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.