Tag Archives: topology

Pseudo-Anosovs with low dilatation: Farb-Leininger-Margalit, and a puzzle

I spent a very enjoyable weekend learning about the dilatation of pseudo-Anosov mapping classes at a workshop organized by Jean-Luc Thiffeault and myself.  The fact that a number theorist and a fluid dynamicist would organize a workshop about an area in low-dimensional topology should indicate, I hope, that the topic is of broad interest!

There are lots of ways to define dilatation, which is a kind of measure of “complexity” of a mapping class.  Here’s the simplest.  Let f be a diffeomorphism from a genus-g Riemann surface S to itself, which is pseudo-Anosov.  Loosely speaking, this means the dynamics of  f are “irreducible” on the surface; for instance, no power of f acts trivially on any subsurface.  (“Most” diffeomorphisms, in any reasonable sense, are pA.)  For any two curves a,b on S, let i(a,b) be the minimal number of intersection points between a and any curve isotopic to b.  (Note that this is typically a lot bigger than the intersection of the homology classes of a and b; the latter measures the number of intersection points counted with sign, which doesn’t change when you isotop the curves.)  It turns out that the quantity

(1/k) log i(f^k(a),b)

approaches a limit as k grows, which strictly exceeds 1;  this limit is called λ(f), the dilatation of f.  It’s invariant under deformation of f; in other words, it depends only on the class of f in the mapping class group of S.  That this limit exists is exciting enough; better still, and indicative of lots of structure I’m passing over in silence, is that λ(f) is an algebraic integer!

(I just remembered that I gave a different description of the dilatation on the blog last year, in connection with an analogy to Galois groups.)

The subject of the conference was pseudo-Anosovs with low dilatation.  The dilatations of pAs in a given genus g are known to form a discrete subset of the interval (1,infinity); thus it makes sense to ask what the smallest dilatation in genus g is.  Lots of progress on this problem has been made in recent years; Joan Birman, Eriko Hironaka, Chia-Yen Tsai, and Ji-Young Ham all talked about results in this vein.  But for general g the answer remains unknown.

A theorem of Penner guarantees that, for any pseudo-Anosov f on a surface of genus g, we have λ(f) > c^(1/g) for some constant c.  So one might call a family f_1, f_2,…. of pAs of varying genera g_1, g_2, …  “low-dilatation” if the quantity λ(f_i)^g_i is bounded.  (One such family, constructed by Hironaka and Eiko Kin, appeared in many of the lectures.)

In this connection, let me advertise the extremely satisfying theorem of Benson Farb, Chris Leininger, and Dan Margalit.  Here’s a natural construction you can do with a pA diffeomorphism f on a surface S.  The diffeo has an invariant foliation which is stretched by f; this foliation has a finite set of singularities.  Remove this to get a punctured surface S^0.  Since the singularities are preserved setwise by f, we have that f restricts to a diffeomorphism of S^0, which is again pA, and which we again call f.  Now we can make a 3-manifold M^0_f by starting with S^0 x [0,1] and gluing S^0  x 0 to S^0 x 1 via f.  By a theorem of Thurston, this will be a hyperbolic 3-manifold; because of the punctures, it’s not compact, but its ends are shaped like tori.

Now here’s the theorem:  suppose f_1, f_2, … is a sequence of pAs which has low dilatation in the sense above.  Then the sequence of 3-manifolds M^0_{f_i} actually consists of only finitely many distinct hyperbolic 3-manifolds.

This has all kinds of marvelous consequences; it tells us that the low-dilatation pAs are in some sense “all alike.”  (For more on the “in some sense” I would need to talk about the Thurston norm and fibered faces and etc. — maybe another post.)  For instance, it immediately implies that in a low-dilatation family of pseudos, the dimension of the subspace of H_1(S_i) fixed by f_i is bounded.

If you’ve read this far, maybe you’d like to see the promised puzzle.  Here it is.  Suppose f_1, f_2, … is a family of pseudos which lie in the Torelli group — that is, f_i acts trivially on H_1(S_i).  Then by the above remark this family can’t be low-dilatation.  Indeed, an earlier theorem of Farb, Leininger, and Margalit tells us that for Torellis we have an absolute lower bound

λ(f) > c

where the constant doesn’t depend on g.

Puzzle: Suppose f_1, f_2, … is a sequence of pseudos in Torelli which has bounded dilatation; this is as strong a notion of “low-dilatation family” as one can ask for.  Is there a “structure theorem” for f_1, f_2, …. as in the general case?  I.E., is there any “closed-form description” of this family?

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McMullen on dilatation in finite covers

Last year I blogged about a nice paper of Thomas Koberda, which shows that every pseudo-Anosov diffeomorphism of a Riemann surface X acts nontrivially on the homology of some characteristic cover of X with nilpotent Galois group.  (This statement is false with “nilpotent” replaced by “abelian.”)  The paper contains a question which Koberda ascribes to McMullen:

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

That question has now been answered by McMullen himself, in the negative, in a preprint released last month.  In fact, he shows that either λ(f) is detected on the homology of a double cover of Σ, or it is not detected by any finite cover at all!

The supremum of the spectral radius of f on the Σ’ is then an invariant of f, which most of the time is strictly bigger than the spectral radius of f on Σ and strictly smaller than λ(f).  Is this invariant interesting?  Are there any circumstances under which it can be computed?

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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.

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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.

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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.

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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 \mathbf{Q}(\sqrt{-d}), as d ranges over squarefree integers in [0..X],  you get a list of about \zeta(2)^{-1} X 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.

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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

\frac{6}{\pi^2}X + O(X^{1/2}) = \zeta(2)^{-1} X + O(X^{1/2}).

(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]?

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

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.

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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}?

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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.

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