Fritz Grunewald, RIP

Fritz Grunewald died unexpectedly this week, just before his 61st birthday.  I never met him but have always been an admirer of his work, and I’d been meaning to post about his lovely paper with Lubotzky, “Linear representations of the automorphism group of the free group.” I’m sorry it takes such a sad event to spur me to get around to this, but here goes.

Let F_n be the free group of rank n, and Aut(F_n) its automorphism group.  How to understand what this group is like?  A natural approach is to study its representation theory.  But it’s actually not so easy to get a handle on representations of this group.  Aut(F_n) acts on F_n^ab = Z^n, so you get one n-dimensional representation; but what else can you find?

The insight of Grunewald and Lubotzky is to consider the action of Aut(F_n) on the homology of interesting finite-index subgroups of F_n.  Here’s a simple example:  let R be the kernel of a surjection F_n -> Z/2Z.  Then R^ab is a free Z-module of rank 2n – 1, and the -1 eigenspace of R^ab has rank n-1.  Now F_n may not act on R, but some finite-index subgroup H of F_n does (because there are only finitely many homomorphisms F_n -> Z/2Z, and we can take H to be the stabilizer of the one in question.)  So H acquires an action on R^ab; in particular, there is a homomorphism from H to GL_{n-1}(Z).  Grunewald and Lubotzky show that this homomorphism has image of finite index in GL_{n-1}(Z).  In particular, when n = 3, this shows that a finite-index subgroup of Aut(F_3) surjects onto a finite-index subgroup of GL_2(Z).  Thus Aut(F_3) is “large” (it virtually surjects onto a non-abelian free group), and in particular it does not have property T.  Whether Aut(F_n) has property T for n>3 is still, as far as I know, unknown.

Grunewald and Lubotzky construct maps from Aut(F_n) to various arithmetic groups via “Prym constructions” like the one above (with Z/2Z replaced by an arbitrary finite group G), and prove under relatively mild conditions that these maps have image of finite index in some specified arithmetic lattice.  Of course, it is natural to ask what one can learn from this method about interesting subgroups of Aut(F_n), like the mapping class groups of punctured surfaces.  The authors indicate in the introduction that they will return to the question of representations of mapping class groups in a subsequent paper.  I very much hope that Lubotzky and others will continue the story that Prof. Grunewald helped to begin.

Distinguished lectures this week: Gunnar Carlsson on persistent homology

Fun week coming up:  Gunnar Carlsson of Stanford will be giving this semester’s Distinguished Lecture Series at Wisconsin.  The talks:

Monday, March 8 and Tuesday, March 9, 4pm, Van Vleck B239:

“Topology and Data”

There is a growing need for mathematical methodologies which can provide understanding of high dimensional data sets. These methods also need certain kinds of robustness, so that they should not be too sensitive to changes of scale and to noise, and they should be applicable to various kinds of unstructured data. In these talks we will discuss methods for adapting idealized notions coming from algebraic topology and homotopy theory to the world of point clouds, and show numerous examples of applications of these methods.

Wednesday, March 10, 1:30pm, 1209 Engineering Hall:

“Functoriality, Generalized Persistence, And Structural Signatures”

See the computational topology at Stanford home page for a good overview of the topics of Gunnar’s lectures.  Nigel Boston, Rob Nowak, and I have been running a learning seminar on the topic:  I’ll try to post again this week about some data that Laura Balzano and I messed around with with persistent homology in mind, and what we learned thereby.

Square pegs, square pegs. Square, square pegs.

Lately I’ve been thinking again about the “square pegs” problem:  proving that any simple closed plane curve has an inscribed square.  (I’ve blogged about this before: here, here, here, here, here.)  This post is just to collect some recent links that are relevant to the problem, some of which contain new results.

Jason Cantarella has a page on the problem with lots of nice pictures of inscribed squares, like the one at the bottom of this post.

Igor Pak wrote a preprint giving two elegant proofs that every simple closed piecewise-linear curve in the plane has an inscribed square.  What’s more, Igor tells me about a nice generalized conjecture:  if Q is a quadrilateral with a circumscribed circle, then every smooth simple closed plane curve has an inscribed quadrilateral similar to Q.  Apparently this is not always true for piecewise-linear curves!

I had a nice generalization of this problem in mind, which has the advantage of being invariant under the whole group of affine-linear transformations and not just the affine-orthogonal ones:  show that every simple closed plane curve has an inscribed hexagon which is an affine-linear transform of a regular hexagon.  This is carried out for smooth curves in a November 2008 preprint of Vrecica and Zivaljevic.  What’s more, the conjecture apparently dates back to 1972 and is due to Branko Grunbaum.  I wonder whether Pak’s methods supply a nice proof in the piecewise linear case.

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

Now I’ll say a little bit about the actual problem treated by the new paper with Venkatesh and Westerland.  It’s very satisfying to have an actual theorem of this kind:  for years now we’ve been going around saying “it seems like asymptotic conjectures in analytic number theory should have a geometric reflection as theorems about stable cohomology of moduli spaces,” but for quite a while it was unclear we’d ever be able to prove something on the geometric side.

The new paper starts with the question: what do ideal class groups of number fields tend to look like?

That’s a bit vague, so let’s pin it down:  if you write down the ideal class group of the quadratic imaginary number fields , as d ranges over squarefree integers in [0..X],  you get a list of about finite abelian groups.

The ideal class group is the one of the most basic objects of algebraic number theory; but we don’t know much about this list of groups!  Their orders are more or less under control, thanks to the analytic class number formula.  But their structure is really mysterious.

The braid group, analytic number theory, and Weil’s three columns

This post is about a new paper of mine with Akshay Venkatesh and Craig Westerland; but I’m not going to mention that paper in the post. Instead, I want to explain why topological theorems about the stable homology of moduli spaces are relevant to analytic number theory.  If you’ve seen me give a talk about this stuff, you’ve probably heard this spiel before.

We start with Weil’s famous quote about “the three columns”:

“The mathematician who studies these problems has the impression of deciphering a trilingual inscription. In the first column one finds the classical Riemannian theory of algebraic functions. The third column is the arithmetic theory of algebraic numbers.  The column in the middle is the most recently discovered one; it consists of the theory of algebraic functions over finite fields. These texts are the only source of knowledge about the languages in which they are written; in each column, we understand only fragments.”

Let’s see how a classical question of analytic number theory works in Weil’s three languages.  Start with the integers, and ask:  how many of the integers between X and 2X are squarefree?  This is easy:  we have an asymptotic answer of the form

(In fact, the best known error term is on order X^{17/54}, and the correct error term is conjectured to be X^{1/4}; see Pappalardi’s “Survey on k-freeness” for more on such questions.)

So far so good.  Now let’s apply the popular analogy between number fields and function fields, going over to Weil’s column 3, and ask: what’s the analogous statement when Z is replaced by F_q[T]?

“Every curve is a Teichmuller curve,” or “Why SL_2(Z) has the congruence subgroup property.”

A Teichmüller curve in M_g, the moduli space of genus-g curves, is an algebraic curve V in M_g such that the inclusion V -> M_g induces an isometry between the constant-curvature metric on V and the restriction of the Teichmüller metric on M_g.

Alternatively:  the cotangent bundle of M_g, considered as a real manifold, admits a natural action of SL_2(R); the orbits are all copies of SL_2(R) / SO(2), or the upper half-plane.  Most of the time, when you project that hyperbolic plane H down to M_g, you get a dense orbit that wanders all over M_g.  But every once in a while, the fibers of the map H -> M_g are a lattice in H, and the image is actually an algebraic curve; that, again, is a Teichmüller curve.

Teichmüller curves are the subject of lots of recent research; for now, let me just say that they are interestingly canonical curves inside M_g.  Matt Bainbridge proved strong results about their intersection numbers in Hilbert modular surfaces.  McMullen classified Teichmuller curves in M_2, giving a very nice algebraic description of the 1-parameter families of genus-2 curves parametrized by Teichmüller curves.  (As far as I know, there’s no such description in higher genus.)  In a recent note, McMullen proved that they are all defined over number fields.

This leads one to ask:  which curves defined over algebraic number fields are Teichmüller curves?  This is the subject of a paper Ben McReynolds and I just posted to the arXiv, “Every curve is a Teichmüller curve.”  The title should be read birationally; what we prove is that every curve X over an algebraic number field is birational (over C) to a Teichmüller curve in some M_g.  (In the posted version, we prove the slightly weaker statement that X is birational to a Teichmüller curve in M_{g,n}), but we’ve since tweaked the argument to get the closed-surface version.)

So why does SL_2(Z) have the congruence subgroup property?  Especially given that it, y’know, doesn’t?

Here’s what I mean.  Let Gamma_{g,n} be the mapping class group of a genus-g surface with n punctures.  Then Gamma_{g,n} acts as a group of outer automorphisms of the fundamental group pi_{g,n} of the surface; and from this, you get an action of Gamma_{g,n} on the finite set

Hom(pi_{g,n},G)/~

where G is a finite group and ~ is conjugacy.

By a congruence subgroup of Gamma_{g,n} let’s mean a stabilizer in this action.  Why this definition?  Well, when g = 1, n = 0, and G = Z/NZ, the stabilizer is just the standard congruence subgroup Gamma_0(N).  And you can easily check that the class of congruence subgroups of Gamma_{1,0} is cofinal with the usual class of congruence subgroups in SL_2(Z).

Now Gamma_{1,1} is also isomorphic to SL_2(Z), but the notion of “congruence subgroup of SL_2(Z)” afforded by this isomorphism is much more general than the usual one.  So much so that one gets the following, which is really the main point of my paper with Ben:

Every finite-index subgroup of Gamma_{1,1} containing the center and contained in Gamma(2) is a congruence subgroup.

It turns out that the finite covers of the moduli space M_{1,1} corresponding to such finite-index subgroups are always Teichmüller curves; since, by Belyi’s theorem, every curve over a number field can be so expressed, we get the desired result.

The italicized assertion above can be thought of as a very strong kind of “congruence subgroup property.”  Of course, CSP usually refers to the property that every finite-index subgroup contains a principal congruence subgroup.  That finite-index subgroups Gamma_{1,1} (and even Gamma_{1,n}) always contain congruence subgroups as defined above is a theorem of Asada, and it’s conjectured to be true for all g,n.  But the statement that every finite-index subgroup of a mapping class group is a congruence subgroup on the nose is substantially stronger, and I imagine it’s true only for (1,1) and the closely related case (0,4), which was proved, in somewhat different language, in the paper “Every curve is a Hurwitz space,” by Diaz, Donagi, and Harbater.  Our argument is very much inspired by theirs — it was to emphasize this debt that we gave our paper more or less the same title.

Do all curves over finite fields have covers with a sqrt(q) eigenvalue?

On my recent visit to Illinois, my colleage Nathan Dunfield (now blogging!) explained to me the following interesting open question, whose answer is supposed to be “yes”:

Q1: Let f be a pseudo-Anosov mapping class on a Riemann surface Sigma of genus at least 2, and M_f the mapping cylinder obtained by gluing the two ends of Sigma x interval together by means of f.  Then M_f is a hyperbolic 3-manifold with first Betti number 1.  Is there a finite cover M of M_f with b_1(M) > 1?

You might think of this as (a special case of) a sort of “relative virtual positive Betti number conjecture.”  The usual vpBnC says that a 3-manifold has a finite cover with positive Betti number; this says that when your manifold starts life with Betti number 1, you can get “extra” first homology by passing to a cover.

Of course, when I see “3-manifold fibered over the circle” I whip out a time-worn analogy and think “algebraic curve over a finite field.”  So here’s the number theorist’s version of the above question:

Q2: Let X/F_q be an algebraic curve of genus at least 2 over a finite field.  Does X have a finite etale cover Y/F_{q^d} such that the action of Frobenius on H^1(Y,Z_ell) has an eigenvalue equal to q^{d/2}?

Koberda on dilatation and finite nilpotent covers

One reason dilatation was on my mind was thanks to a very interesting recent paper by Thomas Koberda, a Ph.D. student of Curt McMullen at Harvard.

Recall from the previous post that if f is a pseudo-Anosov mapping class on a surface Σ, there is an invariant λ of f called the dilatation, which measures the “complexity” of f; it is a real algebraic number greater than 1.  By the spectral radius of f we mean the largest absolute value of an eigenvalue of the linear automorphism of induced by f.  Then the spectral radius of f is a lower bound for λ(f), and in fact so is the spectral radius of f on any finite etale cover of Σ preserved by f.

This naturally leads to the following question, which appears as Question 1.2 in Koberda’s paper:

Is λ(f) the supremum of the spectral radii of f on Σ’, as Σ’ ranges over finite etale covers of Σ preserved by f?

It’s easiest to think about variation in spectral radius when Σ’ ranges over abelian covers.  In this case, it turns out that the spectral radii are very far from determining the dilatation.  When Σ is a punctured sphere, for instance, a remark in a paper of Band and Boyland implies that the supremum of the spectral radii over finite abelian covers is strictly smaller than λ(f), except for the rare cases where the dilatation is realized on the double cover branched at the punctures.   It gets worse:  there are pseudo-Anosov mapping classes which act trivially on the homology of every finite abelian cover of Σ, so that the supremum can be 1!  (For punctured spheres, this is equivalent to the statement that the Burau representation isn’t faithful.)  Koberda shows that this unpleasant state of affairs is remedied by passing to a slightly larger class of finite covers:

Theorem (Koberda) If f is a pseudo-Anosov mapping class, there is a finite nilpotent etale cover of Σ preserved by f on whose homology f acts nontrivially.

Furthermore, Koberda gets a very nice purely homological version of the Nielsen-Thurston classification of diffeomorphisms (his Theorem 1.4,) and dares to ask whether the dilatation might actually be the supremum of the spectral radius over nilpotent covers.  I have to admit I would find that pretty surprising!  But I don’t have a good reason for that feeling.

The entropy of Frobenius

Since Thurston, we know that among the diffeomorphisms of surfaces the most interesting ones are the pseudo-Anosov diffeomorphisms; these preserve two transverse folations on the surface, stretching one and contracting the other by the same factor.  The factor, usually denoted , is called the dilatation of the diffeomorphism and its logarithm is called the entropy. It turns out that , which is evidently a real number greater than 1, is in fact an algebraic integer, the largest eigenvalue of a matrix that in some sense keeps combinatorial track of the action of the diffeomorphism on the surface.  You might think of it as a kind of measure of the “complexity” of the diffeomorphism.  A recent preprint by my colleague Jean-Luc Thiffeault says much about how to compute these dilatations in practice, and especially how to hunt for diffeomorphisms whose dilatation is as small as possible.

A little more about random configurations

In my previous post about the configuration space of hard discs in a box, I neglected to say anything about the main point of Persi’s article!  It’s the following — even though  you don’t know anything about the topology of the space of configurations, you can still do an excellent job of drawing a configuration at random from the natural distribution using Monte Carlo techniques.  And if you’re a physicist trying to model the behavior of a gas or a fluid, you might be more interested in what a random configuration looks like than whether the space of configurations is connected — it might not be so relevant that you can get from something that looks like tight packing 1 to something that looks like tight packing 2, if the probability of doing so is vanishingly small.  Or to put it another way — if the space looks like a bunch of big blobs connected by extremely narrow paths, then from the point of view of physics it might as well be disconnected.

Still, as a topologist, you might ask:  if I can do a good random sample of points from a mystery manifold M, can I compute topological invariants of M with high confidence?  Can you guess whether M is connected?  More generally, can you guess the homology groups of M?

You might think of this as a massive geometric generalization of the age-old problem of “cluster analysis.”  Let’s say you have a bunch of people, and for each person you measure N variables — let’s say, height, weight, and shoe size.  So you have a bunch of points in R^N — in this case R^3.

Maybe you hope that these points are well-modeled as a sample from some multivariate normal distribution.  But under some circumstances, this is a really bad hope!  For instance, if your sample isn’t segregated by gender, you’re going to see two big clusters — one cluster of women where the mean height is around 5’6″, one cluster of men where the mean height is around 5’9″.  You’re not really sampling from a normal distribution — you might be sampling from a superimposition of two different normal distributions with different centers.  Or, alternately, you could think of yourself as sampling points from a manifold in R^3 consisting of just two points — “the ideal woman and the ideal man” — where your measurements are subject to some error that’s distributed normally.

My impression is that statisticians are pretty good at distinguishing between a normal distribution and a superimposition of some small finite set of normal distributions.  But I think it’s much harder to look at a giant cloud of points in R^100 and say “aha — this is actually a random sample from a normal distribution centered on the union of a surface of genus 2 sitting over here, and these ten disjoint circles sitting over there.”

If you were wondering about this while reading the Diaconis article in the Bulletin, you’d be in luck, because flipping forward a few pages you’d get to Gunnar Carlsson’s long survey article on precisely this genre of problem!  More on “topological statistics” once I’ve read Carlsson’s article, but let me point out now that if you’re a young mathematician interested in these matters you might consider going to the CBMS summer school this August, centered on a lecture series by Ghrist.