Shin-Strenner: Pseudo-Anosov mapping classes not arising from Penner’s construction

Balazs Strenner, a Ph.D. student of Richard Kent graduating this year, gave a beautiful talk yesterday in our geometry/topology seminar about his recent paper with Hyunshik Shin.  (He’s at the Institute next year but if you’re looking for a postdoc after that…!)

A long time ago, Robert Penner showed how to produce a whole semigroup M in the mapping class group with the property that all but a specified finite list of elements of M were pseudo-Anosov.  So that’s a good cheap way to generate lots of certified pseudo-Anosovs in the mapping class group.  But of course one asks:  do you get all pA’s as part of some Penner semigroup?  This can’t quite be true, because it turns out that the Penner elements can’t permute singularities of the invariant folation, while arbitrary pA’s can.  But there are only finitely many singularities, so some power of a given pA clearly fixes the singularities.

So does every pA have a power that arises from Penner’s construction?  This is what’s known as Penner’s conjecture.  Or was, because Balazs and Hyunshik have shown that it is falsitty false false false.

When I heard the statement I assumed this was going to be some kind of nonconstructive counting argument — but no, they actually give a way of proving explicitly that a given pA is not in a Penner semigroup.  Here’s how.  Penner’s semigroup M is generated by Dehn twists Q_1, … Q_m, which all happen to preserve a common traintrack, so that there’s actually a representation

$\rho: M \rightarrow GL_n(\mathbf{R})$

such that the dilatation of g is the Perron-Frobenius eigenvalue $\lambda$ of $\rho(g)$.

Now here’s the key observation; there is a quadratic form F on R^n such that F(Q_i x) >= F(x) for all x, with equality only when x is a fixed point of Q_i.  In particular, this shows that if g is an element of M not of the form Q_i^a, and x is an arbitrary vector, then the sequence

$x, g x, g^2 x, \ldots$

can’t have a subsequence converging to x, since

$F(x), F(gx), F(g^2 x), \ldots$

is monotone increasing and thus can’t have a subsequence converging to F(x).

This implies in particular:

g cannot have any eigenvalues on the unit circle.

But now we win!  Because $\rho(g)$ is an integral matrix, so all the Galois conjugates of $\lambda$ must be among its eigenvalues.  In other words, $\lambda$ is an algebraic number none of whose Galois conjugates lie on the unit circle.  But there are lots of pseudo-Anosovs whose dilatations $\lambda$ do have Galois conjugates on the unit circle.  In fact, experiments by Dunfield and Tiozzo seem to show that in a random walk on the braid group, the vast majority of pAs have this property!  And these pAs, which Shin and Strenner call coronal, cannot appear in any Penner semigroup.

Cool!

Anyway, I found the underlying real linear algebra question very appealing.  Two idle questions:

• If M is a submonoid of GL_n(R) we may say a continuous real-valued function F on R^n is M-monotone if F(mx) >= F(x) for all m in M, x in R^n.  The existence of a monotone function for the Penner monoid is the key to Strenner and Shin’s theorem.  But I have little feeling for how it works in general.  Given a finite set of matrices, what are explicit conditions that guarantee the existence of an M-monotone function?  Nonexistence?  (I have a feeling it is roughly equivalent to M containing no element with an eigenvalue on the unit circle, but I’m not sure, and anyway, this is not a checkable condition on the generating matrices…)
• What can we say about the eigenvalues of matrices appearing in the Penner subgroup?  Balazs says he’ll show in a later paper that they can actually get arbitrarily close to the unit circle, which is actually not what I had expected.  He asks:  are those eigenvalues actually dense in the complex plane?

My Thurston memory

My first tenure-track job interview was at Cornell.  During and after my job talk, most people were pretty quiet, but there was one guy who kept asking very penetrating and insightful questions.  And it was very confusing, because I knew all the number theorists at Cornell, and I had no idea who this guy was, or how it was that he obviously understood my talk better than anybody else in the room, possibly including me.

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• If blood found at a crime scene contains a series of genetic markers found in about 1 in a million people, and if you search a database of genetic material from 300,000 people and find just one match, person X, for the blood at the scene, what is the probability that person X is innocent of the crime?  If you said “1 in a million” you might be a prosecutor.  If you said “1 in a million, and I’m barring any expert testimony that says otherwise” you might be a judge.
• Good article in the New York Times about the challenge of teaching teachers to teach.  Deborah Ball of Michigan talks about what math teachers need:
• Working with Hyman Bass, a mathematician at the University of Michigan, Ball began to theorize that while teaching math obviously required subject knowledge, the knowledge seemed to be something distinct from what she had learned in math class. It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from not getting it to mastery. And they need to do this in 45 minutes or less. This was neither pure content knowledge nor what educators call pedagogical knowledge, a set of facts independent of subject matter, like Lemov’s techniques. It was a different animal altogether. Ball named it Mathematical Knowledge for Teaching, or M.K.T. She theorized that it included everything from the “common” math understood by most adults to math that only teachers need to know, like which visual tools to use to represent fractions (sticks? blocks? a picture of a pizza?) or a sense of the everyday errors students tend to make when they start learning about negative numbers. At the heart of M.K.T., she thought, was an ability to step outside of your own head. “Teaching depends on what other people think,” Ball told me, “not what you think.”

The idea that just knowing math was not enough to teach it seemed legitimate, but Ball wanted to test her theory. Working with Hill, the Harvard professor, and another colleague, she developed a multiple-choice test for teachers. The test included questions about common math, like whether zero is odd or even (it’s even), as well as questions evaluating the part of M.K.T. that is special to teachers. Hill then cross-referenced teachers’ results with their students’ test scores. The results were impressive: students whose teacher got an above-average M.K.T. score learned about three more weeks of material over the course of a year than those whose teacher had an average score, a boost equivalent to that of coming from a middle-class family rather than a working-class one. The finding is especially powerful given how few properties of teachers can be shown to directly affect student learning. Looking at data from New York City teachers in 2006 and 2007, a team of economists found many factors that did not predict whether their students learned successfully. One of two that were more promising: the teacher’s score on the M.K.T. test, which they took as part of a survey compiled for the study. (Another, slightly less powerful factor was the selectivity of the college a teacher attended as an undergraduate.)

Ball also administered a similar test to a group of mathematicians, 60 percent of whom bombed on the same few key questions.

• Thurston teams up with the House of Miyake for a Paris runway show loosely based on the fundamental 3-manifold geometries.  Thurston talks fashion:

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 $H_1(\Sigma,\mathbf{R})$ 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 $\lambda$, is called the dilatation of the diffeomorphism and its logarithm is called the entropy. It turns out that $\lambda$, 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.

Thurston on proof and progress in mathematics

I must have read Thurston’s excellent essay “On proof and progress in mathematics,” when it came out, but I don’t have any memory of it.  I re-encountered it the other day while playing with Springer’s eBook service, and flipping through the chapters of the recent collection 18 Unconventional Essays on the Nature of Mathematics.

Thurston makes a passionate case against theorem-proving as the measure of a mathematician’s contribution:

In mathematics,it often happens that a group of mathematicians advances with a certain collection of ideas. There are theorems in the path of these advances that will almost inevitably be proven by one person or another. Sometimes the group of mathematicians can even anticipate what these theorems are likely to be. It is much harder to predict who will actually prove the theorem,although there are usually a few “point people”who are more likely to score. However, they are in a position to prove those theorems because of the collective efforts of the team.The team has a further function,in absorbing and making use of the theorems once they are proven. Even if one person could prove all the theorems in the path single-handedly,they are wasted if nobody else learns them.

There is an interesting phenomenon concerning the “point”people.  It regularly happens that someone who was in the middle of a pack proves a theorem that receives wide recognition as being significant. Their status in the community—their pecking order—rises immediately and dramatically.When this happens,they usually become much more productive as a center of ideas and a source of theorems.Why? First,there is a large increase in self-esteem, and an accompanying increase in productivity. Second, when their status increases,people are more in the center of the network of ideas—others take them more seriously. Finally and perhaps most importantly, a mathematical breakthrough usually represents a new way of thinking,and effective ways of thinking can usually be applied in more than one situation.

This phenomenon convinces me that the entire mathematical community would become much more productive if we open our eyes to the real valuesin what we are doing. Jaffe and Quinn propose a system of recognized roles divided into “speculation”and “proving”. Such a division only perpetuates the myth that our progress is measured in units of standard theorems deduced. This is a bit like the fallacy of the person who makes a printout of the first 10,000 primes. What we are producing is human understanding. We have many different ways to understand and many different processes that contribute to our understanding. We will be more satisfied, more productive and happier if we recognize and focus on this.

Thurston concludes with some very interesting and frank reminiscences, including some regrets, about the way certain parts of topology bent around his gravitational field in the 70s and 80s.

By the way, some libraries have stopped buying new physical books from Springer in favor of access to the e-books.  If you’re at an institution that’s gone this route, tell me about it in comments!

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