Today I gave a talk at the Second Palis-Balzan conference on Dynamical Systems (held at Institut Henri Poincaré, Paris). In fact, I was not supposed to talk in this conference: as I’m serving as a local organizer (together with Sylvain Crovisier), I was planning to give to others the opportunity to speak. However, Jacob Palis insisted that everyone must talk (the local organizers included), and, since he is the main organizer of this conference, I could not refuse his invitation.
Anyhow, my talk concerned a joint work with Alex Wright (currently a PhD student at U. of Chicago under the supervision of Alex Eskin) about the finiteness of algebraically primitive closed -orbits on moduli spaces (of Abelian differentials).
Below the fold, I will transcript my lecture notes for this talk.
1. Preliminaries and statement of main result
The data of a Riemann surface of genus and a (non-trivial) Abelian differential (i.e., holomorphic -form) on determines a structure of translation surface on , that is, by local integration of outside the (finite) set of its zeroes, we obtain an atlas on such that all changes of coordinates are given by translation on . In particular, after cutting along the edges of a triangulation such that the set of zeroes of is exactly the set of vertices of , we can think of as a finite collection of triangles on the plane whose sides are glued by translations.
The set of translation structures on a surface of genus is naturally stratified by the subsets obtained by fixing the multiplicities of the zeroes of . Note that, for each , the number of strata is precisely the number of partitions of : indeed, by Riemann-Hurwitz theorem, the sum of the orders of the zeroes of a non-trivial Abelian differential is always .
In this language, given a partition of , the corresponding stratum of the moduli space of unit area Abelian differentials / translation structures is the set of pairs of Riemann surfaces and Abelian differentials with unit area (that is, the total area of the triangles mentioned above is one) and zeroes of orders modulo biholomorphisms. In terms of translation structures, is the set of translation structures with conical singularities at vertices of the triangles with a total angle of and the total area of the triangles is one modulo cutting and pasting by translations.
From the description of in terms of translation structures (and triangles glued by translations), it is clear that has a natural -action.
This action of on is useful in the investigation of deviations of ergodic averages of interval exchange transformations, translation flows and billiards in rational polygons: for instance, it was recently applied by Delecroix-Hubert-Lelièvre to confirm a conjecture of the physicists Hardy-Weber on the rates of diffusions of typical trajectories in typical realizations of Ehrenfest wind-tree model of Lorentz gases.
In particular, a lot of attention was given to the question of understanding the closure of -orbits and/or ergodic -invariant probability measures. Here, it was widely believed among experts that a Ratner-like result should be true for the -action on because, philosophically speaking, is “very close to a homogenous space” (despite its non-homogeneity). In this direction, Eskin-Mirzakhani showed the following profound result:
Theorem 1 (Eskin-Mirzakhani) Any ergodic -invariant probability measure on any stratum is affine, i.e., it is a density (Lebesgue) measure on certain affine subspaces (in period coordinates) associated to its support.
Of course, this theorem gives hope for a general classification result of all ergodic -invariant probability measures on all strata .
On the other hand, a complete classification of such measures is only available in genus (i.e., for the strata and ) thanks to the works of K. Calta and C. McMullen in 2004. However, as it was pointed out by C. McMullen himself, it is an open problem to generalize his methods to genus .
Nevertheless, some partial progress was made recently for certain types of ergodic -invariant probability measures such as the ones supported in a (single) closed -orbits — also known as Teichmüller curves.
More concretely, M. Möller and his collaborators considered algebraically primitive closed -orbits of genus , that is, Teichmüller curves such that the field generated by the traces of the stabilizers in of any of its points has degree over . Here, they showed that there are only finitely many algebraically primitive Teichmüller curves in the following connected components of stratum:
- the hyperelliptic component consisting of translation surfaces in with an hyperelliptic involution exchanging the two zeroes (cf. this paper of M. Möller in 2008);
- (cf. this paper of M. Bainbridge and M. Möller in 2011);
- the hyperelliptic component of (this is part of a work in progress by Bainbridge, Habegger and Möller).
Remark 1 In genus 2, it follows from the works of Calta and McMullen that there are infinitely many algebraically primitive Teichmüller curves in and only one algebraically primitive Teichmüller curve in .
This scenario motivates the following question: Given a connected component of a stratum in genus , are there only finitely many algebraically primitive Teichmüller curves inside ?
Of course, a positive answer to it hints that, a priori, it could be a good idea to start this classification by listing -invariant probability measures supported on algebraically primitive Teichmüller curves (as there would be only finitely many of them per stratum).
In this post, we want to discuss the following modest contribution to this question:
Before giving the proof of Theorem 2 above, let us make three quick comments.
First, the arguments we use to prove our finiteness result are dynamical in nature (based on the analysis of the so-called Kontsevich-Zorich cocycle), while the methods of Möller and his collaborators were more algebro-geometrical (based on the behavior of algebraically primitive Teichmüller curves near the boundary of Deligne-Mumford’s compactification of moduli spaces).
Interestingly enough, these two methods seem to be “complementary” in the sense that our dynamical method is particularly adapted to the study of algebraically primitive Teichmüller curves in minimal strata , while the algebro-geometrical method of Möller and his collaborators is better adapted to non-minimal strata (and this explains why the result of Bainbridge-Habegger-Möller mentioned above was obtained only very recently compared to the other results).
Secondly, A. Wright and I also have partial results for other (non-minimal) strata, e.g., we can show that algebraically primitive Teichmüller curves can’t form a dense subset of a given connected component of a stratum in genus , but we will restrict our discussion to Theorem 2 because its proof already contain all essential ideas of our forthcoming paper.
Finally, it is likely that our methods might give finiteness of algebraically primitive Teichmüller curves in more strata (and also for some non-algebraically Teichmüller curves) as the knowledge of -invariant closed subsets of strata is improved.
2. Proof of Theorem 2
The basic idea is very simple. Let us fix prime and let us suppose by contradiction that there are infinitely many algebraically primitive Teichmüller curves in some connected component of the minimal stratum .
A recent equidistribution result of A. Eskin, M. Mirzakhani and A. Mohammadi implies that will equidistribute inside the support of an ergodic -invariant probability measure. Actually, for our current purposes, we do not the full strength of this equidistribution result: it is sufficient to know that eventually becomes dense inside in the sense that there exists an integer such that for all and the subset is dense in .
Now, we claim that is equal to the whole connected component of , that is, the sequence of algebraically primitive Teichmüller curves becomes dense in . In fact, as we mentioned earlier, by the results of A. Eskin and M. Mirzakhani (cf. Theorem 1 above), is represented by affine subspaces in period coordinates. As it was shown by A. Wright in this paper here, these affine subspaces representing are defined over some field of degree over , and, moreover, since , one also has the inequality
On the other hand, it is possible to check that the affine subspaces representing a Teichmüller curve (closed -orbit) are defined over a field whose degree over coincide with the degree of the field generated by the traces of the stabilizers of in .
In particular, it follows (from the definition of algebraic primitivity) that the affine subspaces representing the algebraically primitive Teichmüller curves are defined over some fields whose degrees over are . Because for all , we deduce that the degree of divides .
Since is prime (by hypothesis), we have that or . We affirm that : indeed, if this is not true, then ; by inserting this into the inequality (1), we would have ; since consists of entire -orbits (and has dimension ), we get that ; however, this is impossible because contains infinitely many closed -orbits, namely, for .
Once we know that , i.e., , we affirm that , that is, the inequality (1) is an equality. In fact, let us consider the subset consisting of Abelian differentials belonging to after normalization of its total area and let us denote by at some point . By definition, contains the -dimensional subspace . Since is defined over and is defined over a field of degree over , we see that must contain all Galois conjugates . Using this information (and some extra arguments), one can show that the linear subspace has dimension , and, a fortiori, . By combining this with inequality (1), we deduce that .
At this point, we can complete the proof of our claim that by simply counting dimensions: using the period coordinates on , we can identify the tangent space of any point with , a vector space of real dimension ; hence has dimension and, a fortiori, .
In summary, we just proved the following statement (first proved in this paper of A. Wright):
Proposition 3 (A. Wright) If there are infinitely many algebraically primitive Teichmüller curves in some connected component of a minimal stratum with prime, then they must form a dense subset of , that is,
Our plan is to contradict this denseness property of along the following lines.
Let us consider the so-called Kontsevich-Zorich (KZ) cocycle over , that is, we use the -action to move around , and every time that we cut and paste a translation structure obtained from by applying to get a shape “close” to , we keep track of how the homology cycles on were changed. In practice, by selecting an appropriate symplectic basis of (with respect to the usual intersection form on homology), this amounts to “attach” a symplectic matrix to each and .
Generally speaking, the Kontsevich-Zorich cocycle preserves a family of 2-dimensional symplectic plane (called tautological planes). Since this cocycle is symplectic, the symplectic orthogonals of the planes are also preserved, so that the Kontsevich-Zorich cocycle along any -orbit decomposes into a -block corresponding to its restriction to and a -block corresponding to its restriction to . For later use, we will call restricted Kontsevich-Zorich cocycle the restriction of the Kontsevich-Zorich cocycle to . By definition, the restricted KZ cocycle is a -cocycle over -orbits.
As it turns out, the restricted KZ cocycle tends to behave poorly or richly depending on the -orbit we look at.
For example, the restricted KZ cocycle over an algebraically primitive Teichmüller curve in a stratum of genus decomposes into blocks consisting of its restrictions to the non-trivial Galois conjugates of the tautological planes (because these tautological planes are defined over a field of degree over when the Teichmüller curve is algebraically primitive). In particular, the matrices of the restricted KZ cocycle over belong to a “poor” subgroup isomorphic to of the symplectic group .
On the other hand, it is expected that the restricted KZ cocycle over “most” -orbits (in any connected component of any [not necessarily minimal] stratum) has a “rich” behavior: for instance, the simplicity result of Avila-Viana (as well as some numerical experiments) gives hope that the monoid of matrices of the restricted KZ cocycle over a typical -orbit has full Zariski closure .
So, if some family of algebraically primitive Teichmüller curves becomes dense in some connected component of some stratum in genus , we would have to two opposite properties fighting one against the other: on one hand, the restricted KZ cocycle over the dense subset has a very rigid structure, namely, they decompose into blocks of sizes ; on the other hand, the typical -orbit has a “rich” restricted KZ cocycle, say Zariski dense in , and thus it can not be have a very rigid structure.
Therefore, we will contradict the denseness property of in if we can show that:
- (i) some non-trivial piece of information coming from the very rigid structure of the restricted KZ cocycle over “pass to the limit” in the sense that the restricted KZ cocycle over all -orbits in would have some (partial) rigidity property;
- (ii) show that the restricted KZ cocycle over some -orbit in is rich enough so that no rigidity property can occur (not even partially).
Concerning (i), it is tempting to use the decomposition into blocks of sizes as the very rigid property that passes through the limit. However, it is not obvious that the limit of a decomposition into blocks is still a set of blocks: indeed, even though the Grassmanian of planes is compact (and hence we can extract limits), some of these blocks might collapse.
Nevertheless, it possible to check that the restricted KZ cocycle over preserves a very special (projective) subset of Grassmanian of (symplectic) planes that A. Wright and I call Hodge-Teichmüller planes or HT planes for short. More precisely, we say that a plane is a HT plane if the images of its complexification under all matrices of the KZ cocycle intersect non-trivially the and subspaces and of the Hodge filtration of (into holomorphic and anti-holomorphic forms).
Note that HT planes are really very special: since the usual action of on is not holomorphic, the complex structure of changes when we apply the elements , so that the subspaces and move around and hence it is not easy for the images of a plane to keep intersecting the “moving targets” and .
As it turns out, the (rigid) property of preservation of a non-trivial set of HT planes passes through the limit: indeed, this follows from the facts that the HT planes were defined in terms of the intersections of with and and it is known that and vary continuously with .
Also, in the case of the algebraically primitive Teichmüller curves, work of Martin Moller implies that the Galois conjugates of the tautological planes are HT planes and Hodge orthogonal, that is, they are mutually orthogonal with respect to the Hodge inner product in cohomology. Since the Hodge inner product is also known to depend continuously on , we have that the limits of the HT planes associated to lead to Hodge-orthogonal HT planes.
In other words, we showed the following result (that formalizes item (i) above):
Then, the restricted Kontsevich-Zorich cocycle over any -orbit in preserves the non-trivial set of at least HT planes that are mutually Hodge-orthogonal.
The proof of Proposition 5 is based on the following observations. It is not hard to see that the set of at least HT planes preserved by the restricted KZ cocycle is also preserved by its Zariski closure (in ). Moreover, this set of HT planes is a proper compact subset of the Grassmanian of symplectic planes. Therefore, since has no proper compact invariant subset in the Grassmanian of symplectic planes, we have that any -orbit such that the Zariski closure of the restricted KZ cocycle is “at least as rich as ” is likely to do not have HT planes. In other words, the proof of Proposition 5 follows if we construct on each connected component of any stratum in genus some -orbit whose restricted KZ cocycle has a “mild rich Zariski closure”.
In our first attempts, A. Wright and I tried to construct such -orbits by taking adequate ramified covers of certain closed -orbits in genus . In fact, this idea works to produce nice examples in some connected components of some minimal strata (where runs along some arithmetic progression), but if one wishes to construct examples in all connected components of all strata in genus , then one runs into some nasty combinatorial problems that we do not know how to solve.
For this reason, we decided to change our initial strategy outlined in the previous paragraph into the following one. Firstly, we considered the following two square-tiled surfaces (i.e., translation surfaces obtained from finite coverings of the unit torus that are ramified only at the origin ):
These square-tiled surfaces generate closed -orbits in the two connected components and of . Moreover, by computing some matrices of the restricted KZ cocycle over these square-tiled surfaces (using appropriate Dehn twists along the directions indicated above), we can show that the Zariski closure of the restricted KZ cocycle is in both cases. In particular, from our discussion so far, this proves Proposition 5 for the cases of the two connected components of .
Now, the idea to prove Proposition 5 in the general case of a connected component is to find some translation surface (in ) whose restricted KZ cocycle behaves a “little bit” like the corresponding cocycle for or . However, we are not quite able to do this and we proceed in a slightly different way.
Formally, we show Proposition 5 by contradiction and induction using the fact that this proposition was already shown to be true for the connected components of . More concretely, assume that all translation surfaces in some connected component of some stratum have Hodge-orthogonal HT planes. It was shown by Kontsevich-Zorich (in their classification of connected components of strata) that, by carefully collapsing zeroes, and pinching off the handles of like in this picture here
and by deleting the torus component, we reach one of the connected components of . Furthermore, by a close inspection of Kontsevich-Zorich procedure, we can see that, if the initial translation surface has Hodge-orthogonal HT planes and if the pinching procedure is really careful, then each time we pinch off a handle, we kill off at most one of the HT planes. In particular, among the Hodge-orthogonal HT planes of the initial translation surfaces, we know that at least two of them survive by the end of the procedure, and, hence, we obtain some translation surface in some of the connected components of having two Hodge-orthogonal HT planes, a contradiction.