Hello! Today we’ll start the study of a natural -action containing the so-called Teichmüller flow on the moduli space of Abelian differentials. Our ultimate goal today will be the reduction of the study of derivative of Teichmüller flow to a closely related (dynamical) cocycle called Kontsevich-Zorich (KZ for short) cocycle.
1. Dynamics on the moduli space of Abelian differentials
Let be a compact Riemann surface of genus equipped with a non-trivial Abelian differential (that is, a holomorphic 1-form) . In the sequel, we denote by the (finite) set of zeroes of .
1.1. Flat structures
Given any point , let’s select a small path-connected neighborhood of such that . In this setting, the “period” map , obtained by integration along a path inside connecting and is well-defined: indeed, this follows from the fact that any Abelian differential is closed (because they are holomorphic) and the neighborhood doesn’t contain any zeroes of (so that the integral doesn’t depend on the choice of the path inside connecting and ). Furthermore, since (so that ), we see that, by reducing if necessary, this “period” map is a biholomorphism.
In other words, the collection of all such “period” maps provides an atlas of compatible with the Riemann surface structure. Also, by definition, the push-forward of the Abelian differential by any such is precisely the canonical Abelian differential on the complex plane . Moreover, the “local” equality implies that all changes of coordinates have the form where is a constant (since it doesn’t depend on ). Furthermore, since has finite order at its zeroes, it is easy to deduce from Riemann’s theorem on removal singularities that this atlas of can be extended to in such a way that the push-forward of by a local chart around a zero of order is the holomorphic form .
In the literature, a compact surface of genus equipped with an atlas whose changes of coordinates are given by translations of the complex plane outside a finite set of points is called a translation surface structure on . In this language, our previous discussion simply says that any non-trivial Abelian differential on a compact Riemann surface gives rise to a translation surface structure on such that is the pull-back of the canonical holomorphic form on . On the other hand, it is clear that every translation surface structure determines a Riemann surface (since translations are a very particular case of local biholomorphism) and an Abelian differential (by pulling back on : this pull-back is well-defined because is translation-invariant).
In resume, we just saw the proof of the following proposition:
Proposition 1 The set of Abelian differentials on Riemann surfaces of genus is canonically identified to the set of translation structures on a compact (topological) surface of genus .
Example 1 During our Riemann surfaces courses, a complex torus is quite often presented through a translation surface structure: indeed, by giving a lattice , we are saying that the complex torus equipped with the (non-vanishing) Abelian differential is canonically identified with the translation surface structure represented in the picture below (it truly represents a translation structure since we’re gluing opposite parallel sides of the parallelogram determined by and through the translations and ).
Example 2 Consider the polygon of the sole Figure of last post. In this picture, we are gluing parallel opposite sides , , of , so that this is again a valid presentation of a translation surface structure. Let’s denote by the corresponding Riemann surface and Abelian differential. Observe that, by following the sides identifications as indicated in this Figure, we see that the vertices of are all identified to a single point . Moreover, we see that is a special point when compared with any point of because, by turning around , we note that the “total angle” around it is while the total angle around any point of is , that is, a neighborhood of inside resembles “3 copies” of the flat complex plane while a neighborhood of any other point resembles only 1 copy of the flat complex plane. In other words, a natural local coordinate around is , so that , i.e., the Abelian differential has a unique zero of order at . From this, we can infer that is a compact Riemann surface of genus : indeed, by Riemann-Hurwitz theorem, the sum of orders of zeroes of an Abelian differential equals (where is the genus); in the present case, this means , i.e., ; alternatively, one can use Poincaré-Hopf index theorem to the vector field given by the vertical direction on (this is well-defined because these points correspond to regular points of the polygon ) and vanishing at (where a choice of “vertical direction” doesn’t make sense since we have multiple copies of the plane glued together).
Example 3 (Square-tiled surfaces) Consider a finite collection of unit squares on the plane such that the leftmost side of each square is glued (by translation) to the rightmost side of another (maybe the same) square, and the bottom side of each square is glued (by translation) to the top side of another (maybe the same) square. Here, we assume that, after performing the identifications, the resulting surface is connected. Again, since our identifications are given by translations, this procedure gives at the end of the day a translation surface structure, that is, a Riemann surface equipped with an Abelian differential (obtained by pulling back on each square). For obvious reasons, these surfaces are called square-tiled surfaces and/or origamis in the literature. For sake of concreteness, we drew below a L-shaped square-tiled surface derived from 3 unit squares identified as in the picture. By following the same arguments used in the previous example, the reader can easily verify that this L-shaped square-tiled surface with 3 squares corresponds to an Abelian differential with a single zero of order 2 in a Riemann surface of genus 2.
Remark 1 Of course, the description of the square-tiled surfaces maybe generalized (by replacing squares with arbitrary polygons, but keeping sides identifications through translations) in order to produce further examples of translation surfaces/Abelian differentials. The curious reader maybe asking whether all translation surface structures can be recovered by this more general procedure of gluing a finite collection of polygons. In fact, it is possible to prove that any translation surface admits a triangulation such that the zeros of the Abelian differential appear only in the vertices of the triangles (so that the sides of the triangles are saddle connections in the sense that they connect zeroes of the Abelian differential), so that the translation surface can be recovered from this finite collection of triangles. However, if we are “ambitious” and try to represent translation surfaces by side identifications of a single polygon (like in Example~2) instead of using a finite collection of polygons, then we’ll fail: indeed, there are examples where the saddles connections are badly placed so that one polygon never suffices. However, it is possible to prove (with the aid of Veech’s zippered rectangle construction) that all translation surfaces outside a set formed by a countable union of codimension 2 real-analytic suborbifolds can be represented by a sole polygon whose sides are conveniently identified. See Yoccoz’s survey for further details.
Despite its intrinsic beauty, a great advantage of talking about translation structures instead of Abelian differentials is the fact that several additional structures come for free due to the translation invariance of the corresponding structures on the complex plane :
- flat metric: since the usual (flat) Euclidean metric on the complex plane is translation-invariant, its pullback by the local charts provided by the translation structure gives a well-defined flat metric on ;
- area form: again, since the usual Euclidean area form on the complex plane is also translation-invariant, we get a well-defined area form on ;
- canonical choice of a vertical vector-field: as we implicitly mentioned by the end of Example~2, the vertical vertical vector field on can be pulled back to in a coherent way to define a canonical choice of north direction;
- pair of transverse measured foliations: the pullback to of the horizontal and vertical foliations of the complex plane are well-defined and produce a pair of transverse foliations and which are measured in the sense on Thurston: the leaves of these foliations come with a canonical notion of length measures and transversely to them.
Remark 2 It is important to observe that the flat metric introduced above is a singular metric: indeed, although it is a smooth Riemannian metric on , it degenerates when we approach a zero of the Abelian differential. Of course, we know from Gauss-Bonnet theorem that no compact surface of genus admit a completely flat metric, so that, in some sense, if we wish to have a flat metric in a large portion of the surface, we’re obliged to “concentrate” the curvature at tiny places. From this point of view, the fact that the flat metric obtained from translation structures are degenerate at a finite number of points reflects the fact that the “sole” way to produce a “almost completely” flat model of our genus surface is by concentrating all curvature at a finite number of points.
1.2. -action on
Another interesting consequence of the canonical identification between Abelian differentials and translation structures is the fact that it makes transparent the existence of a natural action of on the set of Abelian differentials . Indeed, given an Abelian differential , let’s denote by the maximal atlas on giving the translation structure corresponding to (so that for every index ). Here, the local charts map some open set of to .
Since any matrix acts on , we can post-compose the local charts with , so that we obtain a new atlas on . Observe that the changes of coordinates of this new atlas are also solely by translations, as a quick computation reveals:
In other words, is a new translation structure. The corresponding Abelian differential is, by definition, . Observe that the complex structure of the plane is not preserved by the action of a (typical) matrix. Therefore, the Riemann surface structure with respect to which the 1-form is holomorphic is usually distinct (not biholomorphic) to the Riemann surface structure related to . Here, a notable exception is the group of rotations who may change the Abelian differential without touching the Riemann surface structure (because any rotation preserves the complex structure of the plane).
By definition, it is utterly trivial to see this -action on Abelian differentials given by sides identifications of collections of polygons (as in the previous examples): in fact, given and an Abelian differential related to a finite collection of polygons with parallel sides glued by translations, the Abelian differential corresponds, by definition, to the finite collection of polygons obtained by letting act on the polygons forming as a planar subset and keeping the sides identifications of parallel sides by translations. Notice that the linear action of on the plane evidently respects the notion of parallelism, so that this procedure is well-defined. For sake of concreteness, we drew below some illustrative examples of the actions of the matrices , and on a L-shaped square-tiled surface.
Concerning the structures on introduced in the previous chapter, we notice that this -action preserves each of its strata (), ) and the natural (Lebesgue) measures on them since the underlying affine structures of strata are respected. Of course, this observation opens up the possibility of studying this action via ergodic-theoretical methods applied to . However, it turns out that the strata are a somewhat big: for instance, they are ruled in the sense that the complex lines foliate them. As a result, it is possible to prove that each has infinite mass, so that the use of standard ergodic-theoretical methods is not possible, and although there are some ergodic theorems for systems preserving a measure of infinite mass, they don’t seem to lead us as far as the usual Ergodic Theory.
Anyway, this difficulty can be bypassed by normalizing the area form associated to the Abelian differential. This should be compared with the case of the Euclidean space : indeed, while the Lebesgue measure on has infinite mass, after “killing” the scaling factor and restricting ourselves to the unit sphere , we end up with a finite measure. The details of this procedure are the content of the next subsection.
1.3. -action on and Teichmüller geodesic flow
We denote by the set of Abelian differentials on a genus Riemann surface whose induced area form on has total area . At first sight, one is tempted to say that is some sort of “unit sphere” of . However, since the area form of an arbitrary Abelian differential can be expressed as
where form a canonical basis of absolute periods of , i.e., and is a symplectic basis of (with respect to the intersection form), we see that resembles more a “unit hyperboloid”.
Again, we can stratify by considering and, from the definition of the -action on the plane , we see that and its strata come equipped with a natural -action. Moreover, by disintegrating the natural Lebesgue measure on with respect to the level sets of the total area function , , we can write
where is a natural “Lebesgue” measure on . We encourage the reader to compare this with the analogous procedure to get the Lebesgue measure on the unit sphere by disintegration of the Lebesgue measure of the Euclidean space .
Theorem 2 (H. Masur/W. Veech) The total volume (mass) of .
See also J.-C. Yoccoz survey for a presentation (from scratch) of the proof of this result.
Remark. The computation of the actual values of these volumes took essentially 20 years to be performed and it is due to A. Eskin and A. Okounkov. We may come back to this topic in a future post.
In what follows, given any connected component of some stratum , we call the -invariant probability measure obtained from the normalization of the (restriction to ) of the Masur-Veech measure of .
In this language, the global picture is the following: we dispose of a -action on connected components of strata of the moduli space of Abelian differentials with unit area and a naturally invariant probability (Masur-Veech measure).
Of course, it is tempting to start the study of the statistics of -orbits of this action with respect to Masur-Veech measure, but we’ll momentarily refrain ourselves from doing so (instead we postpone to the next chapter this discussion) because this is the appropriate place to introduce the so-called Teichmüller (geodesic) flow.
The Teichmüller flow on is simply the action of the diagonal subgroup of . The discussions we had so far imply that is the geodesic flow associated to Teichmüller metric (introduced in Chapter 1). Indeed, from Teichmüller’s theorem??, it follows that the path , where and is the underlying Riemann surface structure such that is holomorphic, is a geodesic of Teichmüller metric , and for all (i.e., is the arc-length parameter).
In Figure~1 above, we saw the action of Teichmüller geodesic flow on a Abelian differential associated to a L-shaped square-tiled surface derived from 3 squares. At a first glance, if the reader forgot what the discussion by the end of Chapter 1, he/she will find (again) the dynamics of very uninteresting: the initial L-shaped square-tiled surface gets indefinitely squeezed in the vertical direction and stretched in the horizontal direction, so that we don’t have any hope of finding a surface whose shape is somehow “close” to the initial shape (that is, doesn’t seem to have any interesting dynamical feature such as recurrence). However, as we already mentioned in by the end of Chapter 1 (in the genus 1 case), while this is true in Teichmüller spaces , this is not exactly true in moduli spaces : in fact, while in Teichmüller spaces we can only identify “points” by diffeomorphisms isotopic to the identity, one can profit of the (orientation-preserving) diffeomorphisms not isotopic to identity in the case of moduli spaces to eventually bring deformed shapes close to a given one. In other words, the very fact that we deal with the modular group (i.e., diffeomorphisms not necessarily isotopic to identity) in the case of moduli spaces allows to change names of homology classes of the surfaces as we wish, that is, geometrically we can cut our surface along any closed loop to extract a piece of it, glue back by translation (!) this piece at some other part of the surface, and, by definition, the resulting surface will represent the same point in moduli space as our initial surface. Below, we extracted from Anton Zorich’s survey (Figure 15) a picture illustrating this:
To further analyze the dynamics of Teichmüller flow (and/or -action) on , it is surely important to know its derivative . In the next subsection, we will follow M. Kontsevich and A. Zorich to show that the dynamically relevant information about is captured by the so-called Kontsevich-Zorich cocycle.
1.4. Teichmüller flow and Kontsevich-Zorich cocycle on the Hodge bundle over
We start with the following trivial bundle over Teichmüller space of Abelian differentials :
and the trivial (dynamical) cocycle over Teichmüller flow :
Of course, there is not much to say here: we act through Teichmüller flow in the first component and we’re not acting (or acting trivially if you wish) in the second component of the trivial bundle .
Now, we observe that the modular group acts on both components of by pull-back, and, as we already saw, the action of Teichmüller flow commutes with the action of (since acts by post-composition on the local charts of a translation structure while acts by pre-composition on them). Therefore, it makes sense to take the quotients
and . In the literature, is the (real) Hodge bundle over the moduli space of Abelian differentials and is the Kontsevich-Zorich cocycle (KZ cocycle for short) over Teichmüller flow .
We begin by pointing out that the Kontsevich-Zorich cocycle (unlike its “parent” ) is very far from being trivial. Indeed, since we identify with for any to construct the Hodge bundle and , it follows that the fibers of over and are identified in a non-trivial way if the (standard cohomological) action of on is non-trivial. Alternatively, suppose we fix a fundamental domain of on (e.g., through Veech’s zippered rectangle construction) and let’s say we start with some point at the boundary of , a cohomology class and assume that the Teichmüller geodesic through points towards the interior of . Now, we run Teichmüller flow for some (long) time until we hit again the boundary of and our geodesic is pointing outwards . At this stage, from the definition of Kontsevich-Zorich cocycle, we have the “option” to apply an element of the modular group so that Techmüller flow through points towards “at the cost” of replacing the cohomology class by . In this way, we see that is non-trivial in general. Below we illustrate this “fundamental domain”-based discussion in both genus 1 and cases (the picture in the higher-genus case being idealized, of course, since the moduli space is higher-dimensional).
Here we are projecting the picture from the unit cotangent bundle to the moduli space of torii , so that the evolution of the Abelian differentials are designed by the tangent vectors to the hyperbolic geodesics, while the evolution of cohomology classes is designed by the transversal vectors to these geodesics (compare with the discussion by the end of our first post of this series).
Next, we observe that the is a symplectic cocycle because the action by pull-back of the elements of on preserves the intersection form , a symplectic form on the -dimensional real vector space . This has the following consequence for the Ergodic Theory of KZ cocycle. Given any ergodic Teichmüller flow invariant probabilty , we know from Oseledets theorem that there are real numbers (Lyapunov exponents) and a Teichmüller flow equivariant decomposition at -almost every point such that depends measurably on and
for every and any choice of such that . If we allow ourselves to repeat each accordingly to its multiplicity , we get a list of Lyapunov exponents
Such a list is commonly called Lyapunov spectrum (of KZ cocycle with respect to ). The fact that KZ cocycle is symplectic means that the Lyapunov spectrum is always symmetric with respect to the origin:
that is, for every . Roughly speaking, this symmetry correspond to the fact that whenever appears as an eigenvalue of a symplectic matrix , is also an eigenvalue of (so that, by taking logarithms, we “see” that the appearance of a Lyapunov exponent forces the appearance of a Lyapunov exponent ). Thus, it suffices to study the non-negative Lyapunov exponents of KZ cocycle to determine its entire Lyapunov spectrum.
Also, in the specific case of KZ cocycle, it is not hard to deduce that belong to the Lyapunov spectrum of any ergodic probability . Indeed, by the definition, the family of symplectic planes generated by the cohomology classes of the real and imaginary parts of are Teichmüller flow (and even ) equivariant. Also, the action of Teichmüller flow restricted to these planes is, by definition, isomorphic to the action of the matrices on the usual plane if we identify with the canonical vector and with the canonical vector . Actually, the same is true for the entire -action restricted to these planes (where we replace by the corresponding matrices). Since the Lyapunov exponents of the action on are , we get that belong to the Lyapunov spectrum of KZ cocycle.
Actually, it is possible to prove that (i.e., is always the top exponent), and , i.e., the top exponent has always multiplicity 1, or, in other words, the Lyapunov exponent is always simple. However, since this would take us somewhat far from the current goal, we leave this discussion to another day. Instead, we close this chapter by relating this cocycle with the derivative of Teichmüller flow (which is one of the interests of KZ cocycle).
The local coordinates of (connected components of) strata are given by period maps. Hence, we can compute by composing it with these local charts to obtain linear transformations on where is the set of zeroes singularities of an Abelian differential. It is possible to check that the duals of cycles joining two singularities (“relative part” of cohomology) are create only exponents with multiplicity (=“dimension of relative part”)(morally by the same reason as the exponents of KZ cocycle, except that we can’t construct equivariant relative parts in general; see the remark~3 below). Hence, it remains to understand the restriction of to the absolute cohomology .
By writing and considering the action of on each factor of this tensor product, one can check that acts through the usual action of the matrices on the first factor and it acts through the KZ cocycle on the second factor ! Since the eigenvalues of a tensor product of matrices are the products of eigenvalues of them, and we take logarithms when computing Lyapunov exponents, it follows that the Lyapunov spectrum of the restriction of to has the form where are the Lyapunov exponents of KZ cocycle.
Thus, in resume, the Lyapunov spectrum of Teichmüller flow with respect to an ergodic probability measure supported on a stratum (with ) has the form
where are the non-negative exponents of KZ cocycle with respect to .
Remark 3 Concerning the computation of exponents in the relative part of cohomology (i.e., before passing to absolute cohomology), our job would be easier if admitted a equivariant supplement inside . However, it is possible to construct examples to show that this doesn’t happen in general. See the Appendix B of this article of Yoccoz and myself for more details.
Therefore, the KZ cocycle captures the “main part” of the derivative cocycle , so that, since we’re interested in the Ergodic Theory of Teichmüler flow, we will spend sometime in the next chapters to analyze KZ cocycle (without much reference to ).
For the next post of this series, we will discuss the Ergodic Theory of Teichmuller flow with respect to Masur-Veech measures.