Upper curvature bounds and CAT(K)

I am currently teaching a class at the University of Chicago on hyperbolic groups, and I have just introduced the concept of -hyperbolic (geodesic) metric spaces. A geodesic metrix space is -hyperbolic if for any geodesic triangle , and any there is some with . The quintessential -hyperbolic space is the hyperbolic plane, the unique (up to isometry) simply-connected complete Riemannian 2-manifold of constant curvature . It follows that any simply-connected complete Riemannian manifold of constant curvature is -hyperbolic for some depending on ; roughly one can take .

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 .

Bill Thurston 1946-2012

This morning I heard the awful news that Bill Thurston died last night. Many of us knew that Bill was very ill, but we all hoped (or imagined?) that he would still be with us for a while yet, and the suddenness of this is very harsh. As Sarah Koch put it in an email to me, “Although this was not unexpected, it is still shocking.” On the other hand, I am glad to hear that he was surrounded by family, and died peacefully.

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.

Circle packing – theory and practice

I am spending a few months in Göttingen as a Courant Distinguished Visiting Professor, and talking a bit to Laurent Bartholdi about rational functions — i.e. holomorphic maps from the Riemann sphere to itself. A rational function is determined (up to multiplication by a constant) by its zeroes and poles, and can therefore generically be put in the form where P and Q are polynomials of degree . If then is invertible, and is called a fractional linear transformation (or, sometimes, a Mobius transformation). The critical points are the zeroes of ; note that this is a polynomial of degree (not ) and the images of these points under are the critical values. Again, generically, there will be critical values; let’s call them . Precomposing with a fractional linear transformation will not change the set of critical values.

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)?

Agol’s Virtual Haken Theorem (part 3): return of the hierarchies

Ian gave his second and third talks this afternoon, completing his (quite detailed) sketch of the proof of the Virtual Haken Theorem. Recall that after work of Kahn-Markovic, Wise, Haglund-Wise and Bergeron-Wise, the proof reduces to showing the following:

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’s Virtual Haken Theorem (part 2): Agol-Groves-Manning strike back

Today Jason Manning gave a talk on a vital ingredient in the proof of Agol’s theorem, which is a result in geometric group theory. The theorem is a joint project of Agol-Groves-Manning, and generalizes some earlier work they did a few years ago. Jason referred to the main theorem during his talk as the “Goal Theorem” (I guess it was the goal of his lecture), but I’m going to call it the Weak Separation Theorem, since that is a somewhat more descriptive name. The statement of the theorem is as follows.

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

1. is hyperbolic;
2. is finite; and
3. 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.

Agol’s Virtual Haken Theorem (part 1)

I am in Paris attending a workshop at the IHP where Ian Agol has just given the first of three talks outlining his proof of the Virtual Haken Conjecture and Virtual Fibration Conjecture in 3-manifold topology (hat tip to Henry Wilton at the Low Dimensional Topology blog from whom I first learned about Ian’s announcement last week). I think it is no under overstatement to say that this marks the end of an era in 3-manifold topology, since the proof ties up just about every loose end left over on the list of problems in 3-manifold topology from Thurston’s famous Bulletin article (with the exception of problem 23 — to show that volumes of closed hyperbolic 3-manifolds are not rationally related — which is very close to some famous open problems in number theory). The purpose of this blog post is to say what the Virtual Haken Conjecture is, and some of the background that goes into Ian’s argument. I hope to follow this up with more details in another post (after Agol gives talks 2 and 3 this coming Wednesday). Needless to say this post has been written in a bit of a hurry, and I have probably messed up some crucial details; but if that caveat is not enough to dissuade you, then read on.

Characteristic classes of foliations

I recently learned from Jim Carlson’s blog of the passing of Harsh Pittie on January 16 this year. I never met Harsh, but he is very familiar to me through his work, in particular for his classic book Characteristic classes of foliations. I first encountered this book as a graduate student, late last millenium. The book has several features which make it very attractive. First of all, it is short — only 105 pages. My strong instinct is always to read books cover-to-cover from start to finish, and the motivated reader can go through Harsh’s book in its entirety in maybe 4 hours. Second of all, the exposition is quite lovely: an excellent balance is struck between providing enough details to meet the demands of rigor, and sustaining the arc of an idea so that the reader can get the point. Third of all, the book achieves a difficult synthesis of two “opposing” points of view, namely the (differential-)geometric and the algebraic. In fact, this achievement is noted and praised in the MathSciNet review of the book (linked above) by Dmitri Fuks.

Filling geodesics and hyperbolic complements

Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3-manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flow-invariant way into stable, unstable and flow directions) and contact (i.e. they preserve a contact form — that is, a 1-form for which is a volume form). Their preprint gives some very interesting new constructions of such flows, obtained by surgery along a Legendrian knot (one tangent to the kernel of the contact form) which is transverse to the stable/unstable foliations of the Anosov flow.