Idle question: are Kakeya sets winning?

Jayadev Athreya was here last week and reminded me about this notion of “winning sets,” which I learned about from Howie Masur — originally, one of the many contributions of Wolfgang Schmidt.

Here’s a paper by Curt McMullen introducing a somewhat stronger notion, “absolute winning.”

Anyway:  a winning set (or an absolute winning set) in R^n is “big” in some sense.  In particular, it has to have full Hausdorff dimension, but it doesn’t have to have positive measure.

Kakeya sets (subsets of R^n containing a unit line segment in every direction) can have measure zero, by the Besicovitch construction, and are conjectured (when n=2, known) to have Hausdorff dimension n.  So should we expect these sets to be winning?  Are Besicovitch sets winning?

I have no reason to need to know.  I just think these refined classifications of sets which are measure 0 yet still “large” are very interesting.  And for all I know, maybe there are sets where the easiest way to prove they have full Hausdorff dimension is to prove they’re winning!

 

 

Happy birthday, Dick Gross

Just returned from Dick Gross’s 60th birthday conference, which functioned as a sort of gathering of the tribe for every number theorist who’s ever passed through Harvard, and a few more besides.  A few highlights (not to slight any other of the interesting talks):

• Curt McMullen talked about Salem numbers and the topological entropy of automorphisms of algebraic surfaces (essentially the material discussed in his 2007 Arbeitstagung writeup.)  In particular, he discussed the fact that the logarithm of Lehmer’s number — conjecturally the “simplest” algebraic integer — is in fact the smallest possible positive entropy for an automorphism of a compact complex surface.  Here’s a question that occurred to me after his talk.  If f is a Cremona transformation, i.e. a birational automorphism of P^2, then there’s a way to define the “algebraic entropy” of f, as follows:   the nth iterate of f is given by two rational functions (R_n(x,y),S_n(x,y)), you let d_n be the maximal degree of R_n and S_n, and you define the entropy to be the limit of (1/n) log d_n.  Question:  do we know how to classify the Cremona transformations with zero entropy?  The elements of PGL_3 are in here, as are the finite-order Cremona transformations (which are themselves no joke to classify, see e.g. work of Dolgachev.)  Are there others?
• Serre spoke about characters of groups taking few values, or taking the value 0 quite a lot — this comes up when you want, e.g., to be sure that two varieties have the same number of points over F_p for all but finitely many p, supposing that they have the same number of points for 99.99% of all p.  The talk included the amusing fact that a character taking only the values -1,0,1 is either constant or a quadratic character.  (But, Serre said, there are lots of characters taking only the values 0,3 — what are they, I wonder?)
• Bhargava talked about his new results with Arul Shankar on average sizes of 2-Selmer groups.  It’s quite nice — at this point, the machine, once restricted to counting orbits of groups acting on the integral points of prehomogenous vector spaces, is far more general:  it seems that the group of people around Manjul is getting a pretty good grasp on the general problem of counting orbits of bounded height of the action of G(Z) on V(Z), where G is a group over Z (even a non-reductive group!) and V is some affine space on which G acts.  With the general counting machine in place, the question is:  how to interpret these orbits?  Manjul showed a list of 70 representations to which the current version of the orbit-counting machine applies; each one, hopefully, corresponds to some interesting arithmetic enumeration problem.  It must be nice to know what your next 70 Ph.D. students are going to do…

Dick has a lot of friends — the open mike at the banquet lasted an hour and a half!  My own banquet story was from my college years at Harvard, where Dick was my first-year advisor.  One time I asked him, in innocence, whether he and Mazur had been in graduate school together.  He fixed me with a very stern look.

“Jordan,” he said, “as you can see, I am a very old man.  But I am not as old as Barry Mazur.“

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?

“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.

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.