a genus 2 splitting, reimbedding can always make one side (say Y’) a handlebody. Just as above, a handlebody is always standard, so Y’ is standard, and therefore so is X’, and therefore X, and therefore Y, and therefore P.

It is worth remarking that reimbedding circumvents one natural drawback in a naive approach on the Schoenflies conjecture. Suppose one wanted to show directly that any smooth 3-sphere P was standard, by performing some canonical sequence of simplifying moves on P, ultimately obtaining a standard round 3-sphere. For instance, one could hope to find a flow which gradually straightened out the kinks, making P flatter and flatter until it could be recognized. The existence of such a flow would prove more than just Schoenflies: it would prove not just that the space of smooth (oriented) embeddings of the 3-sphere in the 4-sphere is path-connected (which is another reformulation of Schoenflies) but that its homotopy type was that of SO(4), the space of embeddings of round 3-spheres in the round 4-sphere. By contrast, reimbedding just jumps magically from one point in the space of embeddings to another, and if the Schoenflies conjecture were true, one would know that the two points were joined by a path, but without having to choose an explicit path from one to the other.

Let’s return to Schoenflies. Our original morse function has handles in increasing order, so we can always arrange to find some level 3-sphere with the 0 and 1 handles below, and the 2 and 3 handles above. This 3-sphere splits X into two sides, which are both 4-dimensional handlebodies. Suppose one further knew that the intersection of X with this level 3-sphere was itself a 3-dimensional handlebody. Then X could be represented as a *Heegaard union*. This implies (by a direct argument) that X admits a handle decomposition with only 1 and 2 handles. Is such an X a smooth 4-ball? By looking at the boundary of X in the dual handle decomposition, we see this is equivalent to a 3-dimensional question:

**Generalized Property R Conjecture: **if surgery on an n-component link L in the 3-sphere gives a connect sum of n circle times 2-spheres, can L be transformed into the unlink by handle slides?

It is known that this conjecture would imply Schoenflies for embeddings P with a single minimum or maximum (this follows from a recent result of Agol-Freedman; for details see their preprint) . Unfortunately, it seems likely that this conjecture is *false*: Gompf produced an example of a genus 4 splitting of the 3-sphere in the 4-sphere which gives the fundamental group of X the following presentation of the trivial group:

Generalized Property R would imply that this presentation could be reduced to the trivial presentation of the trivial group by a sequence of *Andrews-Curtis* moves; i.e. Nielsen transformations and conjugation of relators. The Andrews-Curtis Conjecture says that every balanced presentation of the trivial group can be reduced to a trivial presentation (with the same number of generators and relations) by Andrews-Curtis moves, but this conjecture is widely believed to be false, and the presentation above is widely considered to be a premier candidate counterexample.

One can try to weaken this Generalized Property R conjecture by allowing extra kinds of moves, for instance stabilization, corresponding to adding canceling 1-2 handle pairs or canceling 2-3 handle pairs at the level of X. Are these Weak Generalized Property R Conjectures true or false? Let’s find out!

(Update 10/18/13: made a couple of corrections due to Marty Scharlemann)

Tagged: 4-manifolds, Heegaard splittings, Marty Scharlemann, Poincare conjecture, Schoenflies conjecture, smooth topology ]]>

I remember seeing my first cube some time in early 1980; my Dad brought one home from work. He said I could have a play with it if I was careful not to scramble it (of course, I scrambled it). After a couple of hours of frustration trying to restore the initial state, I gave up and went to bed. In the morning the cube had been solved – I remember being pretty impressed with Dad for this (later he admitted that he had just taken the pieces out of their sockets). Within a year, Rubik’s cube fever had taken over – my Mum bought me a little book explaining how to solve the cube, and I memorized a small list of moves. I remember taking part in an “under 10” cube-solving competition; in the heat of the moment, I panicked and got stuck with only two layers done (since there were only two competitors, I came second anyway, and won a prize: a vinyl single of the Barron Knights performing “Mr. Rubik”). The solution in the book was a procedure for completing the cube layer by layer, by judiciously applying in order some sequence of operations, each of which had a precise effect on only a small number of cubelets, leaving the others untouched. In retrospect I find it a bit surprising – in view of how much effort I put into memorizing sequences, reproducing patterns (from the book), and trying to improve my speed – that I never had the curiosity to wonder how someone had come up with this list of “magic” operations in the first place. At the time it seemed a baffling mystery, and I wouldn’t have known where to get started to come up with such moves on my own. So the appearance of my kids playing with a cube 33 years later is the perfect opportunity for me to go back and work out a solution from first principles.

The one useful item I remember from that book was the notation for the cube operations; if we orient the cube in a particular way, and label the faces as up, down, front, back, left, right (in the obvious way), then an anticlockwise twist of one of these faces is denoted by a lower case letter u,d,f,b,l,r and a clockwise twist by the corresponding upper case letter U,D,F,B,L,R. Thus a sequence of moves – and its effect on a cube (in solved initial state) is illustrated in the following figure:

There is nothing special about the sequence RuRLdBBFRulBDD; the idea is just to observe how scrambled the cube can become with the application of a very small number of moves.

The first step of the solution is to “build a layer” – i.e. to get all the cubelets with some given color into the correct position and orientation. This can be done quite easily – first get the “edge cubelets” (those which have two free faces) into place, then the “vertex cubelets”. I think this really is something that can be achieved just by a bit of mucking about, and if you have never played with a cube before, I encourage you to get one, play around with it, and try to build a layer, just to see how easy it is (if a physical cube is hard to come by, you can always play around with the .eps code that generated these figures; see the end of the post). In fact, exactly the same techniques will let you put any four edge cubelets and any four vertex cubelets together in a face, in any orientation, providing you don’t care about the effect on the rest of the cube. This latter observation may not seem particularly useful at this stage, but in fact it is the key to a complete solution; for the sake of notation, let’s refer to this step as *setting up a face*.

Now, having built the first layer, the next step is to build the second layer. There are four edge cubelets that need to be positioned in the second layer; if the first layer is intact, these cubelets are either in the second layer but in the wrong position or orientation, or they are in the third layer. So it suffices to work out how to swap a cubelet from the second layer with one in the third layer – without disturbing the rest of the first or second layers, of course. Well, as an intermediate step, suppose we can swap a cubelet from the second layer with one in the third layer, putting no restrictions on what the effect is on the first or second layer. That’s easy – it’s just the operation of setting up a face. So we can find some sequence of moves that does what we want – call it s – and then survey the result. After performing s, the two edge cubelets that we want to interchange are both in the third layer, and everything else in the third layer was there before performing s. So let’s just twist the third layer (by some power of the “U” move) and replace the cubelet from the second layer with the cubelet from the third layer we want to replace it with. Now here’s the trick: follow that by performing S – the inverse of the operation s. The net result is the operation sUS – a conjugate of U. What is its effect? Well, the operation U itself just permutes the eight cubelets in the top layer (nine including the center, which is fixed of course). So any conjugate of U will also permute just eight cubelets. Which eight? Well, the eight which are in the third layer after performing s – i.e. 7 cubelets from the initial third layer, and the cubelet from the second layer we want to swap. Thus sUS has the effect of swapping one cubelet between the second and third layer, while leaving the remainder of the second and first layers intact, which is exactly what we want. Some experimentation gives a short recipe for an operation of the form s; the result is illustrated in the next figure:

The third layer can be solved by a similar principle. Consider a setting up a face operation s which takes the cubelets in the third layer and scrambles them in a precise way – e.g. by interchanging two edge or vertex cubelets, or changing the orientation of one edge or vertex cubelet. This has some (unpredictable) effect on the first two layers, mucking them up somehow. But the commutator of s and U – i.e. the operation sUSu – will unscramble the first two layers, putting them back as they were, since the support of U is the third layer, and therefore U commutes with any permutation of the first two layers. The effect on the third layer is relatively easy to predict; in the cases described above, it will cyclically permute three edge or vertex cubelets, or change the orientation of two edge or vertex cubelets respectively. These four moves, used in concert, can unscramble the third layer; here’s an explicit example (in this example, one of the moves on edge cubelets affects the vertex cubelets, so the edge cubelets should be put into the correct location and orientation first, and then the vertex cubelets):

There is no claim that these operations are “optimal”; they’re the first thing I came up with when I worked this out last night. Note that these operations do *not* allow you to set the third face up in an arbitrary way while keeping the first two faces fixed; this is because the allowable operations of the Rubik’s cube do not generate the full group of permutations of the oriented cubelets (even conditioned on taking vertex cubelets to vertex cubelets and edge cubelets to edge cubelets). I leave it as an exercise in finite group theory to show that the operations described above allow one to unscramble the cube from any configuration in which it *can* be unscrambled by legal moves.

That’s it! That’s the whole solution. Similar ideas make it easy to solve variations on the cube (e.g. 4x4x4, cubelets with pictures on the faces, tetrahedra, etc.). And it was quite gratifying to see Anna and Lisa so excited to discover the solved cube this morning (and to know that I hadn’t cheated!)

If you want to play with the .eps code that generated these figures, I’ve attached it at the end (yes, I know it’s a hack):

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Tagged: commutators, group theory, Rubik's cube ]]>

The basic object of study in Masur-Minsky is the complex of curves. This object is a kind of nonlinear analogue (in the context of surface topology) of a Bruhat-Tits building, and was introduced by Harvey in the late 70′s; but it was not until the mid-90s that the first interesting theorems about it began to be proved. We fix a surface S and define a simplicial complex C(S), called the *complex of curves*, as follows: an n-simplex in C(S) is a collection of (n+1) isotopy classes of essential embedded curves in S which are pairwise disjoint (this means that any two classes of curves in the collection admit disjoint representatives; it is an elementary but important fact about 2-dimensional topology that this implies that there are a collection of representatives of all the classes which are *simultaneously* pairwise disjoint). The *mapping class group* of S, denoted Mod(S), is the group of isotopy classes of (orientation-preserving) self-homeomorphisms of S. Evidently Mod(S) acts on C(S) by automorphisms. Furthermore, the action is cocompact (i.e. the quotient of C(S) by the action of Mod(S) is compact); however, the action is not *proper*. One way to see this is to think about the stabilizer of a simplex in C(S). A simplex denotes a collection of disjoint curves (up to isotopy); at the very least, the Dehn twists in each of these curves is a commuting family of mapping classes which preserves the collection elementwise. So the stabilizer of an n-simplex contains at least a free abelian group of rank (n+1).

Actually, this lack of properness turns out to be a virtue. If G is a (finitely generated) group, and X is a path metric space on which it acts properly and cocompactly by isometries, then G (in its word metric with respect to any finite generating set) is quasi-isometric to X. The mapping class group is not (for most surfaces S) a hyperbolic group, because it contains lots of free abelian subgroups of high rank (depending on the topology of S), so it can’t be quasi-isometric to a hyperbolic metric space. But because the action of Mod(S) on C(S) is not proper, it leaves open the possibility that C(S) is hyperbolic. This is exactly what Masur-Minsky prove.

The key tool is an operation called *subsurface projection*. Let X be an essential subsurface of S. Define the subsurface projection as follows. For each vertex (i.e. for each isotopy class of essential simple closed curve ) first isotope so that it intersects X in as few components as possible. The definition of the projection falls into several cases:

- if then set ;
- if intersects , then take the boundary of a tubular neighborhood of and throw away the inessential or peripheral components, and set equal to the rest; and
- if then define .

In order for this definition to not be vacuous, we should insist that X has some curves which are essential and non boundary parallel (so that X should be neither an annulus nor a pair of pants). But actually, it is very interesting and important to extend the definition to the case that X is an annulus. If X is an annulus, Mod(X) is just the integers, generated by a Dehn twist in the core circle. One can give a modified definition of C(X) by picking “base points” on the two boundary components, letting the vertices be isotopy classes of proper arcs from one base point to the other, and joining two vertices by an edge if the corresponding isotopy classes are disjoint (except at their endpoints, of course). With this definition, C(X) is just a copy of the real line with vertices at the integers, and Mod(X) acts by (integer) translation. If X is a pair of pants, any mapping class in Mod(X) is a product of Dehn twists in the boundary curves; so these are superfluous from the point of view of understanding subsurface projection. Now, if X is an annulus in S, and is an essential simple closed curve in S, we can choose a hyperbolic structure on S and let be the cover corresponding to the annulus X, and let be the collection of preimages of in . These preimages are disjoint, so if one of them crosses the lift of X to it determines (up to bounded ambiguity) an essential arc in X and therefore an element of C(X).

Now we can describe how subsurface projection is used to control the geometry of C(S). Lets suppose we choose a family of subsurfaces which are permuted by the action of Mod(S). For each index i there is a subsurface projection ; putting these together we get a map from to the product of the (actually, this is not quite true, since for any curve on S, the image of the projection of is empty in C(X) whenever is disjoint from X; what is better behaved is the diameter of the projection of a subset of C(S) to each ). Now, for any two simplices in C(S) we can take their projection to each . Suppose (e.g. by induction) that we understand distance in each better than we do in . Then we can define a sort of “distance” in C(S) between and by adding up the distances between the projections to each . The problem is that each projection involves some choice, and each choice involves a bounded but nonzero amount of ambiguity;* a priori* we might imagine that these infinite bounded choices might add up to a globally uncontrollable ambiguity. Masur-Minsky’s ingenious solution is simply to impose a cutoff: they pick a sufficiently big number K, and then ignore all subsurfaces in which the projections have distance less than K, and then add up the distances in the projections bigger than this. If the family of surfaces is rich enough, the resulting metric is quasi-isometric to the path metric in C(S). If the are hyperbolic, then so is C(S). qed.

Even if the family is not “rich enough”, it is still the case that this construction defines a sort of (approximate) quotient metric on C(S), implicitly defining a hyperbolic space on which Mod(S) acts in an interesting way. If we tailor our choice of surfaces suitably, we can make this action sensitive to certain kinds of information in Mod(S) and not others. What is maybe not obvious is that there are nontrivial variations on this construction that can be obtained by choosing certain kinds of metrics on the — for instance, by choosing *asymmetric* metrics.

Such asymmetric metrics in turn can be chosen to be sensitive to *chiral* information about mapping class groups. The simplest kind of chiral distinction one can make is the distinction between a left-handed and a right-handed Dehn twist. The handedness of a Dehn twist is defined by standing on one boundary component of the annulus supporting the twist, looking inwards, and seeing whether the image of the co-cores (under the twist) twist to the left, or to the right before getting to the other component. Importantly, this does not depend on choosing a side of the annulus to stand on (or, equivalently, an orientation of the curve); although it *does* depend on a choice of orientation on the surface. More generally, if X is a surface with boundary, an element of Mod(X) is *left-veering* (resp. *right-veering*) if, for any point x on the boundary, and any proper arc emanating from x, the image of under the mapping class twists to the left at x relative to (since we are only concerned with isotopy classes of arcs, we mean “the representative which intersects minimally”). The left-veering elements of Mod(X) form a cone (i.e. the product of two left-veering elements is left-veering) and similarly for the right-veering elements; moreover, every conjugate of a left-veering element is left-veering. But it is *not* the case that every nontrivial element is veering one way or the other. The existence of these invariant cones allows one to construct interesting asymmetric metrics on Mod(X) as follows: first choose some (symmetric) generating set for Mod(X). Next, pick some big integer K, and add to the generating set all right-veering elements of word length at most K. The resulting (asymmetric!) generating set defines a new notion of word length and asymmetric “distance” on Mod(X) in which it is harder in some sense to turn left than right. In the case that X is an annulus, so that Mod(X) is just the integers, this is the asymmetric metric in which the distance from N to M is equal to if M is bigger than N, and if N is bigger than M. Since C(X) is also just a copy of the integers when X is an annulus, we can similarly define canonical asymmetric metrics on such C(X). Now define an (asymmetric) metric on C(S) by choosing some Mod(S)-invariant collection of annuli , and for each one define (signed) distance to be the sum of the signed distance of the projections, truncated so that we ignore all projections whose signed distance is sufficiently small (in absolute value).

What is the point of building an asymmetric metric? The key point is that asymmetric metrics on hyperbolic groups (or groups acting on hyperbolic spaces) give rise to *quasimorphisms*, by antisymmetrizing. If G is a group, a function is said to be a quasimorphism if there is some least non-negative real number (called the *defect*) so that for all there is an inequality . A quasimorphism is further said to be *homogeneous* if for all and all integers . If is any quasimorphism, it may be homogenized by defining . Homogenization might increase the defect by (at most) a factor of 2, but it takes quasimorphisms to homogeneous quasimorphisms and furthermore has the property that is bounded. The (real vector) space of homogeneous quasimorphisms on a group G is denoted ; it contains as a subspace the homomorphisms to (i.e. ), which is precisely the subspace on which the defect vanishes. Thus, the defect becomes a norm on , and in fact this quotient space is a *Banach space*. Now, if G is a hyperbolic group, and is an asymmetric (path) metric, then we can define a quasimorphism . It turns out that many classical constructions of quasimorphisms on hyperbolic groups, and groups acting on hyperbolic spaces, are exactly of this kind; for instance the so-called counting quasimorphisms invented by Rhemtulla and rediscovered by Brooks, and the quasimorphisms of Epstein-Fujiwara. Antisymmetrizing asymmetric subsurface projection metrics on mapping class groups gives rise to chiral quasimorphisms which are sensitive to the difference between left and right twisting.

(As an aside, it is interesting to remark that in another talk at the same conference by Zlil Sela, he (Zlil) talked about the problem of solving systems of equations in free semigroups, which he proposed to solve by analogy with his solution for systems of equations in free groups, using the fact that a free semigroup embeds into a free group on the same generators. Many steps of the argument are very similar to the group case. One interesting intermediate step involves analyzing the JSJ decompositions of certain intermediate objects; the subgroups in this decomposition that cause the most trouble are the surface subgroups. In the free group case, one needs to factor out by the action of the mapping class group to get suitable finiteness. In the semigroup case, the surfaces are “decorated” by certain directed structures, coming from the irreversibility of multiplications in semigroups (versus groups); I wondered to Zlil whether the mapping classes preserving such structure would (at least in certain cases?) have something in common with the cones of right-veering (or left-veering) mapping classes described above).

Quasimorphisms on mapping class groups sensitive to chirality should be important for understanding the symplectic geometry of 4-manifolds, especially of surface bundles over surfaces and their cousins. One such chiral invariant (probably the most important) is the signature (for a closed, oriented 4-manifold, the intersection form on middle dimensional homology is symmetric and definite, and the *signature* of the 4-manifold is the signature of this form). Actually, certain connections between signatures and quasimorphisms are reasonably well-known; Wall non-additivity for signatures (i.e. the phenomenon that if 4-manifolds and are glued along subsets of their boundaries to produce also with boundary, then correction term) can be understood as measuring the failure of certain quasimorphisms to be honest homomorphisms; here the correction term is a Maslov triple index, associated to the symplectic rotation quasimorphism on the universal central extension of the symplectic group. Some of this story is well summarized by Gambaudo-Ghys. But as far as I know, the direct construction of quasimorphisms from chirally asymmetric metrics is new, and unexplored; it would be very interesting to see how much 4-manifold topology it can see.

Actually, if I can speculate, it seems to me that the machinery of subsurface projections seems well suited to analyze general symplectic 4-manifolds. Donaldson famously proved that every symplectic manifold admits the structure of a Lefschetz pencil — roughly, a surface bundle outside some (controlled) singular set of codimension 4. In 4 dimensions this gives the symplectic manifold the structure of a surface bundle over a surface outside finitely many “singular fibers” which look like surfaces pinched to a point along some embedded cycle, and the monodromy around this singular fiber is a positive Dehn twist in the pinched cycle. The fibers are Poincare dual to multiples of the cohomology class of the symplectic form (let’s assume we have perturbed and scaled it to be integral), and such fibrations exist for all sufficiently big multiples. Moreover, these fibrations are *unique* (up to isotopy) when the multiple is big enough. So symplectic 4-manifolds are not quite surface bundles. But away from the singular fibers they are, and the monodromy — or at least its projections to subsurfaces avoiding the vanishing cycles — are coarsely well-defined.

I wanted to end this post by pointing out that the “subsurface projection plus cutoff” trick can actually already be seen in a few other geometric contexts in which the connection to quasimorphisms is already explicitly known. One example concerns the random turtles in the hyperbolic plane, discussed in a previous post. One fixes a distance D and an angle A and considers a “turtle” in the hyperbolic plane, who proceeds by repeatedly taking steps of length D and then turning either left or right through angle A (randomly, independently, and with equal probability). When A is sufficiently small (for fixed D), the trajectory of the turtle is a quasigeodesic. However, when A passes some threshold (again, keeping D fixed), the trajectories stop being quasigeodesic, and in fact the winding number of the turtle (as a function of time) is distributed like a Gaussian variable. The variance of the winding number is very sensitive to the difference ; when this difference is small, the variance is very small, and may be estimated as follows. Think of the turtle’s choices of left or right turns as an infinite sequence of Rs and Ls. It might be the case that each step-plus-turn induces an elliptic isometry of the hyperbolic plane through some angle (with N big when is small), so that any string will produce a full left turn, and any string will produce a full right turn. *However*, it is a fundamental feature of hyperbolic geometry that “correlations decay exponentially” — that is, if we have any sequence of Rs and Ls in which there is no substring of at least N consecutive Ls or Rs, the resulting curve is quasigeodesic, and contributed nothing to the winding number at all. The different or substrings are projections to elliptic subgroups centered at different points in the plane; if we sum up all the contributions that exceed the threshold, the result gives the winding number of the turtle.

Here’s another example (very closely related to the turtle, and to the subject of quasimorphisms). Let S be a compact hyperbolic surface with a nonempty geodesic boundary. Let be a “random” bi-infinite geodesic on S (one must be a bit careful how one defines this — for example, one can use Patterson-Sullivan measure on the limit set of the fundamental group). If p is a point in S, we can ask how many times “winds around” p. This means: join p to the boundary of S by a shortest arc, and count the algebraic intersection number of this arc with the geodesic . For any p the winding number (as a function of time) is a Gaussian; but once again, the variance is very sensitive to how close p is to the boundary. If p is very close to the boundary, the only way a geodesic can go “around” it is to have a very long subsegment which is very parallel to the boundary; i.e. it must wind around the boundary many times. If we lift to the universal cover, and consider the projections of the random geodesic to the many different lifts of the boundary components, only those projections whose length is very long will contribute to winding number around p, and all those below a threshold will contribute 0.

One final example is more suggestive than explicit, and I wonder if one can do more with it. Let G denote the group of area preserving diffeomorphisms of the unit disk, supported strictly in the interior (so that each element is fixed on some neighborhood of the boundary). There is a beautiful homomorphism from G to , discovered by Calabi. One definition is as follows. Let be the area form in the disk. This is exact, so we can write . If is a diffeomorphism with then is closed, hence exact, hence for some function . Then the integral of over the disk (with respect to the area form) is a number, and this is Cal(f) (actually, it is possible that there should be a factor of somewhere, depending on the normalization one uses). Another beautiful definition of this homomorphism, discovered by Fathi, is as follows. The group of diffeomorphisms of the disk is contractible, so any can be joined to the identity by a path, unique up to homotopy. Under the track of this path, points in the disk move about; for any pair of points, we can compute the number of times one winds around the other (this is not quite well-defined, so apply the track n times, compute the winding number divided by n, and then take the limit as n goes to infinity). This gives a number which depends on which two points one starts with. So compute the *average* of this number over all pairs of points, where we use the invariant measure on the disk to define a measure on the space of pairs of points. This example was vastly generalized by Gambaudo-Ghys: choose any positive integer M and any quasimorphism on the braid group on M points, and compute the expected value of this quasimorphism on a random M-tuple of points. One can already think of this as a kind of subsurface projection — looking at the diffeomorphism as a homotopy class rel. any finite (approximate) orbit, and recovering a dynamical invariant as a kind of average of projections which are sufficiently “long” to persist under taking powers (actually, I wonder what the result is if one simply truncates the result on pairs of points whose relative winding is less than some fixed cutoff). But one can go further. Entov-Polterovich show the existence of a kind of Calabi quasimorphism on the group of area-preserving diffeomorphisms of the sphere, with the property that for every disk D of area at most 1/2, the restriction to the subgroup supported in D agrees with the (usual) Calabi homomorphism defined above. I wonder if there is a direct way to build such invariants by “projecting” the dynamics of some arbitrary diffeomorphism f of the sphere to some subdisk D (eg by looking at how the tracks of the points cross through D and surgering them with segments of the boundary of D), computing a Calabi homomorphism for each (or “enough”) such subdisk(s) (e.g. enough to cover the sphere), then counting contributions from (maximal) subdisks where this contribution is big enough.

Tagged: asymmetric metric, complex of curves, Lefschetz pencils, left veering, mapping class groups, Masur-Minsky, quasimorphisms, signature, subsurface projection ]]>

The rationale for the workshop (which I had some hand in drafting, and therefore feel comfortable quoting here) was the following:

Recently there has been substantial progress in our understanding of the related questions of which hyperbolic groups are cubulated on the one hand, and which contain a surface subgroup on the other. The most spectacular combination of these two ideas has been in 3-manifold topology, which has seen the resolution of many long-standing conjectures. In turn, the resolution of these conjectures has led to a new point of view in geometric group theory, and the introduction of powerful new tools and structures. The goal of this conference will be to explore the further potential of these new tools and perspectives, and to encourage communication between researchers working in various related fields.

I have blogged a bit about cubulated groups and surface subgroups previously, and I even began this blog (almost 4 years ago now) initially with the idea of chronicling my efforts to attack Gromov’s surface subgroup question. This question asks the following:

**Gromov’s Surface Subgroup Question:** Does every one-ended hyperbolic group contain a subgroup which is isomorphic to the fundamental group of a closed surface of genus at least 2?

The restriction to one-ended groups is just meant to rule out silly examples, like finite or virtually cyclic groups (i.e. “elementary” hyperbolic groups), or free products of simpler hyperbolic groups. Asking for the genus of the closed surface to be at least 2 rules out the sphere (whose fundamental group is trivial) and the torus (whose fundamental group cannot be a subgroup of a hyperbolic group). It is the purpose of this blog post to say that Alden Walker and I have managed to show that Gromov’s question has a positive answer for “most” hyperbolic groups; more precisely, we show that a random group (in the sense of Gromov) contains a surface subgroup (in fact, many surface subgroups) with probability going to 1 as a certain natural parameter (the “length” of the random relators) goes to infinity. **(update April 8:** the preprint is available from the arXiv here.**)**

First let’s start with the precise definition of a random group. There are actually two parameters in the definition — the *density* and the *length* . A random group at density and length is obtained by fixing a finite generating set with at least 2 elements, and adding “random” reduced words of length as relators, where the number of relators to add is governed by the density . Precisely, is the *multiplicative density* of the relators. There are (approximately) cyclically reduced words of length , so we choose subwords, independently and with the uniform measure, as our relators , and then define to be our “random group”.

Gromov introduced random groups and established some of their basic properties. One talks about a random group *at density* , and says that it has a certain property *with overwhelming probability*. What this means is that with fixed , the probability that the property holds goes to 1 as . Gromov showed that there is a remarkable phase transition in this definition. Explicitly, he showed:

**Theorem (Gromov):** A random group has the following properties with overwhelming probability:

1. At the group is either trivial or isomorphic to ;

2. At the group is infinite, hyperbolic, and 2-dimensional; and

3. At the group satisfies the small cancellation condition .

The story at density is more subtle, and it is not so clear what happens, as far as I know. **(****update April 6: **Piotr Przytycki points out that the one-endedness of random groups is actually due to Dahmani-Guirardel-Przytycki. Thanks Piotr!**)**

With this definition, the main theorem Alden and I prove is the following:

**Theorem (Calegari-Walker):** A random group at density contains many quasiconvex surface subgroups, with probability .

In particular, they contain surface subgroups with overwhelming probability. In fact, at the MSRI conference I gave a partial announcement of this theorem, saying only that we could prove the existence of surface subgroups at “some positive density”; I was worried about the fact that at density the group is no longer and therefore not a small cancellation group in the classical sense. However, it turns out that Yann Ollivier developed enough elements of a kind of small cancellation theory for random groups at any that the argument can be pushed all the way.

The proof contains some technical details, but I believe that some of the main ideas of the proof can be given in a blog post. But before I do so, I think it is worth discussing (very) briefly why one might be interested in finding surface subgroups.

For certain classes of hyperbolic groups — for example, fundamental groups of hyperbolic 3-manifolds — finding a surface subgroup was always known to be an important question to give insight into the virtual Haken conjecture. In fact, the Kahn-Markovic construction of such subgroups turned out to be one of the key steps in the eventual proof of that conjecture by Agol. But even beyond 3-manifolds per se, surface subgroups play an important role. At the MSRI conference Vlad Markovic talked about an approach he has to Cannon’s Conjecture — which says if is a hyperbolic group whose boundary is homeomorphic to a 2-sphere, then is virtually isomorphic to a (hyperbolic) 3-manifold group — and his approach depends on being able to find “enough” (quasiconvex) surface subgroups of . I asked Gromov (by email) what had motivated him to pose this question; I don’t think he would mind if I shared his reply, which was:

I do not remember exactly my motivations and heuristic evidence in favor of the existence of “many surface groups in many hyperbolic groups” except for connectedness arguments at the boundaries, but I had (and am having) a feeling that these are essential structural components of hyperbolic groups.

My own view, and my main interest in this question, is stimulated by a belief that surface groups (not necessarily closed, and possibly with boundary) can act as a sort of “bridge” between hyperbolic geometry and symplectic geometry (through their connection to causal structures, quasimorphisms, stable commutator length, etc). Surface groups are the “simplest” kind of hyperbolic groups after free groups, and surfaces themselves are the “simplest” class of symplectic manifold; any route between the two kinds of geometry must surely say a lot about surfaces. In this vein, I should remark that in the world of 3-manifold topology (where these issues are infinitely better understood), surfaces again play the premier role in both worlds: minimal/pleated/shrinkwrapped surfaces in the hyperbolic world, norm minimizing/pseudoholomorphic/convex in the contact/symplectic world. It is worth remarking that for the longest time *embedded* surfaces played a preeminent role in both theories, but that recent breakthroughs (on the hyperbolic side) have depended on developing a deep understanding of *immersed* surfaces. I wonder whether there is an important role for immersed surfaces on the symplectic side (in -manifold topology)? Maybe a reader who is an expert on Heegaard Floer homology can offer an opinion.

OK, let’s move on to the proof of the Random Group Surface Subgroup Theorem. The first step of the proof builds on a construction in our paper Surface subgroups from Linear Programming, where we show that a sufficiently random homologically trivial collection of cyclic words in a free groups can be taken to bound a certain kind of combinatorial object called a *Folded Fatgraph* (this result also underpins the main theorem in my recent related paper Random graphs of free groups contain surface subgroups, joint with Henry Wilton). A fatgraph is just an ordinary graph together with a choice of cyclic ordering on the edges incident to each vertex. Such a graph can be canonically fattened to a compact surface (with boundary) in which it lies as a spine. Stallings famously observed that an immersion (i.e. a locally injective simplicial map) between graphs is injective on fundamental groups; such a map of graphs is said to be* folded*. Thus a folded fatgraph gives an injective surface (with boundary!) subgroup of a free group with prescribed boundary.

The first step in our paper is to make this result more quantitative. A trivalent fatgraph with reduced boundary words is necessarily folded. Our first main result is the following

**Thin Fatgraph Theorem:** If is a sufficiently random homologically trivial collection of cyclically reduced words in a free group , then for any there is some depending only on so that copies of bounds a trivalent fatgraph in which every edge has length at least .

These fatgraphs have very long edges and are trivalent; hence are “thin”. Let me not say anything about the proof except that the first part of it closely models the proof of Thm 8.9 from our SSLP paper linked above, but the last step (which was done by computer in the SSLP paper) depends on an elementary but complicated combinatorial argument (which takes up almost half the paper!). (It is worth remarking that this last combinatorial step has something morally in common with the Kahn-Markovic proof of the Ehrenpreis conjecture via the theory of “good pants homology”, in that we want to cancel some collection of “superfluous” short loops which can be thought of as random excitations on the surface of a (Dirac) sea of perfectly equidistributed loops. I should also remark that some version of this theory — “pants homology” if you will — was earlier developed by me in my paper Faces of the scl norm ball, in which I showed that every homologically trivial immersed collection of geodesics on a hyperbolic surface virtually cobounds an immersed subsurface with a sufficiently large multiple of the boundary.)

By the way, it is natural to wonder just how “random” the collection needs to be for the conclusion of the theorem to hold (technically, we work with a deterministic property called “pseudorandomness” which is a kind of controlled equidistribution at certain scales). One can ask how long a random cyclically reduced (homologically trivial) word needs to be before it bounds a trivalent fatgraph (with, for the sake of concreteness, no constraint on the length of edges). This is a question that can be addressed experimentally by computer. The results are very interesting. For rank 3, we looked at between 100000 and 400000 such words of each even length from 10 to 120. The proportion of such words that bound trivalent fatgraphs is plotted below:

The first time we did this experiment, we only looked at words up to length 50 or so; needless to say, this gives a somewhat misleading idea of the asymptotic picture!

How can one use thin fatgraphs to build surface subgroups? Before tackling a random group, let’s consider a one-relator group with a single (long, random) relator . We can imagine building a (polygonal) surface out of disks, each of which has either or on its boundary, where the disks are glued to each other along mutually inverse subwords of the boundary words. Since a random word will probably not be homologically trivial, we build a surface out of disks labeled and disks labeled , where is as in the Thin Fatgraph Theorem. The 1-skeleton of is a graph, and the way in which it sits in gives it a fatgraph structure.

The first thing one might think therefore is that one should just apply the Thin Fatgraph Theorem to build a fatgraph bounding . One can do this, but why should one expect the resulting surface to be injective? In order for the surface to fail to be injective there must be some essential loop in the 1-skeleton which bounds a van Kampen disk in the group. Without loss of generality, we can assume that this disk has a minimal number of faces; note that each face has either or on its boundary. A (random) 1-relator group is hyperbolic; in fact, it is for any with overwhelming probability, when the length of the relator gets long. So in such a van Kampen diagram there must be very long subwords (of length ) in or which are subwords of . Of course, does contain long subwords of and ; the boundary of the fattening of consists entirely of such words! But in a minimal van Kampen diagram such “boundary” subwords must not occur, and the question is whether contains long subwords in common with or that are not boundary-parallel.

A counting estimate gives the following heuristic answer. By the defining property of a Thin Fatgraph, for any there are paths in of length starting at any point, and only starting points. On the other hand, there are random reduced words of length , and the relator contains at most of them. The difficulty in making this argument rigorous is that the fatgraph is not independent of ; in fact it is constructed “from” in a direct sense! So the trick is to break up into small subwords, and build thin fatgraphs bounding each subword, and then each small thin fatgraph will be independent of the other subwords.

Explicitly, we find what we call a *Bead Decomposition* of ; this is a decomposition of into subwords of length which start and end with mutually inverse subwords of length . The inverse subwords at the start of each are paired, to produce a collection of *beads* of size , separated by intervals of length called *necks*. Each bead on its own will probably be homologically essential, but we can perform a bead decomposition at “the same” locations in the word to get a collection of pairs of inverse beads of length . Taking copies of each pair of beads, we can build a thin fatgraph that bounds it, and then these thin fatgraphs are joined one to the next along necks. By construction, the subwords contained in the spine of the fatgraph bounding a bead are independent of the subwords in for , so with overwhelming probability, they have no long subwords in common. The necks are sufficiently long that whenever a subword passes over a neck, another copy of that subword cannot appear within distance for some (with high probability). But the existence of a van Kampen diagram would give rise to a long string of such coincidences, and therefore we deduce that no van Kampen diagram exists, and the surface is injective.

We now throw in an additional random relators of length independently, and with the uniform measure. Now the naive counting argument above is rigorous, and each additional relator is unlikely to have a long segment in common with a subpath in the spine . In fact, what can be shown is that for each there are of order relators that have of their boundary in common with a subpath of (this common part does not need to be consecutive, but we do need to bound the number of connected components by some constant independent of ; this is where Ollivier’s small cancellation work comes in to bootstrap such “local” small cancellation estimates to “global” ones). From this argument, and some elementary reasoning with van Kampen diagrams, the result follows.

One subtlety is that it is necessary to control the size of the van Kampen diagrams we consider independently of . A path in a hyperbolic group which is not quasigeodesic can be shortened on a segment of size , where is the constant of hyperbolicity. Ollivier shows that is *linear* in , for fixed , and therefore we can obtain estimates on the probability that fails to be injective by considering van Kampen diagrams containing a *bounded* number of disks.

Tagged: ergodic theory, Gromov's surface subgroup question, hyperbolic groups, Random groups, surface subgroups ]]>

https://github.com/dannycalegari/wireframe

and then compiled on any unix machine running X-windows (e.g. linux, mac OSX) with “make”.

The program is quite rudimentary, but I believe it should be useful even in its current state. Users are strenuously encouraged to tinker with it, modify it, improve it, etc. If you use the program and find it useful (or not), please let me know.

A couple of examples of output (which can be created in about 5 minutes) are:

and

(added Feb. 20, 2013): I couldn’t resist; here’s another example:

**(update April 12, 2013:)** Scott Taylor used wireframe to produce a nice figure of a handlebody (in 3-space) having the Kinoshita graph as a spine. He kindly let me post his figure here, as an example. Thanks Scott!

Tagged: software, visualization ]]>

So the “correct” answer to the puzzle is 7 (and the sequence continues 11, 26, ). This is somehow meant to illustrate some profound point; I don’t quite see it myself. Anyway, I would like to suggest that there is a natural sense in which the “real” answer should actually be 8 after all, and it’s the point of this short blog post to describe some connections between this puzzle, the theory of cube complexes (which is at the heart of Agol’s recent proof of the Virtual Haken Conjecture), and the location of the missing 8th region.

Actually, there is no great mystery about where the missing 8th region went. To ensure general position, I first needed to choose lines which were not parallel to each other, so let’s suppose that I have chosen a direction for the lines in advance. As I lay them in the plane one by one, I must also make sure that they don’t intersect an existing crossing. Since I have already chosen a direction for the line, I will either lie to the left or to the right of each existing crossing. After laying two lines, there is one crossing, so there are two choices for how I should lay the third line (given its direction); one choice gives the arrangement above; the other choice gives the following arrangement, which contains the missing 8th region:

Here I am thinking of the two different ways of arranging 3 lines in the plane as being related by a certain kind of “move” which translates the lines but does not turn them. The complementary regions are then specified by knowing on which side of each of the 3 lines they lie. We can think of this as a kind of (binary) code: if we orient each of the three lines, we denote a region to the left of it by L and a region to the right of it by R. Thus each region is coded by a three letter word in the alphabet L,R, so there are 8 possible regions which we can put in bijection with the numbers from 1 to 8 however we like.

The move on configurations of 3 lines is very closely related to a certain kind of move on knot diagrams called the “Reidemeister 3 move”. Think of the lines as shadows cast by strands of string, and think of moving one strand over a crossing of two other strands. The result (on shadows) is the Reidemeister 3 move:

What about 4 lines? We suppose that the 4 lines are ordered by increasing angle from horizontal, and give each complementary region a binary code depending on which side of the lines it’s on, so that there are 16 regions, which we give labels from 0 to 15. There are 8 configurations of 4 lines with given directions, indicated in the figure.

The unbounded regions — 0, 1, 3, 7, 15, 14, 12, 8 — are present in each configuration. As we “cycle” through these 8 configurations, in the order indicated in the figure, one bounded region appears and one disappears, in the cyclic order 10, 2, 6, 4, 5, 13, 9, 11.

For 5 lines there are many more combinatorial possibilities (even up to topological symmetries of the plane), but we can still get between any two configurations by a sequence of Reidemeister 3 moves, and all 32 regions appear in some configuration (actually, in many configurations). For more than 5 lines the story is similar.

There is a nice duality between arrangements of lines (or more generally, arrangements of hyperplanes) and zonohedra. If you recall from a previous post, a zonohedra is a polyhedron obtained as the Minkowski sum of a collection of intervals; that is, if are intervals in some vector space, the zonohedron is the set of points of the form where each . Zonohedra are simply the (linear) projections to lower dimensional spaces of higher dimensional cubes. Given a zonohedron Z in a Euclidean space, there is a corresponding hyperplane arrangement in a projective space of one dimension lower, defined as follows: for each face F of the zonohedron, one considers the collection of supporting hyperplanes for Z that contain F. This is a polyhedron in projective space of dimension dim(Z) – dim(F) -1. So top dimensional faces give rise to points, codimension two faces give rise to segments, and so on. Each *zone* of the zonohedron (that is, each equivalence class of parallel edges, corresponding to one of the ) gives rise to the hyperplane in projective space corresponding to hyperplanes in the Euclidean space containing . What is nice about this correspondence is that complementary regions to the hyperplane arrangement correspond to pairs of opposite *vertices* of the zonohedron. If we consider oriented hyperplanes then we get an arrangement of great spheres on a sphere; in the 2-dimensional case, an arrangement of great circles on the 2-sphere. So the possible regions that can occur (for all configurations) correspond to the vertices of the high dimensional cube, some subset of which project down to become the vertices of the zonohedron. (Note that I have swept under the rug the fact that we are now interested in configurations of great circles on the 2-sphere rather than straight lines in the plane. Three generic great circles on the 2-sphere decompose it into 8 regions, so this is another way of saying where the missing 8th region went: it was hiding round the back of the sphere).

Zones in the zonohedron correspond to *midcubes* in the high dimensional cube that projects to it — that is, the codimension one cubes that slice symmetrically through the center, parallel to a pair of opposite top-dimensional faces. So we can see a direct correspondence between midcubes in a high dimensional cube, and lines in the plane, or great circles in the sphere for that matter. Since we are already considering straight lines in non-Euclidean geometries, let’s ask what happens if we consider generic configurations of (straight) lines in the *hyperbolic* plane. Now things get much more interesting. Two generic lines in the hyperbolic plane might intersect, or they might be disjoint. There is an abstract graph whose vertices correspond to the straight lines in the configuration, and whose edges correspond to the pairs of straight lines which intersect. A complete in this graph — i.e. a configuration of n lines each of which intersects the other — gives rise in a canonical way to an n-dimensional cube, whose shadow is the 3-dimensional zonohedron that parameterizes the combinatorics of the configuration. If some is contained as a subgraph of two distinct , we obtain a complex by gluing together the j-cube and k-cube along their corresponding sub i-cube. The resulting space is a *cube complex* — a combinatorial complex built from cubes by gluings which respect the cubical structure on faces. Unions of midplanes glue together to make combinatorial *hyperplanes* which correspond precisely to the lines in the configuration. If the arrangement of lines was invariant under some group of hyperbolic isometries, then this group acts naturally and combinatorially on the associated cube complex. For example, if we start with a closed hyperbolic surface and a finite configuration of immersed closed geodesics on , the universal cover is the hyperbolic plane with an interesting arrangement of lines which is invariant under the action of the fundamental group .

In fact, it turns out that the key point is not that the arrangement is of lines, but that it is of codimension one objects. If G is a (finitely generated) group, and H is a subgroup, we say that H is *codimension 1* if the quotient of the Cayley graph of G by H has at least two ends. If it does, we can divide the Cayley graph into two H-invariant subsets, so that the frontier has finitely many orbits under the H action; this frontier is a kind of *combinatorial hyperplane* in G. The G translates of this hyperplane might intersect each other in a complicated way in the Cayley graph. As before, we can build a cube complex, where (roughly speaking) the n-cubes are the collections of n translates of the hyperplane each of which intersects the other in an essential way. The details of this construction can be found in a paper of Sageev (who first thought it up) and the end result is that one obtains a natural action of G on a cube complex. In fact, this cube complex is very nice geometrically — it is simply connected, and non-positively curved, so that if we make it a metric space by declaring that every cube is Euclidean with side lengths of edge 1, the result is CAT(0). The construction works just as well with a finite collection of codimension 1 subgroups instead of just one, and under suitable hypotheses, one shows that the group G is isomorphic to the (orbifold) fundamental group of a compact non-positively curved cube complex. This now becomes extremely relevant to Agol’s proof of the VHC — if G is the fundamental group of a hyperbolic 3-manifold, the surface subgroups constructed by Kahn-Markovic (see these blog posts) provide the raw material from which one builds a cube complex on which G acts, and this is the starting point for Agol’s work; see here for an introduction. I don’t know if Sageev was led from combinatorial hyperplane arrangements to cube complexes via zonohedra, but it’s plausible; and in any case thinking of it in these terms (at least for low dimensional examples) helps me to more easily see where the cubes “come from”.

Tagged: cube complexes, hyperplane arrangements, immersed curves, Reidemeister moves, zonohedra ]]>

**Theorem 1.4.2.** For every triangle ABC, the angle bisectors intersect at one point

**Proof.** Verify this for the 64 triangles for which the angle at A and B are one of 10, 20, 30, , 80. Since the theorem is true in these cases it is always true.

We are asked the provocative question: is this proof acceptable? The philosophy of the W-Z method is illustrated by pointing out that this proof is acceptable if one adds for clarity the remark that the coordinates of the intersections of the pairs of angle bisectors are rational functions of degree at most 7 in the tangents of A/2 and B/2; hence if they agree at 64 points they agree everywhere.

Leonid countered with a personal anecdote. Recall that an* altitude* in a triangle is a line through one vertex which is perpendicular to the opposite edge. Leonid related that one day his geometry class (I forget the precise context) were given the problem of showing that the altitudes in a hyperbolic triangle (i.e. a triangle in the hyperbolic plane) meet at a single point — the *orthocenter* of the triangle. After the class had struggled with this for some time, the professor laconically informed them that the result obviously followed immediately from the corresponding fact for Euclidean triangles “by analytic continuation”. Philosophically speaking, this is not too far from the W-Z example, although the details are slightly more shaky — in particular, the class of Euclidean triangles are not Zariski dense in the class of triangles in constant curvature spaces, so a little more remains to be done.

Actually, one might even go back and rethink the W-Z example — how exactly are we to verify that the angular bisectors intersect at a point for the triangles in question without doing a calculation no less complicated that the general case? Let’s raise the stakes further. After some thought, we see that not only will the intersections of pairs of angle bisectors be given by rational functions of the tangents of A/2 and B/2, but the (algebraic) heights of the coefficients of these rational functions can be easily estimated, and one can therefore compute an *effective* lower bound on how far apart the intersections of the angle bisectors would be if they were not equal. We can then literally draw the triangles on a piece of physical paper using a protractor, and verify by eyesight that the angle bisectors appear to coincide to within the necessary accuracy. After rigorously estimating the experimental errors, we can write qed.

While I am off on a tangent, this reminds me of a discussion I once had with Michael Aschbacher about the status of arguments (in topology, say) using diagrams. This could be a computation of the fundamental group of a knot complement by Wirtinger’s algorithm, for example, or a proof that some topological 4-sphere is smoothly standard via Kirby moves. He took what I think is an extreme view, that such arguments are *never* mathematically valid. This is a bit of a fuzzy argument to have if one is not careful to define precisely what one means by a “diagram” — suppose (as is in fact the case) I draw a diagram by writing a (finite) .eps file in ASCII. Then a “diagram” can be taken to be a certain kind of string in a finite alphabet, and the kinds of reasoning about diagrams one is prepared to accept could also be precisely specified and formalized, and could presumably be shown to be consistent with ZFC. This shows (in some very weak sense) that it is possible to conceive of a theory of “reasoning by diagrams” which must be respectable even to Michael Aschbacher. However, in practice one “reasons using diagrams” (just as one reasons in every other context) by a combination of explicit formal rules and pre logical “leaps”: if I extend *this* line indefinitely, it will intersect *that* line here; or, if I pick up *this* strand and pull it behind the *other* strand, it will eliminate* these* three crossings and introduce a new crossing *here*. And so on. If one pursues this line of reasoning too far it starts to degenerate into questions about the reliability of short term memory, or the psychophysics of perception, which throw *any* kind of mathematical reasoning in question. But before reaching that point, one can argue (and Aschbacher* did* argue) that arguments involving diagrams are “special” because of the sheer quantity and sophistication of the pre logical leaps involved. Anyone who has seen how much effort is involved in translating e.g. the Jordan curve theorem into a formal proof system like HOL light might be prepared to concede that Aschbacher has a point.

Anyway, back to hyperbolic orthocenters. If one substitutes spherical for hyperbolic geometry, there is quite a cute proof of the existence of an orthocenter as follows. Let’s fix the unit sphere in 3-space, and let ABC be a Euclidean triangle in a plane tangent to the unit sphere and touching it exactly at the orthocenter O of ABC. Radial projection of the vertices determines a spherical triangle A’B’C’. I claim that the radial projection of the altitudes of ABC become altitudes of A’B’C’, and therefore these altitudes intersect in O, which turns out also to be the (spherical) orthocenter of the (spherical) triangle A’B’C’. To see the validity of the claim, observe first that the radial projection of a straight line in to the sphere is a great circle on the sphere; so if L is any straight line in through O, the radial projection L’ is a great circle through O. Second, note that if M is a straight line in perpendicular to L (as above), the radial projection M’ is a great circle perpendicular to L’; this follows by symmetry: reflection in the plane through the origin containing L takes M to itself and therefore M’ to itself, while fixing L’ pointwise. This proves the claim, and therefore that the (spherical) altitudes of A’B’C’ intersect at O’. By a dimension count, all spherical triangles arise in this way; qed. At this point the appeal to analytic continuation (from spherical to hyperbolic geometry) is more persuasive.

Tagged: analytic continuation, psychology, triangles ]]>

I already know a little bit about square tilings. This is a subject with a history, going back at least to the work of Tutte and his colleagues. The basic problem is just to tile a rectangular region by squares. Easy enough, you say.

Well, yes; if the rectangle has sides which are rationally related, in can be filled up by squares with commensurable side lengths pretty easily. Here a 4 by 8 rectangle is filled with 7 squares of edge length 1, 1 square of edge length 3, and 1 square of edge length 4. It’s more amusing to look for a tiling in which all the squares have different lengths. One well-known tiling, found by Tutte’s colleague Stone, is as follows:

Obviously the problem becomes more interesting and challenging if one starts in advance with the *combinatorics* of a square tiling, and then tries to assign edge lengths to squares in such a way that they fit together nicely. One elegant method, developed by Brooks, Smith, Stone and Tutte, assigns a directed graph to the tiling, with one vertex for each vertical edge (say) and one directed edge for each square. Here’s the graph associated to Stone’s tiling:

The condition that the sum of square lengths on either side of a vertical edge sum to the length of that edge implies that the incoming edge weights and the outgoing edge weights at each vertex sum to the same value (except for at the leftmost and rightmost vertices). On the other hand, the fact that the squares are all square implies that each edge weight (as above) is equal to the length of its projection to a horizontal line; this means that the sum of edge weights around each loop in the graph (with sign changed when the orientation disagrees with the orientation on the loop) is equal to zero. These two conditions are precisely Kirchoff’s two laws for the current flowing through an electrical network where every edge has resistance 1, and the voltage difference between left and right vertices is the width of the rectangle. There is a unique solution; it might have some weights negative, in which case we can reverse the orientation of the edge so that the weights are all positive, and determine a square tiling with slightly different combinatorics. By the way, the uniqueness of the solution has an interesting (and well-known) consequence: since Kirchoff’s laws both impose linear conditions on the edge weights, the space of solutions is a *rational* affine space (in units for which the width is equal to 1). Since this space of solutions consists of a single point, this point has rational coordinates; this implies in particular that the height of the rectangle is a rational multiple of the width, and so are the widths of the squares.

In more homological language, the assignment of weights to edges is a (simplicial) 1-chain. The condition that the incoming and outgoing edge weights at each vertex have equal sum says that this 1-chain is actually a (relative) 1-cycle; i.e. that it is closed. The condition that the sum around every loop is zero says that if we think of this 1-chain as a 1-cochain it is actually a 1-cocycle; i.e. it is co-closed. A (co)-chain which is both closed and co-closed is said to be harmonic, and the uniqueness of a solution corresponds to the uniqueness of a harmonic representative of a (relative co-) homology class.

Incidentally, if we form the graph with one vertex for each ~~vertical~~ horizontal line and one edge for each square, this will be the (planar) dual to the graph above. Edges in one graph correspond to edges in the other, and the closed condition for one set of edge weights becomes the co-closed condition for the other, and vice versa.

Now instead of considering a square tiling of a rectangle, let’s consider a square tilings of a Euclidean torus. A combinatorial tiling gives us a graph, and a harmonic 1-cycle gives us a square tiling with the desired combinatorics. Changing the 1-cycle by rescaling it just rescales the torus and all the squares by the same factor, which is not very interesting. However, there *is* something interesting we can do. The homology of a torus is 2-dimensional, so we can consider a 1-parameter family of homology classes whose projective classes are changing, and a 1-parameter family of harmonic 1-cycles and of square tilings.

Let’s start with the simplest possible example. We fix a graph G embedded in the torus. Since we want G to be able to carry every homology class, we need at least two edges. So let’s take as G the graph with one vertex and two edges, one of which wraps horizontally once around the torus, and one of which wraps vertically around. Any assignment of weights to the edges will be both closed and co-closed, so a 1-parameter family is given by taking weights cos(t), sin(t) for t in the unit circle. The resulting square tilings of the torus have two squares, one of side length cos(t) and one of side length sin(t). The total area of the torus is thus normalized to be 1. The pattern of tilings “rotates” with t as follows:

(click on the image to see it rotate)

OK, how about a more complicated example? Let’s let G be some complicated embedded graph on the torus (so that it can carry any homology class). For the sake of concreteness, let’s let G be the following graph:

G has 10 edges (corresponding to 10 squares in the tiling), 5 vertices and 5 complementary faces. There are 5 vertex conditions and 5 face conditions; however, this system of 10 equations is redundant, and has a 2 dimensional space of solutions.

Weights on the edges of G form a vector space, and there is an inner product on this space which is just the ordinary Euclidean inner product with co-ordinates the weights on each each edge. We want to normalize our weights to have length (i.e. square root of their inner product with themselves) equal to 1, so that the resulting torus will have area 1. All we need to do is find two orthogonal weights M and L which are closed and co-closed, orthogonal to each other (i.e. the inner product of M and L is zero) and of length 1, and then we can form the family cos(t)M + sin(t)L of weights, and the associated square tilings.

The resulting rotating family of tilings is as follows:

(click on image to see it rotate)

Something else is needed to get the “spiraling” evident in Kenyon’s picture. For our square tilings of a torus above, the result of laying down a sequence of squares that winds once around a loop in the torus is to displace the tiling by a translation of the plane; this translation is called the *holonomy* around the loop, and only depends on its homotopy class (actually: on its homology class). Essentially, this is the result of integrating the (dual) 1-form associated to the weight. An educated guess is that in Kenyon’s picture, the holonomy is not a translation, but rather a *dilation* of the plane, centered at some point. At the level of homology, one can think of the dilation factor around a loop as a representation of the fundamental group, and we need to consider (harmonic) 1-cycles with coefficients twisted by this representation.

How to translate this into the language of square tilings and weights? Instead of thinking of a weight on the graph G, let’s let G~ denote the lift of G to the universal cover of the torus; i.e. G~ is a periodic graph in the plane. A twisted weight on G with coefficients in a representation is the same thing as a weight on G~ that transforms according to the given representation. For the sake of simplicity, let’s work with the graph G with one vertex and two edges as in the first example above, so that G~ has one vertex, one horizontal edge, and one vertical edge for each pair of integers. Pick a pair of edges H,V of G~, going to the right and up incoming to the vertex (0,0) respectively and let h,v be the weights on these edges.

If we let A denote the multiplication factor for horizontal translation, and B the multiplication factor for vertical translation, the vertex equation at (0,0) is

The vertex equations at every other vertex are obtained from this one by scaling by power of A and B, so they are satisfied if this one is. The face equation for the face with vertices (-1,-1), (0,-1), (0,0), (-1,0) is

Eliminating h from this pair of equations and dividing out by v gives

In order to enforce spiraling, we would like moving “horizontally” some fixed number of steps to be the same as moving “vertically” some (other) fixed number of steps; this can be imposed by setting for some coprime integers p,q. With these constraints, there is a unique solution h,v in complex numbers, up to scale. The real part of any such solution gives a “spiral” tiling, and the 1-parameter family obtained by multiplying by before taking the real part gives a rotating spiral.

Let’s try an example. Taking p=2,q=1 gives and . There is a totally real solution, giving rise to the following “degenerate” spiral:

Since this solution is totally real, it can’t be “rotated”. Hmm, I wasn’t expecting that. OK, taking p=3,q=1 gives and .

(click on image to see it rotate)

Success!

Getting more squares in the picture is a matter of spiraling slower, which can be achieved by taking p and q bigger. Let’s try p=7,q=1.

(click on image to see it rotate)

If you want to have a play with this yourself, the source of the .eps file that generated these figures is below. To change the amount of spiraling, change the values of A and B, subject to the constraint that . The resulting .eps file can be transformed to a layered .pdf (eg using Preview on a Mac) then to a .gif (eg in gimp). The case q=1 is pretty easy, since then A is the root of with smallest (nonzero) argument, and . Wolframalpha will cough up the values of A and B if you coax it long enough.

(Update January 16): Just for fun, here’s the tiling with p=101, q=1 (warning: the .gif file is quite large!)

(click on image to see it rotate)

%!PS-Adobe-2.0 EPSF-2.0

%%BoundingBox: 0 0 400 400

gsave

400 400 scale

1 20 div setlinewidth

1 setlinejoin

0.5 0.5 translate

/square{4 dict begin

/z exch def

/y exch def

/x exch def

gsave

newpath

rand 10 mod 10 div rand 10 mod 10 div rand 10 mod 10 div setrgbcolor

x y moveto

x z add y lineto

x z add y z add lineto

x y z add lineto

closepath

fill

stroke

grestore

end } def

/simple_edge_squares{4 dict begin

/v exch def

/n v length def

0 1 n 1 sub{

/i exch def

0 0 v i get square

0 v i get translate

} for

end} def

/rcmul{2 dict begin

/t exch def

/z exch def

[ z 0 get t mul z 1 get t mul]

end} def

/ccmul{2 dict begin

/w exch def

/z exch def

[

z 0 get w 0 get mul z 1 get w 1 get mul sub

z 0 get w 1 get mul z 1 get w 0 get mul add

]

end} def

/cconj{1 dict begin

/z exch def

[

z 0 get 0 z 1 get sub

]

end} def

/cnorm{1 dict begin % |z|^2

/z exch def

z 0 get dup mul z 1 get dup mul add

end} def

/ccdiv{2 dict begin % w/z = w*zbar/|z|^2

/z exch def

/w exch def

w z cconj ccmul 1 z cnorm div rcmul

end} def

/ccadd{2 dict begin

/w exch def

/z exch def

[ z 0 get w 0 get add z 1 get w 1 get add ]

end} def

/creal{1 dict begin

/z exch def

z 0 get

end} def

/cimag{1 dict begin

/z exch def

z 1 get

end} def

0 5 355 {

/t exch def

0 srand

gsave

/A [0.71469 -0.870643] def % A is root of x^14+x^8-4x^7+x^6+1=0

/B [0.432505 -0.0429583] def % B = A^-7

% check: A^3B=1

/h [t cos t sin] def

/v h A [-1 0] ccadd ccmul [1 0] B -1 rcmul ccadd ccdiv def %

% h*(A-1)/(1-B)

/Ainv [1 0] A ccdiv def

/Aser [1 0] Ainv ccadd Ainv Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul Ainv ccmul Ainv ccmul ccadd def

/Acom [1 0] Ainv -1 rcmul ccadd def

/htran h Acom ccdiv def

/vtran v Acom ccdiv def

t rotate

htran creal vtran creal -1 mul translate

[h A ccmul creal v B ccmul creal h [-1 0] ccmul creal v [-1 0] ccmul creal] simple_edge_squares

1 1 50{

h -1 rcmul creal v creal translate

/h h A ccdiv def

/v v A ccdiv def

[h A ccmul creal v B ccmul creal h [-1 0] ccmul creal v [-1 0] ccmul creal] simple_edge_squares

} for

showpage

grestore

} for

grestore

%eof

Tagged: discrete complex analysis, graph theory, harmonic functions, quantum mechanics, square tilings ]]>

I had recently bought a video camera, and decided to tape Bill’s talk. I never did anything with it until now (in fact, I don’t think I *ever* re-watched anything that I taped), but it turned out to be not too difficult to transfer the file from tape to computer. Since this seems like an interesting fragment of intellectual history, I thought it might be worthwhile to post the result to YouTube — the video link is here.

Tagged: Bill Thurston, geometrization conjecture, history of mathematics ]]>

Let’s restrict our turtle’s movements to alternating between taking a step of a fixed size S, and turning either left or right through some fixed angle A. Then a (compiled) “program” is just a finite string in the two letter alphabet L and R, indicating the direction of turning at each step. A “random turtle” is one for which the choice of L or R at each step is made randomly, say with equal probability, and choices made independently at each step. The motion of a Euclidean random turtle on a small scale is determined by its turning angle A, but on a large scale “looks like” Brownian motion. Here are two examples of Euclidean random turtles for A=45 degrees and A=60 degrees respectively.

The purpose of this blog post is to describe the behavior of a random turtle in the hyperbolic plane, and the appearance of an interesting phase transition at . This example illustrates nicely some themes in probability and group dynamics, and lends itself easily to visualization.

Let’s work in the Poincaré unit disk model of hyperbolic geometry. In this model, the hyperbolic plane is thought of as the interior of the unit disk in the Euclidean plane, and the hyperbolic metric is related to the Euclidean metric by multiplying distances infinitesimally by at a point whose (Euclidean) distance from the origin is . In this model, the hyperbolic distance between a point at the origin and a point at Euclidean distance away is . So, at the risk of being slightly confusing, let me say that a hyperbolic random turtle has “step size S” if the first step, starting at the origin, lands on the Euclidean circle of radius S.

I wrote a little program called **turtle** to illustrate the motion of a random turtle for various values of S and A; it can be downloaded from my github repository if you want to play with it. The figures below are all produced with it. Let’s look at a few examples.

The phase transition alluded to earlier is very evident in these pictures: for large S and small A, the turtle zooms off in an almost straight line to the boundary, with very little wiggling along the way. For small S and large A, the turtle meanders around aimlessly, filling up lots of space, intersecting its path many times, until eventually wandering off to the boundary in a more or less random direction.

For a given length, what is the critical turning angle? The “worst case” scenario is a turtle which always turns left (or always turns right). For such a turtle there is a critical angle (for a given length) such that the trajectory of the turtle just fails to close up. Technically, the hyperbolic isometry describing the turtle’s motion at each step is *parabolic*, and fixes a unique point at infinity. The segments of the turtle’s trajectory will then osculate an invariant *horocycle* for the parabolic isometry, when the (discrete) atoms of positive turning curvature at the vertices exactly balance the negative curvature of the hyperbolic plane.

A critical turtle trajectory osculates a horocycle

The critical relationship is precisely that , with our convention about the relationship between S and the hyperbolic length of the segments. For angles smaller than this value, the trajectory is a *quasigeodesic* — i.e. it stays within a bounded (hyperbolic) distance of an honest geodesic, and does not wind around at all. For angles bigger than this value, there is a definite probability at every stage that the trajectory will undergo some number of complete full turns, and it might return to some region it has visited before. The trajectory still converges to a point at infinity with probability one (this is a very robust feature of random walk in negatively curved spaces) but it makes deviation of order from this geodesic in the first steps.

One interesting statistic for an immersed path in the plane is the *winding number*. If we trivialize the unit tangent bundle, the derivative can be thought of as a map to the circle, and we can ask how many times it winds around. In the Euclidean plane there is a natural trivialization of the unit tangent bundle via parallel transport, because of the flatness; technically there is a flat orthogonal connection. In the hyperbolic plane any orthogonal connection must have curvature, but there *is* a flat connection with structure group equal to the group of (hyperbolic) isometries, by identifying the unit circle in each tangent bundle with the circle at infinity. Explicitly: every tangent vector is tangent to a unique oriented geodesic which limits to a unique point in the circle at infinity. This identification is global, and respected by the natural action of the isometry group.

For a random turtle in the Euclidean plane, the trajectory turns left or right through angle A at every step, and the winding number after some number of steps is distributed like simple random walk on the integers. That is, if denotes the winding number after steps, then the random variable converges to a normal distribution with mean zero and standard deviation A. The point is that the increments at every stage are independent and identically distributed. On the other hand, for a random turtle in the hyperbolic plane, each step induces an isometry of the hyperbolic plane, and thereby a *projective* transformation of the boundary circle. There is no natural invariant metric on this boundary circle, and therefore it is more subtle to compute winding number from this action.

Let’s abstract the discussion somewhat. Suppose we are given a finite collection of (orientation-preserving) homeomorphisms of the circle. The circle is covered by the line, and the group of orientation-preserving homeomorphisms of the circle is covered by the group of orientation-preserving homeomorphisms of the line that commute with integer translation. Call this covering group , where the tilde denotes central extension. Poincaré’s rotation number is a function from to the real numbers, whose reduction mod the integers is the usual rotation number for a circle homeomorphism. Thinking of our turtle as turning left or turning right continuously implicitly determines a lift of the motion to the universal covering group, so we can suppose that we are given a finite collection of lifts of . Now we consider some random walk where each is drawn independently and uniformly from , and we ask about the distribution of the random variable , which is defined to be the (real valued) rotation number of the composition .

Now, although there is typically no metric/measure on the circle left invariant by there is a natural measure — the so-called *harmonic measure* — which is invariant *on average*. If is a probability measure on the circle, we can define , and then let . The have a subsequence converging to a fixed point for the operator ; such a fixed point is a harmonic measure. Note that such a harmonic measure is quasi-invariant under every . The measure pulls back to a locally finite measure on the real line, and this pullback is harmonic for the action of . We can define a function as follows. For each choose some and define . Then is monotone nondecreasing, and for any and any integer . In particular, the winding number satisfies for any .

Now, by the definition of a harmonic measure, for any and for random , there is an equality (here the notation means the *expectation* of a random function). In particular, is *constant* independent of . We call this constant quantity the *drift* and denote it by . Define a sequence of random variables by . By the calculation above we see that for each , the expectation of conditioned on a particular value of is equal to the given value of . More informally, we could just write and say that at every step, the expected change in the value of is zero. This is a familiar object in probability theory, and is known as a *martingale*. One can think of the values of the martingale as the wealth of a gambler who makes a succession of fair bets. The wealth of such a gambler over time looks roughly like a simple random walk, after reparameterizing time proportional to the rate at which the gambler takes risks (as measured by the variance of the outcomes of each bet). For our random product of homeomorphisms, there are two possibilities: either the martingale converges, as successive “bets” become smaller and smaller, and the winding number converges to some final value (this happens in the case that the length of the turtle’s steps are big compared to the turning angle), or else the position of the point is equidistributed in the circle with respect to , and there is a central limit theorem: converges to a Gaussian.

Returning to our original setup, the left-right symmetry forces the drift to equal zero, and we can identify with the winding number up to a constant. How does the variance of depend on the variables S and A? The following figure shows a graph of the variance as a function of S and A. The red line marks the phase transition from zero variance (i.e. quasigeodesic turtle trajectories) to strictly positive variance.

As one sees from the figure, the phase transition is not something sharp that can be easily seen experimentally, and in fact, the graph looks completely smooth along the phase locus (although we know it can’t be real analytic there). This experimental observation can be theoretically confirmed, as follows.

Consider the behavior of a random turtle, with fixed stepsize, for some turning angle A’ just marginally bigger than the critical angle A. The critical turtle trajectory bounds an infinite polygon with edges of length and external angles A; this polygon can be decomposed into semi-ideal triangles with internal angles and finite side length . As we deform the angle we get a new triangle with angles where , and the angle is opposite a side of fixed length . The hyperbolic law of cosines says in this context that . Since is fixed, and is small, we can approximate ; in other words, the angle is of polynomial (actually, quadratic) order in the difference . Now, suppose for some very large . A turtle trajectory with the property that there is at least one left and at least one right turn in every steps will be quasigeodesic; the only full twists will occur when there is a sequence of at least left turns or right turns in a row. This is a very rare occurrence — it will typically only happen twice in a sequence of steps. Hence the variance of the winding number is of order . In particular, the graph of the variance is tangent to zero to infinite order along the phase locus, as claimed.

(Update:) At Dylan’s request I’ve added a slice of the variance graph, at with angle varying from 0 to 0.2. The vertical axis has been stretched (relative to the 3d graph above) for legibility. The phase transition is at angle 0.1000417 and I must say the graph looks pretty flat there.

Tagged: harmonic measure, Hyperbolic geometry, martingale, phase transition, quasimorphism, random walk, turtles ]]>

This group was studied by Crisp-Sageev-Sapir in the context of their work on right-angled Artin groups, and independently by Feighn (according to Mark Sapir); both sought (unsuccessfully) to determine whether contains a subgroup isomorphic to the fundamental group of a closed, oriented surface of genus at least 2. Sapir has conjectured in personal communication that does not contain a surface subgroup, and explicitly posed this question as Problem 8.1 in his problem list.

After three years of thinking about this question on and off, Alden Walker and I have recently succeeded in finding a surface subgroup of , and it is the purpose of this blog post to describe this surface, how it was found, and some related observations. By pushing the technique further, Alden and I managed to prove that for a fixed free group of finite rank, and for a* random endomorphism* of length (i.e. one taking the generators to random words of length ), the associated HNN extension contains a closed surface subgroup with probability going to 1 as . This result is part of a larger project which we expect to post to the arXiv soon.

The context of this problem is Gromov’s notorious question:

**Question(Gromov):** Does every 1-ended hyperbolic group contain a surface subgroup?

Actually, it is not at all clear if Gromov really asked this question, or what sort of answer he expected. There is a discussion of this in the introduction to a recent paper by Henry Wilton. A positive answer to this question is known in only a few special cases, including

- Coxeter groups (Gordon-Long-Reid)
- Graphs of free groups with cyclic edge groups and (Calegari)
- Fundamental groups of hyperbolic 3-manifolds (Kahn-Markovic)
- Certain doubles and graphs of free groups with cyclic edge groups (Kim-Wilton, Kim-Oum, Kim, Wilton)

(this list is not exhaustive). One strategy to find a surface subgroup is to define a class of groups with the property that every one-ended hyperbolic group contains a subgroup in the class , and then to show that every group in this class further contains a surface subgroup. A reasonable candidate for the class is the class of *one-ended graphs of free groups*. The logic behind this choice is that it is very easy to produce many free subgroups of a one-ended hyperbolic group (in fact, this is more or less the only kind of subgroup one knows how to produce) by Klein’s pingpong argument, and one could perhaps argue that because there are so many such subgroups, that intersect in quite rich and interesting ways, a sufficiently rich collection is one-ended while at the same time has the structure of a graph of groups. On the other hand, the structure of a graph of free groups is similar in some ways to the structure of a Haken 3-manifold, and one knows enough about the components of the graph (i.e. the free factors) that one can try to build a surface subgroup by amalgamating surface-with-boundary subgroups along cyclic subgroups of the edge groups.

Anyway, this is more philosophy than mathematics, but it does partly explain why the class has been widely studied by geometric group theorists interested in Gromov’s question. One important class of graphs of groups are the HNN extensions, whose underlying graphs consist of a single vertex and a single edge joining this vertex to itself. An (injective) endomorphism of a free group thus gives rise to an HNN extension in the class .

Now, suppose is a map from a surface subgroup to . There is a homomorphism sending to 0 and the conjugating element to . The kernel intersected with the image of will determine an infinite cyclic cover of , and one would like to determine whether this map is injective. We can think of as an infinite union of subsurfaces with boundary, where each is attached to and , and contained in a conjugate of the subgroup . If we identify each with for , then we can think of . Let denote the union of the with . Evidently it is sufficient to show that the inclusion of to is injective, since any loop in the kernel of is conjugate into . This is convenient, since we can discuss surface-with-boundary subgroups of a fixed free group, and essentially ignore the endomorphism .

The first thing to check is that each separate inclusion is injective. Each may be represented by a certain kind of diagram, called a *fatgraph*. Basically, a fatgraph is a graph in the usual sense, together with a choice of cyclic ordering of the edges incident to each vertex. A fatgraph embeds canonically as the spine of some surface which itself deformation retracts back to , in such a way that the cyclic order on edges inherited from the embedding agrees with the fatgraph structure. The oriented edges of are labeled with reduced words in in such a way that the labels on opposite sides of an edge of are inverse in . In this way, a fatgraph “represents” a surface-with-boundary mapping to . Here is an example of a (disconnected) fatgraph, whose underlying surface is homeomorphic to the union of two 4-punctured spheres:

Now, the fundamental group of every (component of every) fatgraph is free, but the map to is not necessarily injective. Stallings gave a celebrated criterion for a simplicial map from a graph to a rose (i.e. a standard graph with fundamental group ) to be injective, namely that the map should be *folded*, or equivalently, that the map should be an immersion on the link of every vertex. In terms of fatgraphs, this means that there should be at most one incoming edge at each vertex with each label. The graph pictured above is folded in this sense. Notice if every boundary word is reduced, a 2- or 3-valent vertex is necessarily (locally) folded.

OK, this is a criterion that will certify that an individual might be injective, when represented as a fatgraph. What about the dynamics of ? Notice that the endomorphism has a particularly nice property: if we think of it as representing a self-map of the standard rose to itself, then the map is an *immersion*, in the sense of Stallings. This means that if each of the surfaces is represented by a folded fatgraph , then each will be folded if is. This suggests the following definition:

**Definition.** A fatgraph with associated surface is *-folded* if there is a decomposition of its boundary into and in such a way that (with the opposite orientation), and satisfying the following properties:

- The graph is Stallings folded
- Every -vertex in (i.e. the images under of the vertices of ) is associated to a 2-valent vertex of
- No vertex of is associated to more than one -vertex in
- No vertex of is associated to more than one vertex in

When we talk about a vertex of being “associated” to a vertex of we mean that the vertex of maps to the given vertex of under the deformation retraction of to (this deformation retraction is simplicial when restricted to ).

Now, suppose is -folded. We can glue to by gluing to . Condition 4 implies that the resulting surface is where . In a similar way we can define

and . Now, conditions 2 and 3 imply that every vertex of is obtained by gluing some vertex of to a sequence of 2-valent vertices in various with . In particular, since every vertex of is locally folded, the same is true of every vertex of , and therefore also of . Hence is folded, and thus injective. Since as above, it follows that the suspension of an -folded surface is injective in .

The definition of -folded can be modified for an endomorphism which is not an immersion of . One of the main theorems Alden and I prove is that a “random” endomorphism admits many -folded surfaces in this sense, and therefore the associated HNN extension has (many) surface subgroups. For a random endomorphism of length , the genus of these surfaces will typically be of order at least , but the number will grow at least like for genus .

Now, Sapir’s group is certainly not random in any sense; nevertheless, it is possible to search for an -folded surface. A priori finding an -folded surface with given boundary seems to require trying exponentially many gluings, and is apparently impractical. However, Alden and I are able to show that the search for such a surface can be reduced to a linear programming problem, and thus becomes eminently practical. Sure enough, a computer search rapidly found the following example of an -folded surface in Sapir’s group:

In a bit of detail: the picture above is a fatgraph whose boundary decomposes into three components labeled and one component labeled . There is a 3-fold cover whose boundary decomposes into consisting of three copies of and consisting of three components labeled . The components of are the ones indicated by the blue circles, and one can see that they are embedded, satisfying condition 4. The red dots are the -vertices, and one can check that they are distinct and on 2-valent vertices of the fatgraph. Finally, one can check that the surface is folded in the usual sense of Stallings. It follows that the suspension is an injective surface in Sapir’s group, of genus 31.

(added Thursday, February 21, 2013): Jack Button has just posted a paper to the arXiv making the observation that a random HNN extension of a free group (in the sense of Alden and I, as above) will satisfy the small cancellation condition for any as , with probability , and therefore will be the fundamental group of a special cube complex, by a result of Wise. This is good to know, and underlines the extent to which such HNN extensions resemble 3-manifold groups.

Tagged: f-folded surface, fatgraph, HNN extension, hyperbolic group, Sapir's group, Stallings folding, surface subgroup ]]>

What gives this condition some power is the rich class of examples of spaces which are -hyperbolic for some . One very important class of examples are simply-connected complete Riemannian manifolds with upper curvature bounds. Such spaces enjoy a very strong comparison property with simply-connected spaces of *constant* curvature, and are therefore the prime examples of what are known as CAT(K) spaces.

**Definition:** A geodesic metric space is said to be , if the following holds. If is a geodesic triangle in , let be a *comparison triangle* in a simply connected complete Riemannian manifold of constant curvature . Being a comparison triangle means just that the length of is equal to the length of and so on. For any there is a corresponding point in the comparison edge which is the same distance from and as is from and respectively. The condition says, for all as above, and all , there is an inequality .

The term CAT here (coined by Gromov) is an acronym for Cartan-Alexandrov-Toponogov, who all proved significant theorems in Riemannian comparison geometry. From the definition it follows immediately that any space with is -hyperbolic for some depending only on . The point of this post is to give a short proof of the following fundamental fact:

**CAT(K) Theorem:** Let be a complete simply-connected Riemannian manifold with sectional curvature everywhere. Then with its induced Riemannian (path) metric is .

This theorem is very familiar to people working in coarse geometry, especially geometric group theorists. Because it is really a theorem in Riemannian geometry, rather than coarse geometry per se, its proof is often omitted in expositions of the theory; for example, I don’t believe there is a proof in Gromov-Ballmann-Schroeder or Ballmann (I think it is relegated to the exercises), nor is there a proof in Cheeger-Ebin, although one can piece together an argument from some of the ingredients in this last volume. Therefore I thought it might be a useful exercise to give a more-or-less complete exposition, which is reasonably self-contained and complete (Update: Daniel Groves tells me there is a proof in Bridson-Haefliger, which is good to know).

Part of what makes this a slightly fiddly theorem to prove is that one must somehow connect up the algebraic language of local Riemannian geometry with the metric language of distances, triangles, convexity and so on. The argument breaks up nicely into two parts — an infinitesimal comparison which is proved algebraically, and a global comparison which is derived from the local comparison by a “soft” argument. The first, algebraic part is not very deep, but it does contain an interesting nugget or two, which I will try to explain as I go along.

First, let’s briefly recall some of the ingredients of elementary Riemannian geometry. Given a Riemannian metric, there is a unique connection — the Levi-Civita connection — which is torsion-free, and compatible with the metric. We denote this by , so that denotes the covariant derivative of the vector field along the vector field . For three vector fields one defines the curvature tensor . Geometrically, this measures how rotates as one takes holonomy transport around an infinitesimal *negatively* oriented loop in the - plane. The sectional curvature in the - plane is the ratio

The denominator of this expression is the area of a parallelogram spanned by and , so if are orthogonal and of length 1, it reduces to 1.

If is a point, and is a tangent vector at that point, there is a unique geodesic with and . If is complete, is defined; thus there is an *exponential map* from to taking to . If is the subspace of spanned by a vector , and , then we can define a vector field along by setting it equal to at , for some constant and for all . The exponential map pushes this vector field forward to a vector field on along , called a *Jacobi field*; by its construction, a Jacobi field is tangent to a 1-parameter variation of geodesics. A Jacobi field satisfies the *Jacobi equation* . By abuse of notation, one identifies the frames along by parallel transport, and writes this as .

The easiest way to connect up the notions of curvature and comparison geometry is in the observation that for a manifold of nonpositive curvature, the norm of a Jacobi field is *convex* (as a function along a parameterized geodesic). We compute . Using the Jacobi equation, the second term can be rewritten, so this is equal to . By the hypothesis that curvature is nonpositive, this is . We compute

where the last inequality is just Cauchy-Schwarz.

OK, we are now ready to begin in earnest. Consider a geodesic from to , and a geodesic through making some angle with at . Parameterize by arc length so that , and consider a 1-parameter family of geodesics from to . Note that . If denotes the length of , then the derivative ; in particular, it does not depend on the curvature of the space in question. The curvature manifests itself in second order information. The one-parameter family of geodesics is tangent along to a Jacobi field , where and . Denote the vector field tangent to the s by . The second variation formula (see e.g. Cheeger-Ebin pp. 20-21) says

Now, vanishes at , since vanishes there; moreover at it is tangent to , and therefore vanishes there too. So the first term is zero. Furthermore, the term (since because is tangent to geodesics) and

along , by the Jacobi equation applied to . Hence is *constant* along , and one sees that it contributes a term which depends only on the angle . Lets abbreviate . Another simple calculation (see Cheeger-Ebin pp.24-25) shows that if for any function with then ; this is one of the fundamental (and standard) index lemmas, which say that in a suitable sense, Jacobi fields minimize the form .

We are now ready to compare second derivatives in and in our comparison space . Let and be geodesics as above in a comparison space of constant curvature with the same lengths as and making the same angle at their intersection. Let be the analogous 1-parameter family of geodesics, and let denote the length of . We know that the first derivatives of and agree, and would like to compare second derivatives. Apart from the term that depends only on the angle, this means comparing and . This is basically a special case of the Rauch comparison theorem, and our argument is a simplification of Rauch. Let’s suppose for simplicity that both and are 2-dimensional. Parallel transport along and identifies the tangent spaces along these geodesics with the tangent spaces at and respectively. Choosing an isometry between these tangent spaces which takes to , we can define the “pushforward” to be a vector field along satisfying and . By construction we can write where is tangent to , and where . Thus . On the other hand, at comparable points by definition, and

pointwise by the hypothesis comparing the curvature of and . Hence

and we conclude that the distance function to geodesics is *more convex* in than in the comparison space . This is the desired infinitesimal comparison theorem; it remains to bootstrap it to a global comparison theorem.

Right; let’s look at our comparison triangles and . By the hypothesis that is simply-connected, we can actually map a disk into spanning the geodesic triangle; a minimal area such disk will have intrinsic curvature bounded above by that of , and distances in this disk between points on the boundary will be at least as large as they are in . So without loss of generality, we may assume that is 2-dimensional, and that is spanned by an honest triangular disk. Parameterize the side by length, and let be the point on with . Let be the analogous point on . Define

and .

We know and . We would like to show pointwise. Suppose not, and restrict to a maximal connected interval on which this fails. By the infinitesimal comparison theorem proved above, this interval must have nonempty interior. Let and be the points on and corresponding to the endpoints of the interval. Evidently the triangles and are also comparison triangles; so WLOG we may just take , and so on.

We now employ a trick. Consider a 1-parameter family of comparison triangles in spaces of constant curvature . The CAT(K) Theorem for spaces of *constant* curvature reduces to an explicit calculation, since the function as above can be computed exactly, and we suppose the theorem proved for such spaces. It follows that as increases, the function also increases monotonically. By assumption, for small there is some with . Eventually therefore we get some and some intermediate where and for all points near . But this contradicts the infinitesimal comparison theorem proved above. qed.

The figure above illustrates the meaning of the last step. The blue curve is the graph of , and the red curves are the graphs of for various . As is increased, the red curves move upward in a family. There is some biggest for which the red curve is not entirely above the blue curve, and for that curve, the red and blue curves have a point of tangency. But at that point of tangency we would have , contrary to the infinitesimal comparison theorem which shows with equality iff the curvatures along the corresponding geodesics are pointwise equal, which they are not for .

Tagged: CAT(K), comparison geometry, convexity, Jacobi fields, nonpositive curvature, Riemannian geometry ]]>

I counted Bill as my friend, as well as my mentor, and I have many vivid and happy memories of time I spent with him. I hope that writing down a few of these reminiscences will be cathartic for me, and for others who are coping with this loss.

I remember seeing Bill for the first time when I arrived at Berkeley in 1995; at the start of the academic year, all the incoming graduate students were ushered into the colloquium room to meet some of the senior personnel. Bill was there in his capacity as director of MSRI (the Mathematical Sciences Research Institute). He was wearing jeans with big holes at the knees. He made a speech about MSRI, inviting us all to come up the hill and interact with the visitors there. He also encouraged us to pronounce it as “emissary”, rather than “misery”; it didn’t work — we all called it “misery” (and still do).

I remember actually taking the bus up the hill (maybe a few months later?) in the vague hope of running into Bill and asking him to be my advisor (people had warned me against this, saying that Bill “wasn’t taking students”, because he was too busy running MSRI). I don’t think I had a very clear plan about how this was going to work out. I walked in and saw Bill chatting with Richard Kenyon about entropy of dimer tilings, and hyperbolic volume; at this point I basically froze, turned around and walked out again.

I remember Bill giving a few talks at MSRI during the special program on low-dimensional topology and combinatorics which ran during the academic year 1996-7. He gave one talk explaining his idea of a new proof of (some version of) the Robertson-Seymour theorem on the well quasi-ordering of graphs partially ordered by taking graph minors; he explained this as a kind of compactness result (any class of graphs closed under taking minors is characterized by not containing a certain finite list of excluded minors). A simple version of this compactness concerns the relation on strings (in a finite alphabet), where one string U contains another string V if the letters of V appear in U in the same order, but not necessarily consecutively; I remember Bill explaining this with the example that the string “topology” contains the string “poo”.

I remember Bill at a one-day conference at MSRI on mathematics and the media, with both mathematicians and journalists in attendance. Bill explained some of his ideas about communicating mathematics; he started by drawing a picture (on a big sheet of butcher paper hanging on an easel), explaining the “evolution of mathematical thought”. It was basically a horizontal line; at the left hand side he drew some sort of lizard, and on the right hand side, a monkey and then an upright stick figure representing the modern mathematician.

I remember Bill running the “very informal foliations seminar” at MSRI with Dave Gabai, Joe Christy, and a few other people. This seminar was not advertised; I basically wandered in off the street into the middle of a 3-hour lecture by Bill, explaining his new ideas about universal circles, and how they might be used to approach the geometrization conjecture for 3-manifolds with taut foliations. By the time he was done, I had decided I wanted to work on foliations, and I more or less had my thesis problem.

I remember when Bill moved to Davis. This was the only time I ever saw him in his office at Berkeley — when he was cleaning it out. I remember the little photo that used to be on the door, the one that’s on the cover of “More Mathematical People”, of Bill as a child working at a desk. He saw me watching him carrying his boxes out of his office and looking at the photo, and gave a slightly embarrassed smile.

I remember emailing Bill in early 1998, to explain a few of my tentative ideas about foliations, which had been inspired by his slitherings paper. He invited me to come out to visit him at Davis and talk to him in person. Over the next year or so, I drove out there perhaps a couple of times per month, struggling up the freeway in my third-hand lemon, with the wind rushing in through the bad seals in the door frame. We would have conversations that lasted for hours; stopping occasionally for lunch and coffee. Bill basically became my “unofficial advisor” (my real advisor Andrew Casson was meanwhile going through a tough divorce, and moving to Yale), and perhaps because he did not have many “real” students at Davis at the time, I got a lot of his attention. We spent a lot of time working through the theory of universal circles; I learned a huge amount of mathematics, not only stuff obviously connected to foliations (or even low-dimensional topology), but combinatorics, analysis, group theory, and so on. And yet, Bill listened very carefully to my ideas, and always gave them his full attention and consideration. At the time I don’t think I appreciated how rare this attitude is in a senior mathematician towards a graduate student.

I remember running into Bill in Black Oak Books in Berkeley — in the legendary math book section, of course.

I remember arriving one day a bit early to find Bill waggling his tongue through a gap where one of his teeth had fallen out. He kept making slightly funny expressions on his face thereafter, and it was hard to stay focussed on mathematics for the rest of the day.

I remember when we were trying to work out the details of some construction, Bill got very enthusiastic and we went to the campus store to buy some enormous sheets of paper and a few sets of colored pencils, bringing them all back to Bill’s office and laying the paper out on the floor. Bill was really excited by this episode; he remarked that he used to do this sort of thing “all the time” when he was at Princeton. I got the impression he hadn’t done it for a while.

I remember one day Bill was with a crowd of graduate students, and he was talking about how intimidating it is to start out in mathematics. He thought more senior people contributed to the difficulty, by trying to give the impression that they understood everything, and he wished that people would be more forthright in admitting when they didn’t understand something. I admitted that I had never really learned the details of Galois theory, and Bill exclaimed “that’s great; that’s the sort of thing I mean. Everyone should know that Danny Calegari doesn’t understand Galois theory”. He repeated it several times, to quite a few people. I waited for him to add some things that he had found hard to understand, but that seemed to be it. (Years later in an email he confessed that he had “never really come to grips with the Burau representation” . . .)

I remember working to try to get a project finished in the week before Bill’s daughter was born (we didn’t make it in time). My wife and I were thinking about having kids at the time, and I shyly asked him about the experience. He became very emotional and tender, and talked about what it was like to hold a newborn and have them lie in your arms, trusting you completely.

I remember seeing Bill in 2007 at the Cornell topology conference, and noticing that he looked kind of shaggy, with a few days growth of stubble. I remember giving a talk about immersed curves in surfaces, and mentioning that there are examples of such curves which do not bound an immersed surface, but which “virtually” bound such a surface (i.e. they have a finite cover which bounds). I remember being startled when, after a few minutes, Bill exclaimed, “well, don’t leave us in suspense! what are some examples?” I remember giving Bill the advance copy of my “Foliations” book (it had just arrived) as a late 60th birthday present (I hadn’t gone to his birthday conference earlier that year), and he seemed really pleased to get it, and immediately started looking through it, especially at the pictures. Later on someone told me he had been “showing it to everybody”, which cheered me immensely.

I remember visiting Bill in winter of 2008. At the time my family and I were on a vegan kick, and I remember discussing veganism, and Colin Campbell’s book “The China Study” with Bill, while waiting for the cafeteria people to make us our vegan burritos for lunch. Bill’s wife happened to be very sick that week, and in addition they were moving house, so Bill was very distracted. I remember the class Bill taught one morning that week, in which he gave some beautiful constructions of families of projective structures on some surfaces; and the next class, when he explained how to immerse a projective plane in 3-space with a single triple point. When I left at the end of my visit, Bill apologized for being distracted with so many other things, but hoped that I’d visit again soon. Of course I told him not to apologize, that I’d had a great visit (which was true), and that I hoped I would come again soon when we both had more free time. That was the last time I saw him.

Tagged: Bill Thurston, obituary ]]>