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

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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 ]

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/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

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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 ]]>

The map cannot usually be recovered from (even up to precomposition with a fractional linear transformation); one needs to specify some extra global topological information. If we let denote the preimage of under , and let denote the subset consisting of critical points, then the restriction is a covering map of degree , and to specify the rational map we must specify both and the topological data of this covering. Let’s assume for convenience that 0 is not a critical value. To specify the rational map is to give both and a representation (here denotes the group of permutations of the set ) which describes how the branches of are permuted by monodromy about . Such a representation is not arbitrary, of course; first of all it must be irreducible (i.e. not conjugate into for any ) so that the cover is connected. Second of all, the cover must be topologically a sphere. Let’s call the (branched) cover for the moment, before we know what it is. The Riemann-Hurwitz formula lets one compute the Euler characteristic of from the representation . A nice presentation for has generators represented by small loops around the points , and the relation . For each define to be the number of orbits of on the set . Then

If each is a transposition (i.e. in the generic case), then and we recover the fact that .

This raises the following natural question:

**Basic Question:** Given a set of points in the Riemann sphere, and an irreducible representation satisfying , what are the coefficients of the rational function that they determine (up to precomposition by a fractional linear transformation)?

Note that we would like to recover the coefficients *numerically* (i.e. as numbers with decimal points). And we are really interested in finding a *practical* algorithm, and then implementing it on computer. One obvious (and bad) idea is to just solve for the coefficients of P and Q subject to the constraint that the set of critical values is V (after normalizing so that three of the critical points are 0, 1 and infinity to remove the ambiguity of the precomposition). The problem is that the number of such solutions is exponential in the degree, and although Newton’s method will quickly find *some* solution, it is very, very unlikely to be the solution with the correct combinatorial data.

Another idea — and one that leads to the point of this blog post — is to try to build the branched cover directly as a Riemann surface together with a holomorphic map with the correct critical values and combinatorics, and then uniformize it as to determine the numerical location of the zeros and poles. This sounds more promising, since there is an obvious way to build piecewise from copies of regions in glued together by very explicit maps. The problem is that (numerical) uniformization itself is very slow, at least if one wants any kind of accuracy. On the other hand, we do not need to know the values of the uniformization map everywhere, only the locations of the zeros and poles. So we can try to ask for a fast and approximate method of uniformization which gives sufficiently accurate values of these numbers, that they can then be adjusted quickly to very accurate values by Newton’s method.

One potential idea is to use the method of *circle packing*. A circle packing is a (rigid) configuration of round circles with disjoint interiors and prescribed combinatorial pattern of tangencies. Abstractly, the circle packing determines a triangulation, with one vertex for each circle, one edge for each tangency, and one triangle for each triple of mutually tangent circles. Implicitly, by using the term “round circle”, the domain in which the circles are packed should be a Riemann surface together with a complex projective structure; for example, the Riemann sphere, or the Euclidean or hyperbolic planes. Given a projective surface and a triangulation satisfying mild topological conditions, such a circle packing exists and is unique; this is known as the Circle Packing Theorem (aka the Koebe-Andreev-Thurston theorem). One can also solve for configurations of circles intersecting at prescribed angles; for instance, one can look for a configuration of round circles with prescribed combinatorics, and meeting always at right angles. Such a configuration in the Riemann sphere can be interpreted as the boundaries of a collection of geodesic planes in hyperbolic 3-space meeting in right angles, and cutting out a compact right-angled polyhedron. The existence of such a polyhedron is the base step in Thurston’s inductive proof of geometrization for Haken 3-manifolds.

One can also think of the circle packing as a discrete version of a conformal structure; at a talk at the conference in 1985 celebrating de Branges’ proof of the Bieberbach conjecture, Thurston proposed using circle packings to approximate conformal mappings. One starts with a region U in the complex plane, and packs it nearly tightly with a hexagonal packing of small round circles. Together with the boundary of U one obtains a topological circle packing; the round circle packing with the same combinatorics can be thought of as a packing of the unit disk. One therefore obtains a coarse “map” from U to the disk, taking each round circle in the packing to a round circle in the disk, and one should think of this as a discrete version of a conformal map. As the mesh size goes to zero, Thurston conjectured these maps should converge to the uniformizing map. This conjecture was proved by Rodin-Sullivan.

In the context of our Basic Question we can try to find our desired rational map as follows. Starting with the collection of points V in the Riemann sphere, we build an (almost) round circle packing in such a way that one of the centers is at 0 and infinity and at each point of V. One should probably choose the mesh size quite small near these special points, since the derivative of is going to blow up near V. This determines a topological graph (the 1-skeleton of the triangulation described above). The branching data defines a new graph in an obvious way, and we can find the circle packing associated to that graph. The “preimages” of 0 and infinity are given as the centers of d circles in this new circle packing, and we can take these as our approximate zeros and poles for .

Anyway, this is all preamble to explaining that I wrote a little code to perform circle packing with prescribed combinatorial data, and in case I don’t do anything else with it (which is quite likely) I thought it might be amusing to post the code and some of the pictures it produces. Note that very sophisticated and highly optimized code for circle packing is already available from many other places; for example, Ken Stephenson has an amazing collection of resources (both theoretical and computational) on his website.

The latest code is, and will continue to be, available at github here:

Download the source as a zip file, then unzip and type “make” to make. The program is written in C++, and outputs graphics to the screen using Xlib, and to an .eps file. It can either be run interactively (without an argument) or non-interactively, taking a file name (containing a topological circle packing) as a parameter.

A topological circle packing is written in a (ASCII text) file with the following structure:

number of vertices

valence of vertex 0; list of adjacent vertices in circular order

valence of vertex 1; list of adjacent vertices in circular order (etc.)

valence of vertex n-1; list of adjacent vertices in circular order

initial radius of vertex 0

initial radius of vertex 1 (etc)

initial radius of vertex n-1.

By convention, vertex 0 is the “outer” circle (centered at infinity). The program doesn’t check that the adjacency data that it is given is consistent, or that it gives rise to a topological sphere.

If you try this out, please let me know what you think could be improved or fixed. Feel free to modify or change the code however you like (subject to the usual GPL license conditions). A wishlist would include to add the functionality I sketched above, i.e. to find approximate rational maps with prescribed critical values and branching data; any reader with some time on their hands is warmly invited to do this!

Some screen shots of the X-windows output:

and some sample .eps output (converted to .jpg for wordpress):

The effect of taking a double branched cover over two circles:

Tagged: circle packing, rational maps ]]>

**Theorem (Agol):** Let G be a hyperbolic group acting properly discontinuously and cocompactly on a CAT(0) cube complex X. Then there is a finite index subgroup G’ so that X/G’ is special; in other words, G is *virtually special*.

Agol works with a characterization of virtually special groups, due to Wise, which is more closely tied to the notion of hierarchies.

**Definition:** A hyperbolic group G is QVH if it is obtained inductively by the following procedures:

- the trivial group is QVH;
- If G splits as an amalgam where A and C are QVH, and B is quasiconvex in G, then G is QVH;
- similarly for an HNN extension ; and
- If H is QVH and is contained in G with finite index, then G is QVH.

Wise shows the following:

**Theorem (Wise):** A group is hyperbolic and acts cocompactly on a CAT(0) cube complex with special quotient if and only if it is QVH.

Thus the virtually special groups are the analogue in the world of hyperbolic groups of virtually Haken hyperbolic 3-manifolds in topology; we can say they are the fundamental groups of hyperbolic NPC cube complexes that admit a “quasiconvex virtual hierarchy”. So Ian’s argument works by exhibiting a finite cover of X/G with such a hierarchy.

The groups along which we would like to split in this hierarchy are (finite index in) the fundamental groups of the hyperplanes in X/G. Finding the cover amounts to separating these subgroups in finite covers. We don’t know how to do this directly, but the Weak Separation Theorem of Agol-Groves-Manning (discussed yesterday) shows that these subgroups can be “separated” in infinite covers (in a certain sense). The separation refers to the fact that a finite index subgroup of the hyperplane groups lifts to the cover, so that the hyperplanes in the infinite cover will be compact, 2-sided and embedded. In algebraic language, the Weak Separation Theorem guarantees the existence of a normal subgroup G’ of G so that if we write and then has locally 2-sided embedded compact acylindrical hyperplanes (the acylindricity implies that the fundamental groups of the hyperplanes are malnormal). We can split along these hyperplanes to produce a kind of hierarchy, so if were compact we would be done. The idea is to take the (infinitely many) pieces that are created by the cutting, separate them into finitely many classes, and glue them together in such a way as to create a finite covering of X with a hierarchy of its own.

Ian calls the pieces into which is decomposed by its hyperplanes *cubical polyhedra*, although they are not really polyhedra, but rather cubical subcomplexes of the cubical barycentric subdivision of . The combinatorics of the system of (compact!) hyperplanes in the (noncompact!) is encoded by the so-called *crossing graph* . This graph has as vertex set equal to , the set of hyperplanes of (there is a slightly unfortunate point that Ian uses the terminology “hyperplane” for what Wise calls “walls” when they are embedded; the letter W is supposed to represent the word “walls”; anyway, Ian used the terms hyperplane and walls synonymously in his talk, so if I sometimes accidentally use one word instead of the other, that’s the reason). A pair of hyperplanes (i.e. vertices) share an edge in in two cases, if either:

- the hyperplanes intersect; or
- conjugates of their fundamental groups have an infinite intersection (i.e. they are not acylindrical as a pair)

The group acts on , and moreover the maximum degree of is finite, because hyperplanes are compact and fall into finitely many -orbits, and is locally compact (the hyperbolicity of G guarantees that there are no arbitrarily long essential cylinders running between distinct hyperplanes, so hyperplanes which are sufficiently far away from each other will not contribute an edge as in case 2 above) . Let k be the maximum degree of the vertices of .

Suppose we could color the graph with finitely many colors in such a way that adjacent vertices have different colors, and the coloring is invariant under some finite index subgroup of . Then the quotient would be compact with a quasiconvex hierarchy, and we would be done. Another way of saying that the coloring should be invariant under a finite index subgroup of is to say that we have a finite set of colorings of which are permuted by the action of . Let denote the set of all colorings of by the numbers such that adjacent vertices get different colors. The set can be topologized with the topology of convergence of colorings on finite subgraphs, making it into a compact (totally disconnected) space, and the group acts on this space by homeomorphisms. Translating the statement above into this language, if we could find an invariant finite set on this space, we would be done. Instead, Ian finds an invariant *probability measure*. This is a completely general statement, and applies to all locally finite graphs with cocompact group actions.

**Theorem:** Let be a graph with bounded valence k, and G a group acting cocompactly on . Then there is a G-invariant probability measure on the space of (k+1)-colorings of .

Ian gave a very elegant proof of this theorem; after working it out, Lewis Bowen informed him that the theorem is a consequence of known work of Kechris-Solecki-Todorocevic on Borel colorings of Borel graphs (the kind that arise in the theory of measure equivalence of group actions). But Ian’s proof is so elegant that I can’t resist reproducing it here.

Pick orbit classes of the edges of . We call an assignment of colors to the vertices which does not necessarily assign different colors to adjacent vertices a *labeling*. The weight of a labeling is the number of whose vertices get the same color. Weight extends linearly to the space of probability measures on labelings; a G-invariant probability measure on labelings of weight 0 will be a G-invariant probability measure on colorings.

Now, a random labeling with n colors will have weight m/n. There is a function from labelings with n colors to labelings with (n-1) colors whenever that takes each vertex labeled n to the smallest number which is not a label on an adjacent vertex. Extend this operation by linearity to probability measures on labelings. This operation is G-equivariant and does not increase weight. So start with a random G-invariant probability measure on labelings with n colors (call it ) and reduce the number of colors one by one to . This gives a measure on G-invariant labelings with colors whose weight is at most m/n (the weight of the random labeling). Take a weak limit of the as ; this is a G-invariant probability measure on -labelings whose weight is 0; that is, it is a probability measure on -colorings, as desired. qed

OK, we now have a -invariant probability measure on colorings of . The colors 1 to (k+1) correspond to the order in which to cut along the hyperplanes in a hierarchy, so we can think of this probability measure as a kind of “superposition” of hierarchies of , which want to be pulled back from some hierarchy on a finite cover of . When we cut along hyperplanes and then try to glue them back up, we need to remember the labels on all the cut open facets meeting the hyperplanes we are gluing. So it is important not just to remember the colors on vertices, but also all the colors on adjacent vertices we have cut up earlier, and the colors on vertices adjacent to them that we have cut up earlier, and so on.

Given a graph and a coloring of the vertices by numbers from 1 to (k+1), each vertex determines a “descending link”, which is the union of all simplicial paths emanating from that vertex along which the numbers decrease. The *supercolor* of a vertex is the structure of its descending link as a colored graph. Since the valence is bounded, there are only finitely many supercolors. Supercoloring determines an equivalence relation finer than coloring, and therefore an equivalence relation on where if there is some g in so that gv=w and the supercolor of v with respect to c is equal to the supercolor of w with respect to (remember that denotes the set of hyperplanes). Similarly we can define equivalence relations on where denotes the collection of faces of in the barycentric subdivision dual to edges, by saying that if where is the hyperplane containing , and is the hyperplane containing . Finally we can define an equivalence relation on where denotes cubical polyhedra, by if for corresponding faces F,E of P,Q.

A non-negative -invariant real valued function

satisfies the *gluing equations* if for every face in the boundary of polyhedra , and for every equivalence class we have where the sum is taken over equivalence classes restricting on F to equivalence classes which are equivalent to (i.e. the supercolors agree on the given face). Note that this is a *finite* sum, since there are only finitely many supercolors and orbits of faces or cubical polyhedra.

The point of the measure is that it defines a solution to the gluing equations, by setting to be equal to the measure of the set of colors inducing the given supercolor on P. Since there are only finitely many gluing equations, and they have integer coefficients, the existence of one nontrivial solution implies the existence of a nontrivial *integer* solution; i.e. we can find an taking integer values.

I think it’s time for me to take a break now, so I’m posting this with the intention of coming back to add more details about how to use this integer solution to the gluing equations to get a hierarchy. Let me just say cryptically that this solution lets one glue up a finite collection of pieces which immerses into our (partially glued up) hierarchy in such a way that at the next step what needs to be glued are a finite collection of pieces which cover the same compact hyperplane with equal total degrees. It is at this point that Wise’s Malnormal Special Quotient Theorem lets one find finite covers in which the pieces can be matched in pairs and glued up along the hyperplane in question. More later (I hope). For the moment, here’s Ian explaining a detail to some doofus.

OK, after a break and another full day of lectures, I’m suitably rested, and ready to (briefly!) describe the endgame.

We imagine that we have already glued everything up to get a quasiconvex hierarchy, and then we inductively split along hyperplanes in the order of their colors . The result will be a collection of cubical polyhedra whose boundaries are decorated with what is known as a boundary pattern, which keeps track of where the cuts where made.

If V is a finite cube complex obtained by (perhaps partially) cutting open along a quasiconvex hierarchy, we will denote its boundary pattern by . It is glued back to a hierarchy by first gluing up ; this gives a new finite cube complex V’ with a new boundary pattern which are the image of the boundary pattern in V.

So Ian’s inductive gluing procedure starts with with boundary pattern which consists simply of a collection of cubical polyhedra. The number of colored polyhedra of each type is the corresponding coefficient of our (integer) solution to the gluing equations, and the facets in the given boundary pattern are those with the corresponding color.

There is no obstruction to gluing up the facets in pairs, since this is exactly what the gluing equation guarantees we can do. This gives rise to with boundary pattern . Now the components of are more complicated, and it is not immediately clear how to glue them up. It will turn out that the components of all cover certain boundary components (with respect to a particular boundary pattern) of a particular compact cube complex in such a way that the sum of the degrees of the covers on either side of the component agrees. We would therefore like to take a finite cover of in which the components of map to corresponding components of the boundary of in a way which can be matched up and then glued. In Ian’s paper he describes a method to find such a finite cover inductively, using a method in an appendix of Agol-Groves-Manning; however he pointed out in his talk on Wednesday that the cover can be found in one step by the MSQT. Anyway, I won’t say any more about it here.

OK, in this way we glue up in a finite cover to get , and then the components of immerse into covering the same object from two sides with the same degree, so we can pass to a further cover (by the MSQT) where they can be paired, and glued up to get , and so on. Eventually we have glued up everything in a finite cover, obtained a hierarchy, applied Wise’s theorem that QVH is equivalent to virtually special, and then it’s time to break out the champagne.

So what’s ? In fact there is a for each j; it is a disjoint union of hyperplanes of the original complex split open along the hyperplanes they intersect of smaller color, quotiented out by the action of the stabilizer in of the associated equivalence class. This complex has the property that each equivalence class of face for which the color of F in the coloring c is j has a unique representative in the complex . It is this fact, together with the fact that the set of polyhedra in satisfies the gluing equations (because inductively it is a cover of a partially glued union of polyhedra which as a set satisfy the gluing equations) which implies that the map is an immersion, and then the gluing equations say that the degrees on either side have the same sum, and now the previous paragraph makes sense (ahem!).

Well, that’s it for my summary. I will post a few photos of the blackboard taken by Patrick Massot and Alden Walker when I get a chance (one such photo by Patrick is above). If you want more details, then I believe Ian intends to post his preprint before too long, so keep watching the ~~skies~~ arXiv.

**Update (April 13):** Ian’s preprint is now available on the arXiv here.

Tagged: CAT(0) cube complex, gluing equations, hierarchy, malnormal special quotient theorem, virtually special ]]>

**Weak Separation Theorem (Agol-Groves-Manning):** Let G be a hyperbolic group, let H be a subgroup of G which is quasiconvex, and isomorphic to the fundamental group of a virtually special NPC cube complex, and let g be an element of G which is not contained in H. Then there is a surjection so that

- is hyperbolic;
- is finite; and
- is not contained in .

In the remainder of this post I will try to explain the proof of this theorem, to the extent that I understand it. Basically, this amounts to my summarizing Manning’s talk (or the part of it that I managed to get down in my notes); again, any errors, foolishness, silly blog post titles etc. are due to me.

Recall from my previous post that an NPC (non-positively curved) cube complex is a compact quotient of a CAT(0) cube complex by a group (in this case H) acting properly discontinuously, and that the complex is virtually special if it has a finite (orbi-)cover satisfying the Haglund-Wise conditions (i.e. hyperplanes are embedded, two-sided, and there are no self- or interosculations). The fundamental group of a virtually special cube complex is a linear group (i.e. embeds in for some n), and is therefore *residually finite*; this means that the intersection of all finite index normal subgroups consists only of the identity element. The fact that all finitely generated linear groups are residually finite is known (to topologists, anyway) as *Selberg’s Lemma*. Roughly, the idea is to consider the ring A of matrix entries in a faithful linear representation, and then the desired finite index subgroups are the kernels of maps to for suitable prime ideals in (see e.g. here for more details). More generally, if G is a group, a subgroup H is said to be *separable* if for any g not in H there is a homomorphism from G to a finite group so that the image of g is disjoint from the image of H. Groups in which every finitely generated subgroup is separable are said to be LERF; it is a consequence of Agol’s proof of the VHC that every hyperbolic 3-manifold group is LERF, but we are getting ahead of ourselves here.

The Weak Separation Theorem is in the direction of showing that the subgroup H is separable; if the quotient could be taken to be finite, that is exactly what it would show. But finding a quotient where is finite turns out to be good enough for Agol’s purposes.

An important special case of the theorem is when H is *almost malnormal*. A subgroup H of G is normal if conjugation in G sends H to itself. H is *malnormal* if for all , where superscript denotes conjugation, and H is *almost malnormal* if is *finite* for all .

Bowditch showed that if G is hyperbolic and H is quasiconvex and almost malnormal in G, then the pair is *relatively hyperbolic*. The concept of relative hyperbolicity generalizes the fundamental group of a noncompact complete negatively curved manifold of finite volume; the fundamental group is not (necessarily) a hyperbolic group (although it is in some cases!) but the lack of hyperbolicity is concentrated in the noncompact *cusp* of the manifold; the fundamental group of the cusp itself might or might not be hyperbolic. It is a *parabolic* subgroup of the fundamental group, and the *pair* is relatively hyperbolic. Abstractly, a pair is relatively hyperbolic, where is a collection of conjugacy classes of subgroups of , if the space obtained by attaching “horoballs” to the conjugates of the subgroups in in (the Cayley graph of) G is hyperbolic (see here for more details). Note that if the subgroups are themselves hyperbolic, then is also hyperbolic (in the absolute sense).

Relatively hyperbolic groups invite relatively hyperbolic Dehn filling, by analogy with Thurston’s hyperbolic Dehn surgery for (noncompact) hyperbolic 3-manifolds with cusps. Suppose is relatively hyperbolic, where . For each i choose some normal subgroup of . The quotient of by the normal closure (in G) of all the is called a *filling* of G by the , and is denoted . If each of the is finite index in , we call it a *peripherally finite* filling. The fundamental theorem of hyperbolic Dehn surgery, due (in this form) originally to Osin, is as follows:

**Theorem (Osin):** Let F be a finite subset of G, and let be relatively hyperbolic. Then there is a finite subset B of G, so that for any filling for which B does not intersect any of the , one has the following:

- for each i;
- the image pair in the quotient is relatively hyperbolic
- the restriction of to is injective.
- for all i

Actually, the fourth condition is not proved by Osin, but can be deduced with “a bit of work”, according to Jason. Notice that if the filling is peripherally finite, so that all are finite, then these quotients of the parabolic groups are trivially hyperbolic (since every finite group is hyperbolic), and therefore the quotient group is also hyperbolic.

If H is almost malnormal, Osin’s surgery theorem implies the Weak Separability Theorem, as follows. Since H is almost malnormal and quasiconvex, and G hyperbolic, by Bowditch the pair is relatively hyperbolic. We let F (as in the statement of Osin’s theorem) consist only of the element , and let be the finite set of “bad” fillings the theorem guarantees. Since H is assumed to be virtually special, it is residually finite, and therefore contains a finite index normal subgroup N missing B. Taking the quotient of G by the normal closure of N gives a surjection to a group in which the image of H is finite, and disjoint from the image of g, as desired.

What if H is not malnormal? Then one must induct on an invariant called the *height*, which measures the failure of the group to be almost malnormal. The idea of height was introduced by Gitik-Mitra-Rips-Sageev.

**Definition: **If H is quasiconvex in a hyperbolic group G, the height of H is the least integer n so that if there are elements so that are distinct, then the intersection of conjugates is finite.

H has height 0 if and only if it is finite. It has height 1 if and only if it is almost malnormal and infinite. One can define a complex whose k-simplices are the -fold infinite intersections of distinct conjugates of H, and height is then the dimension of this complex plus one. Gitik-Mitra-Rips-Sageev prove that every quasiconvex subgroup of a hyperbolic group has *finite* height. They also show that for any k there are only finitely many H-conjugacy classes of infinite groups of the form (this is vacuous for k as big as the height or bigger). The minimal such infinite intersections are not far from being malnormal, and after a suitable modification, will give rise to a suitable relatively hyperbolic family.

Start with , the collection of H-conjugacy classes of minimal infinite intersections of the form (these are conjugacy classes of subgroups in H). These subgroups are intersections of quasiconvex subgroups, and are easily seen to be quasiconvex themselves. Replace each D in by its commensurator in H (note that each D is finite index in ), and then choose one such subgroup per H-conjugacy class. This produces a new collection of conjugacy classes of subgroups in H. Observe that the elements of are almost malnormal — in fact is an almost malnormal collection, meaning that nontrivial conjugates of one subgroup intersect any other subgroup in the collection in a finite set — so that is relatively hyperbolic (Bowditch’s criterion for relative hyperbolicity generalizes to almost malnormal collections of quasiconvex subgroups).

Now, replace each D in by its commensurator in G; again each D is finite index in . This produces a collection of subgroups of G; choose one subgroup for each G-conjugacy class to arrive finally at an almost malnormal collection of quasiconvex subgroups of G, and deduce (by Bowditch again) that is relatively hyperbolic.

**Definition:** A filling with each in some in as above is an *H-filling* if whenever is infinite for some in , then is contained in D.

An H-filling by definition induces a filling of H, i.e. a quotient . With this terminology, Agol-Groves-Manning prove

**Theorem (Agol-Groves-Manning): **Let G be hyperbolic, let H be quasiconvex in G of height at least 1, and let and be as above, and let g be in . Then for all “sufficiently long” peripherally finite H-fillings ,

- is isomorphic to the induced filling of H;
- is quasiconvex in ;
- is not contained in ; and
- the height of is strictly less than the height of H.

Let’s not worry too much about what the condition “sufficiently long” means here; suffice it to say that such fillings can be found if H is residually finite.

Now, in the setup of the Weak Separation Theorem, the subgroup H is the fundamental group of a virtually special NPC complex, and is therefore residually finite. So we can apply this filling theorem of AGM to reduce the height of H. But now one is stuck, because the resulting image , while of strictly lower height, might not be residually finite. Here is where Wise’s Malnormal Special Quotient Theorem (alluded to at the end of my previous post) comes in. The statement of the MSQT is as follows:

**Malnormal Special Quotient Theorem (Wise): **Let H be hyperbolic, let be relatively hyperbolic, where the in are almost malnormal and quasiconvex. Suppose H is the fundamental group of a virtually special NPC complex. Then there are finite index subgroups in the so that if is any peripherally finite filling with contained in the , then is the fundamental group of a virtually special NPC complex.

The MSQT implies that, providing we are careful for our H-filling to kill subgroups contained in the , the image of H will be virtually compact special, and the induction can be continued, reducing the height of (the image of) H until the Weak Separation Theorem is proved.

Tagged: height, hyperbolic Dehn surgery, hyperbolic groups, malnormal groups, quasiconvex subgroup, subgroup separation, virtually special cube complexes ]]>