The recent breakthrough article of A. Eskin and M. Mirzakhani sheds some light about the geometric structure of -invariant probability measures on moduli spaces of Abelian differentials. In a nutshell, they showed the following analog of the celebrated Ratner’s measure classification theorem in the non-homogenous setting of moduli spaces of Abelian differentials: any ergodic -invariant probability measure on these moduli spaces is an affine measure fully supported on some affine suborbifold.
In their (long) proof of this result, A. Eskin and M. Mirzakhani use several arguments inspired by the low entropy method of M. Einsiedler, A. Katok and E. Lindenstrauss, the exponential drift argument of Y. Benoist and J.-F. Quint and, as a preparatory step for the exponential drift argument, they show the semisimplicity of the Kontsevich-Zorich cocycle.
In Eskin-Mirzakhani’s article, the proof of the semisimplicity property of the Kontsevich-Zorich cocycle is based on the work of G. Forni and the study of symplectic and isotropic -invariant subbundles of the Hodge bundle.
It is interesting to point out that, while symplectic -invariant subbundles of the Hodge bundle occur in several known examples (see, e.g., these articles here), the existence of some example of isotropic -invariant subbundle is not so clear.
Indeed, the question of the existence of non-trivial isotropic -invariant subbundles of the Hodge bundle was posed by Alex Eskin and Giovanni Forni (independently) and they were partly motivated by the fact that the non-existence of such subbundles would allow to “forget” about isotropic -invariant subbundles and thus, simplify (at least a little bit) some arguments in Eskin-Mirzakhani paper.
In this note here, Gabriela Schmithüsen and I answered this question of A. Eskin and G. Forni by exhibiting a square-tiled surface of genus with squares such that the Hodge bundle over the -orbit of has non-trivial isotropic -invariant subbundles.
Fortunately, the basic idea of this example is simple enough to fit into a (short) blog post, and, for this reason, we will spend the rest of this post explaining the general lines of the construction of (leaving a few details to the our note with Gabi).
For the sake of convenience of the reader, we reproduce below Section 2 of our note with Gabi where some key facts about translation surfaces are recalled. Of course, this section is far from being an appropriate introduction to this subject and the reader might want to consult A. Zorich’s survey, and/or these posts here for a gentle expositions on the moduli spaces of Abelian differentials. Also, the reader may find useful to consult the introduction of our article with J.-C. Yoccoz for further comments on the relationship between the Kontsevich-Zorich cocycle and the action on homology of affine diffeomorphisms of translation surfaces.
A translation surface is the data of a non-trivial Abelian differential on a Riemann surface . This nomenclature comes from the fact that the local primitives of outside the set of its zeroes provides an atlas on whose changes of coordinates are all translations of the plane . In the literature, these charts are called translation charts and an atlas formed by translation charts is called translation atlas or translation (surface) structure. For later use, we define the area of as .
The Teichmüller space of unit area Abelian differentials of genus is the set of unit area translation surfaces of genus modulo the natural action of the group of orientation-preserving homeomorphisms of isotopic to the identity. The moduli space of unit area Abelian differentials of genus is the set of unit area translation surfaces of genus modulo the natural action of the group of orientation-preserving homeomorphisms of . In particular, where is the mapping class group (of isotopy classes of orientation-preserving homeomorphisms of ).
The point of view of translation structures is useful because it makes clear that acts on the set of Abelian differentials : indeed, given , we define as the translation surface whose translation charts are given by post-composing the translation charts of with . This action of descends to and . The action of the diagonal subgroup of is the so-called Teichmüller (geodesic) flow.
Remark 1 By collecting together unit area Abelian differentials with orders of zeroes prescribed by a list of positive integers with , we obtain a subset of called stratum in the literature. From the definition of the -action on , it is not hard to check that the strata are -invariant.
The Hodge bundle over is the quotient of the trivial bundle by the natural action of the mapping-class group on both factors. In this language, the Kontsevich-Zorich cocycle is the quotient of the trivial cocycle
by the mapping-class group . In the sequel, we will call as KZ cocycle for short.
For our current purposes, let us restrict ourselves to the class of translation surfaces covering the square flat torus equipped with the Abelian differential induced by on . In the literature, these translation surfaces are called square-tiled surfaces or origamis because is tiled by the (open) squares given by the pre-image of the open unit square . In particular, by labeling these open squares , we see that a square-tiled surface determines a pair of permutations coding the successive appearances of squares along the horizontal and vertical directions, and vice-versa.
The stabilizer — also known as Veech group — of a square-tiled surface with respect to the action of is commensurable to , and its -orbit is a closed subset of isomorphic to the unit cotangent bundle of the hyperbolic surface .
The Veech group consists of the “derivatives” (linear parts) of all affine diffeomorphisms of , that is, the orientation-preserving homeomorphisms of fixing the set of zeroes of whose local expressions in the translation charts of are affine maps of plane. The group of affine diffeomorphisms of is denoted by and it is possible to show that is precisely the subgroup of elements of stabilizing in . The Veech group and the affine diffeomorphisms group are part of the following exact sequence
where, by definition, is the subgroup of automorphisms of , i.e., the subgroup of elements of whose linear part is trivial (i.e., identity).
In this language, the KZ cocycle on the Hodge bundle over the -orbit of is intimately related to the action on homology of . Indeed, since is the stabilizer of in , we have that the KZ cocycle is the quotient of the trivial cocycle
For later use, we observe that, given a square-tiled surface (where is a finite cover ramified precisely over ), the KZ cocycle, or equivalently , preserves the decomposition
where and .
Closing this preliminary section, we recall that, given a finite ramified covering of Riemann surfaces, the ramification data of a point is the list of ramification indices of all pre-images of counted with multiplicities.
2. Forni’s subbundle
Before trying to construct examples of isotropic -invariant subbundles of the Hodge bundle, we need to have a clue where they can possibly be found. Here, we are in good shape because, by Theorem A.6 and A.4 in Appendix A of Eskin-Mirzakhani’s paper, we have:
Theorem 1 Let be an isotropic -invariant subbundle of the Hodge bundle. Then, all Lyapunov exponents of the restriction of the Kontsevich-Zorich cocycle to vanish and, furthermore, the Kontsevich-Zorich cocycle acts isometrically on with respect to an adequate (Hodge) norm.
In other words, this theorem says that all isotropic -invariant subbundles of the Hodge bundle must live inside the maximal -invariant subbundle where the Kontsevich-Zorich cocycle acts isometrically. In the literature, the subbundle is called Forni’s subbundle and some of its properties were studied in these two articles here.
In particular, it is worth to look first at examples where Forni’s subbundle is not trivial before searching for isotropic -invariant subbundles. As it turns out, such examples are known: the Eierlegende Wollmichsau origami, the Ornithorynque origami and, more generally, certain square-tiled cyclic covers are some examples where the Forni subbundle is not trivial.
However, by a closer inspection of these examples, one can check that, even though Forni’s subbundle is not trivial in these examples, there are no isotropic -invariant subbundles simply because there are no proper -invariant subbundles (inside Forni’s subbundle of these examples) at all! For instance, the fact that this happens for the Eierlegende Wollmilchsau and Ornithorynque was shown in this article here.
In other words, despite the non-triviality of Forni’s subbundle in these examples, the fact that Forni’s subbundle is –irreducible makes that we don’t have a chance to find isotropic -invariant subbundles for the Eierlegende Wollmilchsau (say).
3. Isotropic -invariant subbundles
We just saw that the Eierlegende Wollmichsau and some other square-tiled cyclic covers are not the examples we are looking because the Kontsevich-Zorich cocycle acts irreducibly on their (non-trivial) Forni subbundles. On the other hand, the results in this article here indicate that these examples are not very far from being the desired ones. Indeed, it is shown in this article that the Kontsevich-Zorich cocycle acts on the Forni subbundle of the Eierlegende Wollmilchsau via a finite group (computed explicitly in the paper). In particular, by taking adequate finite covers of , the Kontsevich-Zorich cocycle will act trivially on the corresponding piece of Forni subbundle and thus one can eventually get rid the irreducibility issue.
This idea is the key tool in our note with Gabi and, as it turns out, it is sufficiently simple so that one can even construct explicit examples with isotropic -invariant subbundles of the Hodge bundle.
More precisely, we start with the Eierlegende Wollmilchsau . The Forni subbundle in this case is a -dimensional symplectic subbundle of the Hodge bundle over the orbit of and the elements of the affine group (or equivalently the Kontsevich-Zorich cocycle) with linear part (derivative) in the congruence subgroup
and fixing the zeroes of act trivially on . See, e.g., our article with J.-C. Yoccoz for a proof of these facts.
Therefore, if we can construct a finite cover such that all affine diffeomorphisms of “descend” to an affine diffeomorphisms of (in the sense that ) with linear part in and fixing the zeroes of , then is our desired example. Indeed, in this setting, it follows that the action of any affine diffeomorphism of on the -dimensional subbundle occurs through the action on of some affine diffeomorphism of with linear part in . On the other hand, as we mentioned above, the action of any such is trivial. So, we deduce that the action of any affine diffeomorphism of on the -dimensional subbundle is trivial. In particular, any line (-dimensional subspace) of defines an isotropic subbundle invariant under the whole affine group, i.e., any line of induces an isotropic -invariant subbundle.
Finally, it remains only to know what are some conditions so that a finite cover has the property that all affine diffeomorphisms “descend” to affine diffeomorphisms with derivative in fixing the zeroes of . Intuitively, is composed of several copies of glued in some way. Thus, what could go “wrong” when trying to “descend” an affine diffeomorphism is that the equation might not define a meaningful object because mixes up the several copies of in some strange way. In order to avoid this problem, the idea is to construct so that the ramification data over certain special points are different: this makes that these points are geometrically different from each other and so they can’t be mixed together by affine diffeomorphisms; in particular, these points prevent the copies of containing them to mix up in strange ways. More concretely, in Proposition 2 of our note with Gabi, we show that it suffices prescribe different ramification data for the covering over some -torsion. By doing so in a somewhat “minimalist” way (such as in Lemma 3.1 of our note with Gabi), one ends up with the following concrete example of square-tiled surface of genus 15 and 512 squares with isotropic -invariant subbundles: