Introduction to Teichm\"uller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school"Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory"(from 4 to 16 July, 2011). In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichm\"uller and moduli space of translation surfaces, the Teichm\"uller flow and the SL(2,R)-action on these moduli spaces and the Kontsevich-Zorich cocycle over the Teichm\"uller geodesic flow. We sketch two applications of the ergodic properties of the Teichm\"uller flow and Kontsevich-Zorich cocycle, with respect to Masur-Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichm\"uller flow and the Kontsevich-Zorich cocycle work as \emph{renormalization dynamics} for interval exchange transformations and translation flows. In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich-Zorich cocycle with respect to invariant measures other than the Masur-Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich-Zorich cocycle are very different from the case of Masur-Veech measures. Finally, we end these notes by constructing some examples of closed SL(2,R)-orbits such that the restriction of the Teichm\"uller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary SL(2,R)-representations have arbitrarily small spectral gap (and in particular it has complementary series).


Quick review of basic elements of Teichmüller theory
The long-term goal of these lecture notes is the study of the so-called Teichmüller geodesic flow and its noble cousin Kontsevich-Zorich cocycle, and some of its applications to interval exchange transformations, translation flows and billiards. As any respectable geodesic flow, Teichmüller flow acts naturally in a certain unit cotangent bundle. More precisely, the phase space of the Teichmüller geodesic flow is the unit cotangent bundle of the moduli space of Riemann surfaces.
For this initial section, we'll briefly recall some basic results of Teichmüller theory leading to the conclusion that the unit cotangent bundle of the moduli space of Riemann surfaces (i.e., the phase space of the Teichmüller flow) is naturally identified to the moduli space of quadratic differentials. As we'll see later in this text, the relevance of this identification resides in the fact that it makes apparent the existence of a natural SL(2, R) action on the moduli space of quadratic differentials such that the Teichmüller flow corresponds to the action of the diagonal subgroup g t := e t 0 0 e −t of SL(2, R). In any event, the basic reference for this section is J. Hubbard's book [40].
1.1. Deformation of Riemann surfaces: moduli and Teichmüller spaces of curves. Let us consider two Riemann surface structures S 0 and S 1 on a fixed (topological) compact surface of genus g ≥ 1. If S 0 and S 1 are not biholomorphic (i.e., they are "distinct"), there is no way to produce a conformal map (i.e., holomorphic map with non-vanishing derivative) f : S 0 → S 1 .
However, we can try to produce maps f : S 0 → S 1 which are as "nearly conformal" as possible.
To do so, we need a reasonable way to "measure" the amount of "non-conformality" of f . A fairly standard procedure is the following one. Given a point x ∈ S 0 and some local coordinates around x and f (x), we write the derivative Df (x) of f at x as Df (x)u = ∂f ∂z (x)u + ∂f ∂z (x)u, so that Df (x) sends infinitesimal circles into infinitesimal ellipses of eccentricity ∂f ∂z (x) + ∂f ∂z (x) ∂f ∂z (x) − ∂f ∂z (x) where k(f, x) := | ∂f ∂z (x)| | ∂f ∂z (x)| . This is illustrated in the figure below: |a| + |b| |a| − |b| Df (x)u = au + bu We say that K(f, x) is the eccentricity coefficient of f at x, while is the eccentricity coefficient of f . Note that, by definition, K(f ) ≥ 1 and f is a conformal map if and only if K(f ) = 1 (or equivalently k(f, x) = 0 for every x ∈ S 0 ). Hence, K(f ) accomplishes the task of measuring the amount of non-conformality of f . We call f : S 0 → S 1 quasiconformal whenever K = K(f ) < ∞.
In the next subsection, we'll see that quasiconformal maps are useful to compare distinct Riemann structures on a given topological compact surface S. In a more advanced language, we consider the moduli space M(S) of Riemann surface structures on S modulo conformal maps and the Teichmüller space T (S) of Riemann surface structures on S modulo conformal maps isotopic to the identity. It follows that M(S) is the quotient of T (S) by the so-called modular group (or mapping class group) Γ(S) := Γ g := Diff + (S)/Diff + 0 (S) of isotopy classes of diffeomorphisms of S (here Diff + (S) is the set of orientation-preserving diffeomorphisms and Diff + 0 (S) is the set of orientation-preserving diffeomorphisms isotopic to the identity). Therefore, the problem of studying deformations of Riemann surface structures corresponds to the study of the nature of the moduli space M(S) (and the Teichmüller space T (S)). | ∂f ∂z (x)| . Since we are dealing with Riemann surfaces (and we used local charts to perform calculations), k(f, x) doesn't provide a globally defined function on S 0 . Instead, by looking at how k(f, x) transforms under changes of coordinates, one can check that the quantities k(f, x) can be collected to globally define a tensor µ(x) (of type (−1, 1)) via the formula: In the literature, µ(x) is called a Beltrami differential. Note that µ L ∞ < 1 when f is an orientation-preserving quasiconformal map. The intimate relationship between quasiconformal maps and Beltrami differentials is revealed by the following profound theorem of L. Ahlfors and L. A direct consequence of this striking result to the deformation of Riemann surface structures is the following proposition (whose proof is left as an exercise to the reader): Proposition 2. Let X be a Riemann surface and µ a Beltrami differential on X. Given an atlas φ i : U i → C of X, denote by µ i the function on V i := φ i (U i ) ⊂ C defined by Then, there is a family of mappings ψ i (µ) : V i → C solving the Beltrami equations ∂ψ i (µ) ∂z = µ i ∂ψ i (µ) ∂z such that ψ i (µ) are homeomorphisms from V i to ψ i (µ)(V i ).
Moreover, ψ i • φ i : U i → C form an atlas giving a well-defined Riemann surface structure X µ in the sense that it is independent of the initial choice of the atlas φ i : U i → C and the choice of φ i verifying the corresponding Beltrami equations.
In other words, the measurable Riemann mapping theorem of Alhfors and Bers implies that one can use Beltrami differentials to naturally deform Riemann surfaces through quasiconformal mappings. Of course, we can ask to what extend this is a general phenomena: namely, given two Riemann surface structures S 0 and S 1 , can we relate them by quasiconformal mappings? The answer to this question is provided by the remarkable theorem of O. Teichmüller: Theorem 3 (O. Teichmüller). Given two Riemann surfaces structures S 0 and S 1 on a compact topological surface S of genus g ≥ 1, there exists a quasiconformal mapping f : S 0 → S 1 minimizing the eccentricity coefficient K(g) among all quasiconformal maps g : S 0 → S 1 isotopic to the identity map id : S → S. Furthermore, whenever a quasiconformal map f : S 0 → S 1 minimizes the eccentricity coefficient in the isotopy class of a given orientation-preserving diffeomorphism h : S → S, we have that the eccentricity coefficient of f at any point x ∈ S 0 is constant, i.e., K(f, x) = K(f ) except for a finite number of points x 1 , . . . , x n ∈ S 0 . Also, quasiconformal mappings minimizing the eccentricity coefficient in a given isotopy class are unique modulo (pre and post) composition with conformal maps.
In the literature, any such minimizing quasiconformal map in a given isotopy class is called an extremal map. Using the extremal quasiconformal mappings, we can naturally introduce a distance between two Riemann surface structures S 0 and S 1 by where f : S 0 → S 1 is an extremal map isotopic to the identity. The metric d is called Teichmüller metric. The major concern of these notes is the study of the geodesic flow associated to Teichmüller metric on the moduli space of Riemann surfaces. As we advanced in the introduction, it is quite convenient to regard a geodesic flow living on the cotangent bundle of the underlying space. The discussion of the cotangent bundle of T (S) is the subject of the next subsection.
1.3. Quadratic differentials and the cotangent bundle of the moduli space of curves.
The results of the previous subsection show that the Teichmüller space is modeled by the space of Beltrami differentials. Recall that Beltrami differentials are measurable tensors µ of type (−1, 1) such that µ L ∞ < 1. It follows that the tangent bundle to T (S) is modeled by the space of measurable and essentially bounded (L ∞ ) tensors of type (−1, 1) (because Beltrami differentials form the unit ball of this Banach space). Hence, the cotangent bundle to T (S) can be identified with the space Q(S) of integrable quadratic differentials on S, i.e., the space of (integrable) tensors q of type (2, 0) (that is, q is written as q(z)dz 2 in a local coordinate z). In fact, we can determine the cotangent bundle once one can find an object (a tensor of some type) such that the pairing µ, q = S qµ is well-defined; when µ is a tensor of type (−1, 1) and q is a tensor of type (2, 0), we can write qµ = q(z)µ(z)dz 2 dz dz = q(z)µ(z)dz dz = q(z)µ(z)|dz| 2 in local coordinates, i.e., we obtain a tensor of type (1,1), that is, an area form. Therefore, since µ is essentially bounded, we see that the requirement that this pairing makes sense is equivalent to ask that the tensor q of type (2, 0) is integrable.
Next, let's see how the geodesic flow associated to the Teichmüller metric looks like after the identification of the cotangent bundle of T (S) with the space Q(S) of integrable quadratic differentials. Firstly, we need to investigate more closely the geometry of extremal quasiconformal maps between two Riemann surface structures. To do so, we recall another notable theorem of O.

Teichmüller:
Theorem 4 (O. Teichmüller). Given an extremal map f : S 0 → S 1 , there is an atlas φ i on S 0 compatible with the underlying complex structure such that • the changes of coordinates are all of the form z → ±z + c, c ∈ C outside the neighborhoods of a finite number of points, • the horizontal (resp., vertical) foliation { φ i = 0} (resp., { φ i = 0}) is tangent to the major (resp.minor) axis of the infinitesimal ellipses obtained as the images of infinitesimal circles under the derivative Df , and • in terms of these coordinates, f expands the horizontal direction by the constant factor of √ K and f contracts the vertical direction by the constant factor of 1/ √ K.
An atlas φ i satisfying the property of the first item of Teichmüller theorem above is called a halftranslation structure. In this language, Teichmüller's theorem says that extremal maps f : S 0 → S 1 (i.e., deformations of Riemann surface structures) can be easily understood in terms of halftranslation structures: it suffices to expand (resp., contract) the corresponding horizontal (resp., vertical) foliation on S 0 by a constant factor equal to e d(S0,S1) in order to get a horizontal (resp., vertical) foliation of a half-translation structure compatible with the Riemann surface structure of S 1 . This provides a simple way to describe the Teichmüller geodesic flow in terms of half-translation structures. Thus, it remains to relate half-translation structures with quadratic differentials to get a pleasant formulation of this geodesic flow. While we could do this job right now, we'll postpone this discussion to the third section of these notes for two reasons: • Teichmüller geodesic flow is naturally embedded into a SL(2, R)-action (as a consequence of this relationship between half-translation structures and quadratic differentials), so that it is preferable to give a unified treatment of this fact later; • for pedagogical motivations, once we know that quadratic differentials is the correct object to study, it seems more reasonable to introduce the fine structures of the space Q(S) before the dynamics on this space (than the other way around).
In particular, we'll proceed as follows: for the remainder of this subsection, we'll briefly sketch the bijective correspondence between half-translation structures and quadratic differentials; after that, we make some remarks on the Teichmüller metric (and other metric structures on Q(S)) and we pass to the next subsection where we work out the particular (but important) case of genus 1 surfaces; then, in the spirit of the two items above, we devote Section 2 to the fine structures of Q(S), and Section 3 to the dynamics on Q(S).
Given a half-translation structure φ i : U i → C on a Riemann surface S, one can easily construct a quadratic differential q on S by pulling back the quadratic differential dz 2 on C through φ i : indeed, this procedure leads to a well-defined global quadratic differential on S because we are assuming that the changes of coordinates (outside the neighborhoods of finitely many points) have the form z → ±z + c. Conversely, given a quadratic differential q on a Riemann surface S, we take an atlas φ i : U i → C such that q| Ui = φ * i (dz 2 ) outside the neighborhoods of finitely many singularities of q. Note that the fact that q is obtained by pulling back the quadratic differential dz 2 on C means that the associated changes of coordinates z → z send the quadratic differential dz 2 to (dz ) 2 . Thus, our changes of coordinates outside the neighborhoods of the singularities of q have the form z → z = ±z + c, i.e., φ i is a half-translation structure.
Remark 5. Generally speaking, a quadratic differential on a Riemann surface is either orientable or non-orientable. More precisely, given a quadratic differential q, consider the underlying halftranslation structure φ i and define two foliations by { φ i = c} and { φ i = c} (these are called the horizontal and vertical foliations associated to q). We say that q is orientable if these foliations are orientable and q is non-orientable otherwise. Alternatively, we say that q is orientable if the changes of coordinates of the underlying half-translation structure φ i outside the singularities of q have the form z → z + c. Equivalently, q is orientable if it is the global square of a holomorphic 1-form, i.e., q = ω 2 , where ω is a holomorphic 1-form, that is, an Abelian differential. For the sake of simplicity of the exposition, from now on, we'll deal exclusively with orientable quadratic differentials q, or, more precisely, we'll restrict our attention to Abelian differentials. The reason to doing so is two-fold: firstly, most of our computations below become more easy and clear in the orientable setting, and secondly, usually (but not always) some results about Abelian differentials can be extended to the non-orientable setting by a double cover construction, that is, one consider a (canonical) double cover of the initial Riemann surface equipped with a non-orientable quadratic differential q such that a global square of the lift of q is well-defined. In the sequel, we denote the space of Abelian differentials on a compact surface S of genus g by H(S) or H g . Remark 6. In Subsection 2.1, we will come back to the correspondence between quadratic differentials and half-translation structures in the context of Abelian differentials: more precisely, in this subsection we will see that Abelian differentials bijectively correspond to the so-called translation structures.
We close this subsection with the following comments.
Remark 7. The Teichmüller metric is induced by the family of norms on the cotangent bundle Q(S) of Teichmüller space T (S) given by the L 1 norm of quadratic differentials (see Theorem 6.6.5 of [40]). However, this family of norms doesn't depend smoothly on the base point in general, so that it doesn't originate from a Riemannian metric. In fact, this family of norms defines only a Finsler metric, i.e., it is a family of norms depending continuously on the base point.  We refer to the excellent introduction of the paper [12] (and references therein) of K. Burns, H.
Masur and A. Wilkinson for a nice account on the Weil-Petersson metric. Ending this remark, we note that the basic difference between the Teichmüller metric and the Weil-Petersson metric is the following: as we already indicated, the Teichmüller metric is related to flat (half-translation) structures, while the Weil-Petersson metric can be better understood in terms of hyperbolic structures.
1.4. An example: Teichmüller and moduli spaces of elliptic curves (torii). The goal of this subsection is the illustration of the role of the several objects introduced previously in the concrete case of genus 1 surfaces (elliptic curves). Indeed, we'll see that, in this particular case, one can do "everything" by hand.
We begin by recalling that an elliptic curve, i.e., a Riemann surface of genus 1, is uniformized by the complex plane. In other words, any elliptic curve is biholomorphic to a quotient C/Λ where Λ ⊂ C is a lattice. Given a lattice Λ ⊂ C generated by two elements w 1 and w 2 , that is, Λ = Zw 1 ⊕Zw 2 , we see that the multiplication by 1/w 1 or 1/w 2 provides a biholomorphism isotopic to the identity between C/Λ and C/Λ(w), where Λ(w) := Z ⊕ Zw is the lattice generated by 1 and w ∈ H ⊂ C (the upper-half plane of the complex plane). In fact, w = w 2 /w 1 or w = w 1 /w 2 here.
Next, we observe that any biholomorphism f between C/Λ(w ) and C/Λ(w) can be lifted to an automorphism F of the complex plane C. This implies that F has the form F (z) = Az + B for some A, B ∈ C. On the other hand, since F is a lift of f , we can find α, β, γ, δ ∈ Z such that Expanding these equations using the fact that F (z) = az + b, we get w = αw + β γw + δ Also, since we're dealing with invertible objects (f and F ), it is not hard to check that αδ − βγ = 1 (because it is an integer number whose inverse is also an integer). In other words, recalling that we see that the torii C/Λ(w) and C/Λ(w ) are biholomorphic if and only if w ∈ SL(2, Z) · w.
For example, we show below the torii C/Λ(i) (on the left) and C/Λ(1 + i) (in the middle).
Since 1 + i is deduced from i via the action of T = 1 1 0 1 ∈ SL(2, Z) on H, we see that these torii are biholomorphic and hence they represent the same point in the moduli space M 1 (see the right hand side part of the figure above). On the other hand, they represent distinct points in the Teichmüller space T 1 (because T = id).
Our discussion so far implies that the Teichmüller space complex-analytic manifolds while moduli spaces are orbifolds 1 . In any case, it is natural to consider both spaces from the topological point of view because Teichmüller spaces are simply connected so that they are the universal covers of moduli spaces. Finally, closing this section, we note that our discusssion above also shows that, in the genus 1 case, the mapping class group Γ 1 is SL(2, Z).

Some structures on the set of Abelian differentials
We denote by L g the set of Abelian differentials on a Riemann surface of genus g ≥ 1, or more precisely, the set of pairs (Riemann surface structure on M, ω) where M is a compact topological surface of genus g and ω is a 1-form which is holomorphic with respect to the underlying Riemann surface structure. In this notation, the Teichmüller space of Abelian differentials is the quotient T H g := L g /Diff + 0 (M ) and the moduli space of Abelian differentials is the quotient H g := L g /Γ g . 1 In general, the mapping class group doesn't act properly discontinuously on Teichmüller space because some Riemann surfaces are "more symmetric" (i.e., they have larger automorphisms group) than others. In fact, we already saw this in the case of genus 1: the modular curve H/SL(2, Z) isn't smooth near the points w = i and w = e πi/3 because the (square and hexagonal) torii corresponding to these points have larger automorphisms groups when compared with a typical torus C/Λ(w).
Here Diff + 0 (M ) and Γ g := Diff + (M )/Diff + 0 (M ) (the set of diffeomorphisms isotopic to the identity and the mapping class group resp.) act on the set of Riemann surface structure in the usual manner, while they act on Abelian differentials by pull-back.
In order to equip L g , T H g and H g with nice structures, we need a more "concrete" presentation of Abelian differentials. In the next subsection, we will see that the notion of translation structures provide such a description of Abelian differentials.
2.1. Abelian differentials and translation structures. Given any point p ∈ M − Σ, let's select U (p) a small path-connected neighborhood of p such that U (p) ∩ Σ = ∅. In this setting, the "period" map φ p : U (p) → C, φ p (x) := x p ω obtained by integration along a path inside U (p) connecting p and x is well-defined: indeed, this follows from the fact that any Abelian differential is closed (because they are holomorphic) and the neighborhood U (p) doesn't contain any zeroes of ω (so that the integral z p ω doesn't depend on the choice of the path inside U (p) connecting p and x). Furthermore, since p ∈ M − Σ (so that ω(p) = 0), we see that, by reducing U (p) if necessary, this "period" map is a biholomorphism.
In other words, the collection of all such "period" maps φ p provides an atlas of M −Σ compatible with the Riemann surface structure. Also, by definition, the push-forward of the Abelian differential ω by any such φ p is precisely the canonical Abelian differential (φ p ) * (ω) = dz on the complex plane C. Moreover, the "local" equality x p ω = x q ω + q p ω implies that all changes of coordinates have the form φ q • φ −1 p (z) = z + c where c = p q ω ∈ C is a constant (since it doesn't depend on z). Furthermore, since ω has finite order at its zeroes, it is easy to deduce from Riemann's theorem on removal singularities that this atlas of M − Σ can be extended to M in such a way that the push-forward of ω by a local chart around a zero p ∈ Σ of order k is the holomorphic form z k dz.
In the literature, a compact surface M of genus g ≥ 1 equipped with an atlas whose changes of coordinates are given by translations z → z + c of the complex plane outside a finite set of points Σ ⊂ M is called a translation surface structure on M . In this language, our previous discussion simply says that any non-trivial Abelian differential ω on a compact Riemann surface M gives rise to a translation surface structure on M such that ω is the pull-back of the canonical holomorphic form dz on C. 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 dz on C: this pull-back ω is well-defined because dz is translation-invariant).
In resume, we just saw the proof of the following proposition: Proposition 10. The set L g of Abelian differentials on Riemann surfaces of genus g ≥ 1 is canonically identified to the set of translation structures on a compact (topological) surface M of genus g ≥ 1.
Example 11. During Riemann surfaces courses, a complex torus is quite often presented through a translation surface structure: indeed, by giving a lattice Λ = Zw 1 ⊕ Zw 2 ⊂ C, we are saying that the complex torus C/Λ equipped with the (non-vanishing) Abelian differential dz 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 w 1 and w 2 through the translations z → z + w 1 and z → z + w 2 ).

Figure 2. A genus 2 translation surface
In this picture, we are gluing parallel opposite sides v j , j = 1, . . . , 4, of P , so that this is again a valid presentation of a translation surface structure. Let's denote by (M, ω) 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 P are all identified to a single point p. Moreover, we see that p is a special point when compared with any point of P − {p} because, by turning around p, we note that the "total angle" around it is 6π while the total angle around any point of P − {p} is 2π, that is, a neighborhood of p inside M resembles "3 copies" of the flat complex plane while a neighborhood of any other point q = p resembles only 1 copy of the flat complex plane. In other words, a natural local coordinate around p is ζ = z 3 , so that ω = dζ = d(z 3 ) = 3z 2 dz, i.e., the Abelian differential ω has a unique zero of order 2 at p. From this, we can infer that M is a compact Riemann surface of genus 2: indeed, by Riemann-Hurwitz theorem, the sum of orders of zeroes of an Abelian differential equals 2g −2 (where g is the genus); in the present case, this means given by the vertical direction on M − {p} (this is well-defined because these points correspond to regular points of the polygon P ) and vanishing at p (where a choice of "vertical direction" doesn't make sense since we have multiple copies of the plane glued together).
Example 13 (Rational billiards). Let P be a rational polygon, that is, a polygon whose angles are all rational multiples of π. Consider the billiard on P : the trajectory of a point in P in a certain direction consists of straight lines until we hit the boundary ∂P of the polygon; at this moment, we apply the usual reflection laws (saying that the angle between the outgoing ray with ∂P is the same as the angle between the incoming ray and ∂P ) to prolongate the trajectory. See the figure below for an illustration of such an trajectory.
In the literature, the study of general billiards (where P is not necessarily a polygon) is a classical subject with physical origins (e.g., mechanics and thermodynamics of Lorenz gases). In the particular case of billiards in rational polygons, an unfolding construction (due to R. Fox and R. Keshner [35], and A. Katok and A. Zemlyakov [46]) allows to think of billiards on rational polygons as translation flows on translation surfaces. Roughly speaking, the idea is that each time the trajectory hits the boundary ∂P , instead of reflecting the trajectory, we reflect the table itself so that the trajectory keeps in straight line: The group G generated by the reflections with respect to the edges of P is finite when P is a rational polygon, so that the natural surface X obtained by this unfolding procedure is compact.
Furthermore, the surface X comes equipped with a natural translation structure, and the billiard dynamics on P becomes the translation (straight line) flow on X. In the picture below we drew the translation surface (Swiss cross) obtained by unfolding a L-shaped polygon, unfolding and in the picture below we drew the translation surface (regular octagon) obtained by unfolding a triangle with angles π/8, π/2 and 3π/8. In general, a rational polygon P with k edges and angles πm i /n i , i = 1, . . . , N has a group of reflections G of order 2N and, by unfolding P , we obtain a translation surface X of genus g where In particular, it is possible to show that the only genus 1 translation surfaces obtained by the unfolding procedure come from the following polygons: a square, an equilateral triangle, a triangle with angles π/3, π/2, π/6, and a triangle with angles π/4, π/2, π/4 (see the figure below).
For more informations about translation surfaces coming from billiards on rational polygons, see this survey of H. Masur and S. Tabachnikov [54]. Usually, we obtain an i.e.t. as a return map of a translation flow on a translation surface.
Conversely, given an i.e.t. T , it is possible to "suspend" it (in several ways) to construct translation flows on translation surfaces such that T is the first return map to an adequate transversal to the translation flow. For instance, we illustrated in the figure below a suspension construction due to H. Masur [51] applied to the third i.e.t. of Figure 3.
Here, the idea is that: • the vectors ζ i have the form ζ i = λ i + √ −1τ i ∈ C R 2 where λ i are the lengths of the intervals permuted by T ; • then, we organize the vectors ζ i on the plane R 2 in order to get a polygon P so that by going upstairs we meet the vectors ζ i in the usual order (i.e., ζ 1 , ζ 2 , etc.) while by going downstairs we meet the vectors ζ i in the order determined by T , i.e., by following the combinatorial receipt (say a permutation π of d elements) used by T to permute intervals; • by gluing by translations the pairs of sides label by vectors ζ i , we obtain a translation surface whose vertical flow has the i.e.t. T as first return map to the horizontal axis R × {0} (e.g., in the picture we drew a trajectory of the vertical flow starting at the interval B on the "top part" of R × {0} and coming back at the interval B on the bottom part of R × {0}); • finally, the suspension data τ i is chosen "arbitrarily" as soon as the planar figure P is not degenerate: formally, one imposes the condition j<i τ j > 0 and π(j)<i τ j < 0 for every Of course, this is not the sole way of suspending i.e.t.'s to get translation surfaces: for instance, in this survey [72] of J.-C. Yoccoz, one can find a detailed description of an alternative suspension procedure due to W. Veech (and nowadays called Veech's zippered rectangles construction).
Example 15 (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 M equipped with an Abelian differential (obtained by pulling back dz 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 in Figure 4 below a L-shaped square-tiled surface derived from 3 unit squares identified as in the picture (i.e., pairs of sides with the same marks are glued together by translation). 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 16. So far we produced examples of translation surfaces/Abelian differentials from identifications by translation of pairs of parallel sides of a finite collection of polygons. The curious reader maybe asking whether all translation surface structures can be recovered by this procedure.
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 12) 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 [72] 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 C: • flat metric: since the usual (flat) Euclidean metric dx 2 + dy 2 on the complex plane C is translation-invariant, its pullback by the local charts provided by the translation structure gives a well-defined flat metric on M − Σ; • area form: again, since the usual Euclidean area form dx · dy on the complex plane C is also translation-invariant, we get a well-defined area form dA on M ; • canonical choice of a vertical vector-field : as we implicitly mentioned by the end of Example 12, the vertical vertical vector field ∂/∂y on C can be pulled back to M − Σ in a coherent way to define a canonical choice of north direction; • pair of transverse measured foliations: the pullback to M − Σ of the horizontal {x = constant} and vertical {y = constant} foliations of the complex plane C are well-defined and produce a pair of transverse foliations F h and F v which are measured in the sense on Thurston: the leaves of these foliations come with a canonical notion of length measures |dy| and |dx| transversely to them.
Remark 17. It is important to observe that the flat metric introduced above is a singular metric: indeed, although it is a smooth Riemannian metric on M − Σ, it degenerates when we approach a zero p ∈ Σ of the Abelian differential. Of course, we know from Gauss-Bonnet theorem that no compact surface of genus g ≥ 2 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 g ≥ 2 surface is by concentrating all curvature at a finite number of points.
Once we know that Abelian differentials and translation structures are essentially the same object, we can put some structures on L g , T H g and H g .
2.2. Stratification. Given a non-trivial Abelian differential ω on a Riemann surface of genus g ≥ 1, we can form a list κ = (k 1 , . . . , k σ ) by collecting the multiplicities of the (finite set of) zeroes of ω. Observe that any such list κ = (k 1 , . . . , k σ ) verifies the constraint σ l=1 k l = 2g − 2 in view of Poincaré-Hopf theorem (or alternatively Gauss-Bonnet theorem). Given an unordered list κ = (k 1 , . . . , k σ ) with σ l=1 k l = 2g −2, the set L(κ) of Abelian differentials whose list of multiplicities of its zeroes coincide with κ is called a stratum of L g . Since the actions of Diff + 0 (M ) and Γ g respect the multiplicities of zeroes, the quotients T H(κ) := L(κ)/Diff + 0 (M ) and H(κ) := L(κ)/Γ g are welldefined. By obvious reasons, T H(κ) and H(κ) are called stratum of T H g and H g (resp.). Notice that, by definition, In other words, the sets L g , T H g and H g are naturally "decomposed" into the subsets (strata) L(κ), T H(κ) and H(κ). However, at this stage, we can't promote this decomposition into disjoint subsets to a stratification because, in the literature, a stratification of a set X is a decomposition k l = 2g − 2. Given an Abelian differential ω ∈ T H(κ), we denote by Σ(ω) the set of zeroes of ω. It is possible to prove that, for every ω 0 ∈ T H(κ), there is an open 2 set ω 0 ∈ U 0 ⊂ T H(κ) such that, after identifying, for all ω ∈ U 0 , the cohomology H 1 (M, Σ(ω), C) with the fixed complex vector space H 1 (M, Σ(ω 0 ), C) via the Gauss-Manin connection (i.e., through identification of the integer lattices H 1 (M, Σ(ω), Z⊕iZ) and H 1 (M, Σ(ω 0 ), Z⊕iZ)), the period map Θ : given by is a local homeomorphism. A sketch of proof of this fact (along the lines given in this article [45] of A. Katok) goes as follows. We need to prove that two closed 1-forms η 0 and η 1 with the same relative periods, i.e., Θ(η 0 ) = Θ(η 1 ), are isotopic as far as they are close enough to each other. The idea to construct such an isotopy is to apply a variant of the so-called Moser's homotopy trick.
More precisely, one considers η t = (1 − t)η 0 + tη 1 , and one tries to find the desired isotopy φ t by In this direction, let's see what are the properties satisfied by a solution φ t . By taking the derivative, Assuming that φ t is the flow of a (non-autonomous) vector field X t , we get where L Xt is the Lie derivative along the direction of X t .
By hypothesis, Θ(η 0 ) = Θ(η 1 ). In particular, η 0 and η 1 have the same absolute periods, so that In other words, we can find a smooth family U t of functions with dU t =η t . By inserting this into the previous equation, it follows that On the other hand, by Cartan's magic formula, By inserting this into the previous equation, we get Here we're considering the natural topology on strata T H(κ) induced by the developing map. More precisely, given ω ∈ L(κ), fix p 1 ∈ Σ(ω), an universal cover p : f M → M and a point P 1 ∈ f M over p 1 . By integration of p * ω from P 1 to a point Q ∈ f M , we get, by definition, a developing map Dω : ( f M , P 1 ) → (C, 0) completely determining the translation surface (M, ω). In this way, the injective map ω → Dω allows us to see L(κ) as a subset of the space At this point, we see how one can hope to solve the original equation φ * t η t = η 0 : firstly we fix a smooth family U t with dU t =η t = η 1 − η 0 (unique up to additive constant), secondly we define a vector field X t such that i Xt η t = −U t , so that U t + i Xt η t = 0 and a fortiori d(U t + i Xt η t ) = 0, and finally we let φ t be the isotopy associated to X t . Of course, one must check that this is well-defined: for instance, since η t have singularities (zeroes) at the finite set Σ = Σ(η 0 ) = Σ(η 1 ), we need to know that one can take U t = 0 at Σ, and this is possible because U t (p j ) − U t (p i ) = pj piη t = pj pi (η 1 − η 0 ), and η 0 and η 1 have the same relative periods. We leave the verification of the details of the definition of φ t as an exercise to the reader (whose solution is presented in [45]).
For an alternative proof of the fact that the period maps are local coordinates based on W.
Remark 19. Concerning the possibility of using Veech's zippered rectangles to show that the period maps are local homeomorphisms, we vaguely mentioned this particular strategy because the zippered rectangles are a fundamental tool: indeed, besides its usage to put nice structures on T H(κ), it can be applied to understand the connected components of H(κ), make volume estimates for µ κ , study the dynamics of Teichmüller flow on H(κ) through combinatorial methods (Markov partitions), etc. In other words, Veech's zippered rectangles is a powerful tool to study global geometry and dynamics on H(κ). Furthermore, it allows to connect the dynamics on H(κ) to the interval exchange transformations and translation flows, so that it is also relevant in applications of the dynamics of Teichmüller flow. However, taking into account the usual limitations of space and time, and the fact that this tool is largely discussed in Yoccoz's survey [72], we will not give here more details on Veech's zippered rectangles and applications.
In resume, by using the period maps as local coordinates, T H(κ) has a structure of affine complex manifold of complex dimension 2g + σ − 1 and a natural (Lebesgue) measure λ κ . Also, it is not hard to check that the action of the modular group Γ g := Diff + (M )/Diff + 0 (M ) is compatible with the affine complex manifold structure on T H(κ), so that, by passing to the quotient, one gets that the stratum H(κ) of the moduli space of Abelian differentials has the structure of a affine complex orbifold and a natural Lebesgue measure µ κ .
Remark 20. Note that, as we emphasized above, after passing to the quotient, H(κ) is an affine complex orbifold at best. In fact, we can't expect H(κ) to be a manifold since, as we saw in Subsection 1.4, even in the most simple example of genus 1, H(∅) is an orbifold but not a manifold.
In particular, the fact that the period maps are local homeomorphisms is intimately related to the fact that we were talking specifically about the Teichmüller space of Abelian differentials T H(κ) (and not of its cousin H(κ)).
The following example shows a concrete way to geometrically interpret the role of period maps as local coordinates of the strata T H(κ).
Example 21. Let Q be a polygon and (M, ω 0 ) a translation surface like in Example 12. As we saw in this example, M is a genus 2 Riemann surface and ω 0 has an unique zero (of order 2) at the point p ∈ M coming from the vertices of Q (because the total angle around p is 6π), that is, ω 0 ∈ T H(2). Also, it can be checked that the four closed loops α 1 , α 2 , α 3 , α 4 of M obtained by projecting to M the four sides v 1 , v 2 , v 3 , v 4 from Q are a basis of the absolute homology group H 1 (M, Z). It follows that, in this case, the period map in a small neighborhood U 0 of ω 0 is where V 0 is a small neighborhood of (v 1 , v 2 , v 3 , v 4 ). Consequently, we see that any Abelian differential ω ∈ H(2) sufficiently close to ω 0 can be obtained geometrically by small arbitrary perturbations (indicated by dashed red lines in the figure below) of the sides v 1 , v 2 , v 3 , v 4 of our initial polygon Q (indicated by blue lines in the figure below).
Remark 22. The introduction of nice affine complex structures in the case of non-orientable quadratic differentials is slightly different from the case of Abelian differentials. Given a non-orientable quadratic differential q ∈ Q(κ) on a genus g Riemann surface M , there is a canonical (orienting) double-cover π κ : M → M such that π * κ (q) = ω 2 , where ω is an Abelian differential on M . Here, M is a connected Riemann surface because q is non-orientable (otherwise it would be two disjoint copies of M ). Also, by writing κ = (o 1 , . . . , o τ , e 1 , . . . , e ν ) where o j are odd integers and e j are even integers, one can check that π κ is ramified over singularities of odd orders o j while π κ is regular (i.e., it has two pre-images) over singularities of even orders e j . It follows that In particular, Riemann-Hurwitz formula implies that the genus g of M is g = 2g − 1 + τ /2.
Denote by σ : M → M the non-trivial involution such that π κ • σ = π κ (i.e., σ interchanges the sheets of the double-covering π κ ). Observe that σ induces an involution σ * acting on the , Σ κ is the set of singularities of q and p 1 , . . . , p η are the (simple) poles of q. Since σ * is an involution, we can decompose is the subspace of σ * -anti-invariant cycles. Observe that the Abelian differential ω is σ * -anti-invariant: indeed, since ω 2 is π κ (q), we have that σ * ( ω) = ± ω; if ω were σ * -invariant, it would follow that π * κ ( ω) 2 = q, i.e., q would be the square of an Abelian differential, and hence an orientable quadratic differential (a contradiction). From this, it is not hard to believe that one can prove that a small neighborhood of q can be identified with a small neighborhood of the σ * -anti-invariant Abelian differential ω inside the anti-invariant subspace H 1 − ( M , Σ κ , C). Again, the changes of coordinates are locally affine, so that Q(κ) also comes equipped with a affine complex structure and a natural Lebesgue measure.
At this stage, the terminology stratification used in the previous subsection is completely justified by now and we pass to a brief discussion of the topology of our strata.

2.4.
Connectedness of strata. At first sight, it might be tempting to say that the strata H(κ) are always connected, i.e., once we fix the list κ of orders of zeroes, one might conjecture that there are no further obstructions to deform an arbitrary Abelian differential ω 0 ∈ H(κ) into another arbitrary Abelian differential ω 1 ∈ H(κ). However, W. Veech [71] discovered that the stratum H(4) has two connected components. In fact, W. Veech distinguished the connected components of H(4) by means of a combinatorial invariants called extended Rauzy classes 3 . Roughly speaking, W.
Veech showed that there is a bijective correspondence between connected components of strata and extended Rauzy classes, and, using this fact, he concluded that H(4) has two connected components because he checked (by hand) that there are precisely two extended Rauzy classes associated to this stratum. By a similar strategy, P. Arnoux proved that the stratum H(6) has three connected components. In principle, one could think of pursing the strategy of describing extended Rauzy classes to determine the connected components of strata, but this is a hard combinatorial problem: for instance, when trying to compute extended Rauzy classes associated to connected components 3 A slightly modified version of Rauzy classes, a combinatorial invariant (composed of pair of permutations with d symbols with d = 2g + s − 1, where g is the genus and s is the number of zeroes) introduced by G. Rauzy [65] in his study of interval exchange transformations. of strata of Abelian differentials of genus g, one should perform several combinatorial operations with pairs of permutations on an alphabet of d ≥ 2g letters. 4 Nevertheless, M. Kontsevich and A. Zorich [48] managed to classify completely the connected components of strata of Abelian differentials with the aid of some invariants from algebraic and geometrical nature: technically speaking, there are exactly three types of connected components of strata -hyperelliptic, even spin and odd spin. The outcome of Kontsevich-Zorich classification are the following results: Theorem 23 (M. Kontsevich and A. Zorich). Fix g ≥ 4.
The classification of the connected components of strata of Abelian differentials of genus g = 2 and 3 are slightly different: Theorem 24 (M. Kontsevich and A. Zorich). In genus 2, the strata H(2) and H(1, 1) are connected. In genus 3, both of the strata H(4) and H(2, 2) have two connected components, while the other strata in genus 3 are connected.
For more details, we strongly recommend the original article by M. Kontsevich and A. Zorich.

Dynamics on the moduli space of Abelian differentials
Let (M, ω) be a compact Riemann surface M of genus g ≥ 1 equipped with a non-trivial Abelian differential (that is, a holomorphic 1-form) ω ≡ 0. In the sequel, we denote by Σ the (finite) set of zeroes of ω. 4 Just to give an idea of how fast the size of Rauzy classes does grow, let's mention that the cardinality of the largest Rauzy classes in genera 2, 3, 4 and 5 are 15, 2177, 617401 and 300296573 resp. (can you guess the next number for genus 6? :)) For more informations on how these numbers can be computed see V. Delecroix's work [15].
3.1. GL + (2, R)-action on H g . The canonical identification between Abelian differentials and translation structures makes transparent the existence of a natural action of GL + (2, R) on the set of Abelian differentials L g . Indeed, given an Abelian differential ω, let's denote by {φ α (ω)} α∈I the maximal atlas on M −Σ giving the translation structure corresponding to ω (so that φ α (ω) * dz = ω for every index α). Here, the local charts φ α (ω) map some open set of M − Σ to C.
Since any matrix A ∈ GL + (2, R) acts on C = R ⊕ iR, we can post-compose the local charts φ α (ω) with A, so that we obtain a new atlas {A • φ α (ω)} on M − Σ. Observe that the changes of coordinates of this new atlas are also solely by translations, as a quick computation reveals: In other words, {A • φ α (ω)} is a new translation structure. The corresponding Abelian differential is, by definition, A · ω. Observe that the complex structure of the plane is not preserved by the action of a (typical) GL + (2, R) matrix. Therefore, the Riemann surface structure with respect to which the 1-form A·ω is holomorphic is usually distinct (not biholomorphic) to the Riemann surface structure related to ω. Here, a notable exception is the group of rotations SO(2, R) ⊂ GL + (2, R) 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 GL + (2, R)-action on Abelian differentials given by sides identifications of collections of polygons (as in the previous examples): in fact, given A ∈ GL + (2, R) and an Abelian differential ω related to a finite collection of polygons P with parallel sides glued by translations, the Abelian differential A · ω corresponds, by definition, to the finite collection of polygons A · P obtained by letting A act on the polygons forming P as a planar subset and keeping the sides identifications of parallel sides by translations. Notice that the linear action of A on the plane evidently respects the notion of parallelism, so that this procedure  Of course, this observation opens up the possibility of studying this action via ergodic-theoretical methods applied to λ κ . However, it turns out that the strata H(κ) are a somewhat big: for instance, they are ruled in the sense that the complex lines C · ω foliate them. As a result, it is possible to prove that each λ κ 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 A associated to the Abelian differential. This should be compared with the case of the Euclidean space R n : indeed, while the Lebesgue measure on R n has infinite mass, after "killing" the scaling factor and restricting ourselves to the unit sphere S n−1 ⊂ R n , we end up with a finite measure. The details of this procedure are the content of the next subsection.   Abelian differentials ω on a genus g Riemann surface M whose induced area form dA(ω) on M has total area A(ω) := M dA(ω) = 1. At first sight, one is tempted to say that H (1) g is some sort of "unit sphere" of H g . However, since the area form A(ω) of an arbitrary Abelian differential ω can be expressed as where A j , B j form a canonical basis of absolute periods of ω, i.e., A j = αj ω, B j = βj ω and {α j , β j } g j=1 is a symplectic basis of H 1 (M, R) (with respect to the intersection form), we see that H (1) g resembles more a "unit hyperboloid". Again, we can stratify H and, from the definition of the GL + (2, R)-action on the plane C R 2 , we see that H (1) g and its strata H (1) (κ) come equipped with a natural SL(2, R)-action. Moreover, by disintegrating the natural Lebesgue measure on λ κ with respect to the level sets of the total area function A : H g → R + , ω → A(ω), we can write κ is a natural "Lebesgue" measure on H (1) (κ). We encourage the reader to compare this with the analogous procedure to get the Lebesgue measure on the unit sphere S n−1 by disintegration of the Lebesgue measure of the Euclidean space R n .
Of course, from the "naturality" of the construction, it follows that λ (1) κ is a SL(2, R)-invariant measure on H(κ). The following fundamental result was proved by H. Masur [51] and W. Veech [68]: Remark 27. 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 [25]. We will make some comments on this later (in Section D). For the sake of reader's convenience, we present here the following intuitive argument of H.
Masur [52] explaining why λ (1) κ has finite mass. In the genus 1 case (of torii), we saw in Subsection 1.4 that the moduli space H For the discussion of the general case, we will need the notion of maximal cylinders of translation surfaces. In simple terms, given a closed regular geodesic γ in a translation surface (M, ω), we can 5 Actually, since H 1 ) = π 2 /6. However, we will not insist on this explicit computation because it is not easy to generalize it for moduli spaces of higher genera Abelian differentials. form a maximal cylinder C by collecting all closed geodesics of (M, ω) parallel to γ not meeting any zero of ω. In particular, the boundary of C contains zeroes of ω. Given a maximal cylinder C, we denote by w(C) its width (i.e., the length of its waist curve γ) and by h(C) its height (i.e., the distance across C). For example, in the figure below we illustrate two closed geodesics γ 1 and γ 2 (in the horizontal direction) and the two corresponding maximal cylinders C 1 and C 2 of the L-shaped square-tiled surface of Example 15. In this picture, we see that C 1 has width 2, C 2 has width 1, and both C 1 and C 2 have height 1.
Continuing with the argument for the finiteness of the mass of λ Next, we recall that λ (1) κ was defined via the relative periods. In particular, the set of translation surfaces (M, ω) ∈ H (1) (κ) of genus g ≥ 2 with diameter ≤ C(g) has finite λ (1) κ -measure. Hence, it remains to estimate the λ (1) κ -measure of the set of translation surfaces (M, ω) with diameter ≥ C(g). Here, we recall that there is a maximal cylinder C ⊂ (M, ω) with height h ∼ diam(M, ω).
Since (M, ω) has area one, this forces the width w of C, i.e., the length of a closed geodesic (waist curve) γ in C to be small. By taking γ and a curve ρ across C as a part of the basis of the relative homology of (M, ω), we get two vectors v = γ ω and u = ρ ω with |v ∧ u| ≤ 1 and v small. In other words, we can think of the cusp of H (1) (κ) corresponding to translation surfaces (M, ω) with v small as a subset of the set {(v, u) ∈ R 2 × R 2 : |v ∧ u| ≤ 1}. This ends the sketch of proof of Theorem 26 because the λ (1) κ -measure of cusps is then bounded by In what follows, given any connected component C of some stratum H (1) (κ), we call the SL(2, R)invariant probability measure µ C obtained from the normalization of the (restriction to C) of λ In this language, the global picture is the following: we dispose of a SL(2, R)-action on connected components C of strata H (1) (κ) of the moduli space of Abelian differentials with unit area and a naturally invariant probability µ C (Masur-Veech measure).
Of course, it is tempting to start the study of the statistics of SL(2, R)-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 section this discussion) because this is the appropriate place to introduce the so-called Teichmüller (geodesic) flow.
The Teichmüller flow g t on H (1) g is simply the action of the diagonal subgroup g t := e t 0 0 e −t of SL(2, R). The discussions we had so far imply that g t is the geodesic flow of the Teichmüller metric (introduced in Section 1). Indeed, from Teichmüller's theorem 4, it follows that the path {(M t , ω t ) : t ∈ R}, where ω t = g t (ω 0 ) and M t is the underlying Riemann surface structure such that ω t is holomorphic, is a geodesic of Teichmüller metric d, and d((M 0 , ω 0 ), (M t , ω t )) = t for all t ∈ R (i.e., t is the arc-length parameter).
In Figure 5 above, we saw the action of Teichmüller geodesic flow g t on a Abelian differential ω ∈ H(2) associated to a L-shaped square-tiled surface derived from 3 squares. At a first glance, if the reader forgot the discussion by the end of Section 1, he/she will find (again) the dynamics of g t 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, g t doesn't seem to have any interesting dynamical feature such as recurrence). However, as we already mentioned in by the end of Section 1 (in the genus 1 case), while this is true in Teichmüller spaces T H(κ), this is not exactly true in moduli spaces H(κ): 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 Γ g = Diff + (M )/Diff + 0 (M ) (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 included a picture illustrating this:  We start with the following trivial bundle over Teichmüller space of Abelian differentials T H (1) g : and the trivial (dynamical) cocycle over Teichmüller flow g t : 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 H 1 g . Now, we observe that the modular group Γ g acts on both components of H 1 g by pull-back, and, as we already saw, the action of Teichmüller flow g t commutes with the action of Γ g (since g t acts by post-composition on the local charts of a translation structure while Γ g acts by pre-composition on them). Therefore, it makes sense to take the quotients In the literature, H 1 g is the (real) Hodge bundle over the moduli space of Abelian differentials H g = T H g /Γ g and G KZ t is the Kontsevich-Zorich cocycle 6 (KZ cocycle for short) over Teichmüller flow g t .
We begin by pointing out that the Kontsevich-Zorich cocycle G KZ t (unlike its "parent" G KZ t ) is very far from being trivial. Indeed, since we identify (ω, [c]) with (ρ * (ω), ρ * ([c])) for any ρ ∈ Γ g to construct the Hodge bundle and G KZ t , it follows that the fibers of H 1 g over ω and ρ * (ω) are identified in a non-trivial way if the (standard cohomological) action of ρ on H 1 (M, R) is non-trivial.
Alternatively, suppose we fix a fundamental domain D of Γ g on T H g (e.g., through Veech's zippered rectangle construction) and let's say we start with some point ω at the boundary of D, a cohomology class [c] ∈ H 1 (M, R) and assume that the Teichmüller geodesic through ω points towards the interior of D. Now, we run Teichmüller flow for some (long) time t 0 until we hit again the boundary of D and our geodesic is pointing outwards D. At this stage, from the definition of Kontsevich-Zorich cocycle, we have the "option" to apply an element ρ of the modular group Γ g so that Teichmüller flow through ρ * (g t0 ω) points towards D "at the cost" of replacing the cohomology  , so that this ambiguity problem doesn't concern generic orbits. In any event, as far as Lyapunov exponents are concerned, this ambiguity is not hard to solve. By Hurwitz theorem, #Aut(M, ω) ≤ 84(g − 1) < ∞, so that one can get rid of the ambiguity by taking adequate finite covers of KZ cocycle (e.g., by marking horizontal separatrices of the translation surfaces). See, e.g., [58] for more details and comments on this.
Here we are projecting the picture from the unit cotangent bundle H(∅) = SL(2, R)/SL(2, Z) to the moduli space of torii M 1 = H/SL(2, Z), so that the evolution of the Abelian differentials g t (ω) 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.  know from Oseledets theorem that there are real numbers (Lyapunov exponents) λ µ 1 > · · · > λ µ k and a Teichmüller flow equivariant decomposition H 1 (M, R) = E 1 (ω) ⊕ · · · ⊕ E k (ω) at µ-almost every point ω such that E i (ω) depends measurably on ω and and any choice 7 of . such that log + G KZ ±1 dµ < ∞. If we allow ourselves to repeat each λ µ i accordingly to its multiplicity dimE i (ω), we get a list of 2g Lyapunov exponents λ µ 1 ≥ · · · ≥ λ µ 2g . Such a list is commonly called Lyapunov spectrum (of KZ cocycle with respect to µ). The fact that KZ cocycle is symplectic means that the Lyapunov spectrum is always symmetric with respect to 7 We will see in Remark 31 below that one can choose the so-called Hodge norm here. the origin: Roughly speaking, this symmetry correspond to the fact that whenever θ appears as an eigenvalue of a symplectic matrix A, θ −1 is also an eigenvalue of A (so that, by taking logarithms, we "see" that the appearance of a Lyapunov exponent λ forces the appearance of a Lyapunov exponent −λ). 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 ±1 belong to the Lyapunov spectrum of any ergodic probability µ. Indeed, by the definition, the family of symplectic planes for the entire SL(2, R)-action restricted to these planes (where we replace g t by the corresponding matrices). Since the Lyapunov exponents of the g t action on R 2 are ±1, we get that ±1 belong to the Lyapunov spectrum of KZ cocycle.
Actually, it is possible to prove that λ µ 1 = 1 (i.e., 1 is always the top exponent), and λ µ 1 = 1 > λ µ 2 , i.e., the top exponent has always multiplicity 1, or, in other words, the Lyapunov exponent λ µ 1 is always simple. However, since this requires some machinery (variational formulas for the Hodge norm on the Hodge bundle), we postpone this discussion to Subsection 3.5 below, and we close this subsection by relating this cocycle with the derivative of Teichmüller flow g t (which is one of the interests of KZ cocycle).  Lyapunov exponents ±λ µ i , 1 ≤ i ≤ g of the KZ cocycle. Thus, in resume, the Lyapunov spectrum of Teichmüller flow g t with respect to an ergodic probability measure µ supported on a stratum H(κ) (with κ = (k 1 , . . . , k σ )) has the form Therefore, the KZ cocycle captures the "main part" of the derivative cocycle Dg t , so that, since we're interested in the Ergodic Theory of Teichmüller flow, we will spend sometime in the next sections to analyze KZ cocycle (without much reference to Dg t ).

3.4.
Hodge norm on the Hodge bundle over H (1) g . By definition, the task of studying Lyapunov exponents consists precisely in understanding the growth of norm of vectors. Of course, the particular choice of norm doesn't affect the values of Lyapunov exponents (essentially because two norms on a finite-dimensional vector space are equivalent), but for the sake of our discussion it will be convenient to work with the so-called Hodge norm.
Let M be a Riemann surface. The Hodge (intersection) form (., .) on H 1 (M, C) is given The Hodge form is positive-definite on the space H 1,0 (M ) of holomorphic 1-forms on M , and negative-definite on the space H 0,1 (M ) of anti-holomorphic 1-forms on M . For instance, given a holomorphic 1-form α = 0, we can locally write α(z) = f (z)dz, so that Since dx ∧ dy is an area form on M and |f (z)| 2 ≥ 0, we get that (α, α) > 0. Again, this induces an inner product (., .) and a norm . on the real Hodge bundle H 1 g (R) = H 1 g = (T H (1) g × H 1 (M, R))/Γ g over H (1) g . In the literature, (., .) is the Hodge inner product and . is the Hodge norm on the real Hodge bundle.
Observe that, in general, the subspaces H 1,0 and H 0,1 are not equivariant with respect to the (natural complex version of the) KZ cocycle (on H 1 g (C)), and this is one of the reasons why the Hodge norm . is not preserved by the KZ cocycle in general. In the next subsection, we will study first variation formulas for the Hodge norm along the KZ cocycle and its applications to the Teichmüller flow.   g . Denote by α 0 the holomorphic 1-forms with c = α 0 . By applying the Teichmüller flow g t to ω, we endow M with a new Riemann surface structure such that ω t = g t (ω) is an Abelian differential. In particular, c = α t where α t is a holomorphic 1-form with respect to the new Riemann surface structure associated to ω t .
Of course, by definition, KZ cocycle acts by parallel transport on the Hodge bundle, so that the cohomology classes c are not "changing". However, since the representatives α t we use to "measure" the "size" (Hodge norm) of c are changing, it is an interesting (and natural) problem to know how fast the Hodge norm changes along KZ cocycle, or, equivalently, to compute the first variation of the Hodge norm along KZ cocycle: is the projection in the second factor of the Hodge bundle and . ωt is the Hodge norm with respect to the Riemann surface structure induced by ω t .
In this subsection we will calculate this quantity by following the original article [27]. By working locally outside the zeroes of ω t , we can choose local holomorphic coordinates z t with ω t = dz t . Now, we note that, by definition of the Teichmüller flow, dz t = ω t := e t dx + ie −t dy, where dz = dx + idy, By writing α t = f t ω t , and by taking derivatives, we have locally In particular, since (∂u t /∂z t )dz t + (∂u t /∂z t )dz t := du t , we find that ∂u t /∂z t =ḟ t + f t .
Proof. The first statement of this corollary follows from the main formula in Theorem 29, while the second statement follows from an application of Cauchy-Schwarz inequality: Remark 31. This corollary implies that the KZ cocycle is log-bounded with respect to the Hodge norm, that is, log G KZ ±1 (ω, c) g±1(ω) ≤ 1 for all c ∈ H 1 (M, R) with c ω = 1. Hence, given any finite mass measure µ on H (1) g , we have that Corollary 32. Let µ be any g t -invariant ergodic probability on H g . Then, λ µ 2 < 1 = λ µ 1 .
Proof. By Corollary 30, we have that λ µ 1 ≤ 1. Moreover, since the Teichmüller flow g t (ω) : by Corollary 30. Hence, by integration, By Oseledets theorem and Birkhoff's theorem, for µ-almost every ω ∈ H g , we obtain that This reduces the task of proving that λ µ 2 < 1 to show that Λ + (ω) < 1 for every ω ∈ H (1) g . Here, we proceed by contradiction. Assume that Λ + (ω) = 1 for some Abelian differential ω. By definition, this means that In other words, by looking at the proof of Corollary 30, we have a case of equality in an estimate derived from Cauchy-Schwarz inequality. It follows that, by denoting α(h) = 0 the (M, ω)-holomorphic 1-form with h = [ α(h)], the functions u(h) := α(h)/ω and u(h) = α(h)/ω differ by a multiplicative constant a ∈ C, i.e., Since u(h) is a meromorphic function and, a fortiori, u(h) is an anti-meromorphic function, this is At this stage, one could work more to derive further applications of the Hodge norm to Teichmüller dynamics: for instance, using the Hodge norm it is possible to show some uniform hyperbolicity and quantitative recurrence estimates for the Teichmüller flow g t with respect to any compact set K ⊂ H (1) g , and this information was used by J. Athreya and G. Forni [1] to study deviations of ergodic averages for billiards on rational polygons. However, we will refrain ourselves from doing so because we prefer to give in the next section some interesting applications of the facts derived in this subsection.

Ergodic theory of Teichmüller flow with respect to Masur-Veech measure
Let C be a connected component of a stratum H (1) (κ) of Abelian differentials with unit area, and denote by µ C the corresponding Masur-Veech probability measure. It is not hard to see that an i.e.t. T is determined by a metric data, i.e., lengths of the connected components of I − D T , and combinatorial data, i.e., a permutation π indicating how the connected components of I − D T are "rearranged" after applying T to them. For instance, in the example of picture above where 4 intervals are exchanged, the combinatorial data is the permutation π : {1, 2, 3, 4} → {1, 2, 3, 4} with (π(1), π(2), π(3), π(4)) = (4, 3, 2, 1).
In particular, it makes sense to talk about "almost every" i.e.t.: it means that a certain property holds for almost every choice of metric data with respect to the Lebesgue measure.
By applying their result (Theorem 26), H. Masur [51] and W. Veech [68] deduced that: http://en.wikipedia.org/wiki/Mandelbrot set]). In broad terms, the idea is that given a paramter family of dynamical systems and an appropriate renormalization procedure (defined at least for a significant part of the parameters), one can often infer properties of the dynamical system for "typical parameters" by studying the dynamics of the renormalization.
For the case at hand, we can describe this idea in a nutshell as follows. An i.e.t. T can be "suspended " (in several ways) to a translation surface (M, ω): the two most "popular" ways are Masur's suspension construction and Veech's zippered rectangles construction (cf. Example 14 above). For example, in Figure 11 below, we see a genus 2 surface (obtained by glueing the opposites sides of the polygon marked with the same letter A, B, C or D by translation) presented as a (Masur's) suspension of an i.e.t. with combinatorial data (π(1), π(2), π(3), π(4)) = (4, 3, 2, 1).
To see that this is the combinatorial data of the i.e.t., it suffices to "compute" the return map of vertical translation flow to the special segment in the "middle" of the polygon.
By definition, T is the first return time map of the vertical translation flow of the Abelian differential ω to an appropriate horizontal separatrix associated to some singularity of ω. Here, the vertical translation flow φ ω t associated to a translation surface (M, ω) is the flow obtained by following (with unit speed) vertical geodesics of the flat metric corresponding to ω. In particular, since the flat metric has singularities (in general), φ ω t is defined almost everywhere (as vertical trajectories are "forced" to stop when they hit singular points [zeroes] of ω)! See Figure 11 below for an illustration of these objects. There one can see an orbit through a point q hitting a singularity in finite time (and hence stopping by then) and an orbit through a point p whose orbit never hits a singularity (and hence it can be prolonged forever). In particular, we can study orbits of T by looking at orbits of the vertical flow on (M, ω). Here, the idea is that long orbits of the vertical flow can wrap around a lot on (M, ω), so that a natural procedure is to use Teichmüller flow g t = diag(e t , e −t ) to make the long vertical orbit shorter and shorter (so that it wraps less and less), thus making it reasonably easier to analyse. I.e., one uses Teichmüller flow to renormalize the dynamics of the vertical flow on translation surfaces (and/or i.e.t.'s). Of course, the price we pay is that this procedure changes the shape of (M, ω) (into (M, g t (ω))). But, if the Teichmüller flow g t has nice recurrence properties (so that the shape (M, g t (ω)) is very close to (M, ω) for appropriate choice of large t), one can hope to bypass the difficulty imposed by the change of shape.
In the case of showing unique ergodicity of almost every i.e.t., H. Masur and W. Veech observed that this can be derived from Poincaré's recurrence theorem applied to Teichmüller flow endowed with Masur-Veech measure. Of course, for this application of Poincaré recurrence theorem, it is utterly important to know that Masur-Veech measure is a probability (i.e., it has finite mass), a fact ensured by Theorem 26.
Evidently, this is a very rough sketch of the proof of Theorem 34. For more details, see J.-C.
Notice that the same kind of reasoning as above indicates that the unique ergodicity property must also be true for "almost every" translation flow in the sense that the vertical translation flow on µ C almost every translation surface structure (M, ω) ∈ C is uniquely ergodic. Indeed, the following theorem (again by H. Masur [51] and W. Veech [68]) says that this is the case:  [14].
In the sequel, we will present a sketch of proof of this result based on the recurrence of Teichmüller flow, and the simplicity of the top exponent 1 = λ µ C 1 > λ µ C 2 (see Corollary 32 above). We start by assuming that the vertical translation flow φ ω t of our translation surface (M, ω) is minimal, that is, every orbit defined for all times t ≥ 0 are dense: this condition is well-known to be related to the absence of saddle connections (see, e.g., J.-C. Yoccoz survey [72]), and the last property has full measure (since the presence of saddle connections for (M, e iθ ω) corresponds to a countable set of directions θ ∈ R, and the Masur-Veech measure µ C is natural). Now, given an ergodic φ ω t -invariant probability µ, consider x ∈ M a µ-typical point, and T ≥ 0. Let γ T (x) ∈ H 1 (M, R) be the homology class obtained by "closing" the piece of (vertical) trajectory T ]} with a bounded (usually small ) segment connecting x to φ ω T (x). A well-known theorem of Schwartzman [66] says that In the literature, ρ(µ) is called Schwartzman asymptotic cycle. By Poincaré duality, the Poincaré dual of ρ(µ) gives us a class c(µ) ∈ H 1 (M, R) − {0}. Geometrically, c(µ) is related to the flux of φ ω t through transverse closed curves γ with respect to µ. More precisely, given γ a closed curve transverse to φ ω t , the flux is For the sake of the subsequent discussion, we recall that any φ ω t -invariant probability µ induces a transverse measure µ on pieces of segments δ transverse to φ ω t : indeed, we define µ(δ) by the flux through δ, i.e., lim φ ω t (δ))/t. Since φ ω t is simply a translation along the leaves of the vertical foliation of ω, we see that µ can be locally written as Leb × µ in any "product" open set of the form φ ω s (δ) not meeting singularities of φ ω t (where δ is a transverse segment).
We claim that the map µ → c(µ) is injective. Indeed, given two ergodic φ ω t -invariant probabilities µ 1 and µ 2 with c(µ 1 ) = c(µ 2 ), we observe that the transverse measures µ 1 and µ 2 induced by them on a closed curve γ transverse to φ ω t differ by the derivative of a continuous function U on γ. Indeed, U can be obtained by integration: by fixing an "origin" 0 ∈ γ and an orientation on γ, we is the segment of γ going from 0 to x in the positive sense (with respect to the fixed orientation). Of course, the fact that U is well-defined 9 is guaranteed by the assumption c(µ 1 ) = c(µ 2 ). Now, we note that U is invariant under the return map induced by φ ω t , so that, by minimality of φ µ t , we conclude that the continuous function U must be constant. Therefore, µ 1 = µ 2 , i.e., µ 1 and µ 2 have the same transverse measures. Since µ 1 and µ 2 are the Lebesgue measure along the flow direction, we obtain that µ 1 = µ 2 , so that the claim is proved.
Next, we affirm that c(µ) (or equivalently ρ(µ)) decays exponentially fast like e −t under KZ cocycle whenever the Teichmüller flow orbit g t (ω) of ω is recurrent. Indeed, let us fix t ≥ 0 such that g t (ω) is very close to ω, and we consider the action of KZ cocycle G KZ t on ρ(µ). Since, by On the other hand, since g t contracts the vertical direction by a factor of e −t and γ T (x) is essentially a vertical trajectory (except for a bounded piece of segment connecting x to φ ω T (x)), we get where . θ is the stable norm on H 1 (M, R) with respect to θ (obtained by measuring the length of [primitive] closed curves [i.e., elements of H 1 (M, Z)] using the flat structure induced by θ and extending this "by linearity"). In the previous calculation, we implicitly used the fact that g t (ω) is very close to ω, so that the stable norms . gt(ω) and . ω are comparable by definite factors, and thus the factor of 1/T can "kill " eventual (bounded ) error terms coming from the "closing" procedure used to define γ T (x). Therefore, our affirmation is proved.
Finally, we note that the fact 1 = λ µ C 1 > λ µ C 2 (i.e., simplicity of the top KZ cocycle exponent, see Corollary 32 above) means that there is only one direction in H 1 (M, R) which is contracted like e −t ! (namely, R · [Im(ω)]) Therefore, given ω with minimal vertical translation flow and recurrent Teichmüller flow orbit, any φ ω t -invariant ergodic probability µ satisfies c(µ) ∈ R · [Im(ω)]. Since φ ω t preserves the Lebesgue measure Leb (flat area induced by ω), we obtain that any φ ω t -invariant ergodic probability µ is a multiple of Leb, and, a fortiori, µ = Leb. Thus, φ ω t is uniquely ergodic for such ω's. Since we already saw that µ C almost everywhere the vertical translation flow is minimal, we have only to show that µ C -almost every ω is recurrent under Teichmüller flow to complete the proof of Theorem 35, but this is immediate from Poincaré's recurrence theorem (since Teichmüller flow preserves the Masur-Veech measure µ C , a finite mass measure). 9 I.e., it produces the same value for U (0) when we go around γ.

Ergodicity of Teichmüller flow.
In the fundamental papers [51] and [69], H. Masur and W. Veech independently showed the following result: Theorem 37 (H. Masur, W. Veech). The Teichmüller geodesic flow g t is ergodic, and actually mixing, with respect to µ C .
Concerning the first part of the statement, we observe that the ergodicity of the Teichmüller flow g t is essentially a consequence of the simplicity of the top exponent 1 = λ µ C 1 > λ µ C 2 and the existence of nice ("long") stable and unstable manifolds for g t . Indeed, as we already know, the simplicity of the top exponent λ µ C 1 implies that, except for the zero Lyapunov exponent coming from the flow direction, the Teichmüller flow g t has no other zero exponent (since 1 − λ µ 2 > 0 is the second smallest non-negative exponent). In other words, the Teichmüller flow is non-uniformly hyperbolic in the sense of the Pesin theory. This indicates that Hopf 's argument may apply in our context. Recall that Hopf's argument starts by observing that ergodic averages are constant along stable and unstable manifolds: more precisely, given a point x such that the ergodic average exists for a (uniformly) continuous observable ϕ : C → R, then the ergodic averages ϕ(y) := lim t→+∞ 1 t t 0 ϕ(g s (y))ds exists and ϕ(y) = ϕ(x) for any y in the stable manifold W s (x) of x. Actually, since y ∈ W s (x), we have lim s→+∞ d(g s (y), g s (x)) = 0, so that, by the uniform continuity of ϕ, the desired claim follows.
Of course, a similar result for ergodic averages along unstable manifolds holds if we replace t → +∞ by t → −∞ in the definition of ϕ. Now, the fact that we consider "future" (t → +∞) ergodic averages along stable manifolds and "past" (t → −∞) ergodic averages along unstable manifolds is not a major problem since Birkhoff's ergodic theorem ensures that these two "types" of ergodic averages coincide at µ C almost every point.
In particular, since the ergodicity of µ C is equivalent to the fact that ϕ is constant at µ C almost every point, if one could access any point y starting from any point x using pieces of stable and unstable manifolds like in Figure 12 below, we would be in good shape (here, we're skipping some details because Hopf's argument needs that the intersection points appearing in Figure 12 to satisfy Birkhoff's ergodic theorem; in general, this is issue is strongly related to the so-called absolute continuity property of the stable and unstable manifolds, but this is not a problem in our context since Pesin's theory ensures absolute continuity of W s and W u ).
However, it is a general fact that Pesin theory of non-uniformly hyperbolic systems only provides the existence of short stable and unstable manifolds. Even worse, the function associating to a typical point the size of its stable/unstable manifolds is only measurable. In particular, the nice W S W S W U x y Figure 12. A point y which is accessible from x by stable and unstable manifolds. scenario drew below may not happen in general (and actually the best Hopf's argument [alone] can do is to ensure the presence of a countable number of ergodic components [at most]).
Fortunately, in the specific case of Teichmüller flow, one can determine explicitly the stable and unstable manifolds: since g t acts on ω by multiplying [Re(ω)] by e t and Im(ω) by e −t , we infer In particular, we see that these invariant manifolds are "large" subsets corresponding to affine subspaces in period coordinates. Therefore, the potential problem pointed out in the previous paragraph doesn't exist, and one can proceed with Hopf's argument to eventually derive the ergodicity of Teichmüller flow with respect to Masur-Veech measure µ C .
Concerning the second part of the statement of this theorem, we should say that the mixing property of Teichmüller flow is a consequence of its ergodicity and the mere existence of the SL(2, R)-action: indeed, while ergodicity alone doesn't imply mixing in general (e.g., irrational rotations of the circle are ergodic but not mixing), the fact that Teichmüller flow is part of a whole SL(2, R)-action permits to derive mixing from ergodicity in view of the nice representation theory of SL(2, R). We discuss this together with the exponential mixing property of Teichmüller flow in the next subsection.

4.3.
Exponential mixing (and spectral gap of SL(2, R) representations). Generally speaking, we say that a flow (φ t ) t∈R on a space X is mixing with respect to an invariant probability µ for every f, g ∈ L 2 (X, µ). Of course, the mixing property always implies ergodicity of (φ t ) t∈R but the converse is not always true (e.g., irrational translation flows on the torus T 2 = R 2 /Z 2 are ergodic but not mixing). However, as we're going to see in a moment, when the flow (φ t ) t∈R is part of a larger SL(2, R) action, it is possible to show that ergodicity implies mixing.
More precisely, suppose that we have a SL(2, R) action on a space X preserving a probability measure µ, and let (φ t ) t∈R be the flow on X corresponding to the action of the diagonal subgroup diag(e t , e −t ) of SL(2, R). In this setting, one has: Of course, the Teichmüller flow g t on a connected component C of a stratum of the moduli space of Abelian differentials equipped with its natural (Masur-Veech) probability measure µ C is a prototype example of flow verifying the assumptions of the previous proposition.
As we pointed out above, the proof of this result uses knowledge of the representation theory of SL(2, R). We strongly recommend reading Livio Flaminio's notes in this volume for a nice discussion of this subject. For sake of convenience, we quickly reviewed some results on this topic in Appendix A below. In particular, we will borrow the notations from this Appendix.
We begin by observing that the SL(2, R) action on (X, µ) induces an unitary representation is the Hilbert space of L 2 functions of (X, µ) with zero mean. In particular, from the semisimplicity of SL(2, R), we can write H as a integral of irreducible unitary SL(2, R) representations H ξ : The fact that (φ t ) t∈R is µ-ergodic implies that the SL(2, R) action is µ-ergodic, that is, the trivial representation doesn't appear in the previous integral decomposition. By Bargmann's classification, every nontrivial unitary irreducible SL(2, R) representation belongs to one of the following three classes (or series): principal series, discrete series and complementary series. See Livio Flaminio's notes in this volume and/or Appendix A below for more discussion.
By M. Ratner's work [63], we know that, for every t ≥ 1 and for every v, w ∈ H ξ with H ξ in the principal or discrete series, where C > 0 is an universal constant. Of course, we're implicitly using the fact that, by hypothesis, φ t is exactly the action of the diagonal subgroup diag(e t , e −t ) of SL(2, R) on X. Also, for every t ≥ 1 and for every v, w ∈ H ξ C 3 vectors (see Appendix A for more details) with H ξ in the complementary series, one can find a parameter s = s(H ξ ) ∈ (0, 1) (related to the eigenvalue where C s > 0 is a constant depending only on s and u C 3 is the C 3 norm of a C 3 vector u along the SO(2, R) direction. In the notation of Appendix A, . Furthermore, C s can be taken uniform on intervals of the form s ∈ [1 − s 0 , s 0 ] with 1/2 < s 0 < 1.
Putting these informations together (and using the classical fact that C 3 vectors are dense), one obtains that |C t (v, w)| → 0 as t → ∞ (actually, it goes "exponentially fast" to zero in the sense explained above) for: • all vectors v, w ∈ H ξ when H ξ belongs to the principal or discrete series; • a dense subset (e.g., C 3 vectors) of vectors v, w ∈ H ξ when H ξ belongs to the complementary series.
Using this (and the integral decomposition Hence, (φ t ) t∈R is µ-mixing and the proof of Proposition 38 is complete.
Once the Proposition 38 is proved, a natural question concerns the "speed"/"rate" of convergence of C t (f, g) to zero (as t → ∞). In a certain sense, this question was already answered during the proof of Proposition 38: using Ratner's results [63], one can show that C t (f, g) converges exponentially fast to zero for all f, g in a dense subet of L 2 0 (X, µ)(e.g., f, g C 3 vectors) if and only if the unitary SL(2, R) representation H = L 2 0 (X, µ) has spectral gap, i.e., there exists s 0 ∈ (0, 1) such that, when writing H as an integral H = H ξ dλ(ξ) of unitary irreducible SL(2, R) represenations, no H ξ in the complementary series has parameter s = s(H ξ ) ∈ (s 0 , 1). Actually, it is possible to show that the spectral gap property is equivalent to the nonexistence of almost invariant vectors: recall that a representation of a Lie group G on a Hilbert space H has almost vector when, for all compact subsets K and for all ε > 0, there exists an unit vector v ∈ H such that gv − v < ε for all g ∈ K.
In general, it is a hard task to prove the spectral gap property for a given unitary SL(2, R) Yoccoz showed the following theorem: Theorem 39 (A.Ávila, S. Gouëzel, J.-C. Yoccoz). The Teichmüller flow g t on C is exponentially mixing with respect to µ C (in the sense that C t (f, g) → 0 exponentially as t → ∞ for "sufficiently smooth" f, g), and the unitary SL(2, R) representation L 2 0 (C, µ C ) has spectral gap.
In the proof of this result,Ávila, Gouëzel and Yoccoz [5] proves firstly that the Teichmüller geodesic flow (i.e., the action of the diagonal subgroup A = {a(t) : t ∈ R} on the moduli space Q g of Abelian differentials) is exponentially mixing with respect to Masur-Veech measure (indeed this is the main result of their paper) and they use a reverse Ratner estimate to derive the spectral gap property from the exponential mixing (and not the other way around!). Here, the proof of the exponential mixing property with respect to Masur-Veech measure is obtained by delicate (mostly combinatorial) estimates on the so-called Rauzy-Veech induction.
More recently,Ávila and Gouëzel [4] developed a more geometrical (and less combinatorial) approach to the exponential mixing of algebraic SL(2, R)-invariant probabilities.
Roughly speaking, an algebraic SL(2, R)-invariant measure µ is a probability measure supported on an affine suborbifold supp(µ) of C (in the sense that supp(µ) corresponds, in local period coordinates, to affine subspaces in relative homology) such that µ is absolutely continuous (wrt the Lebesgue measure on the affine subspaces corresponding to supp(µ) in period charts) and its density After the celebrated works of K. Calta [13] and C. McMullen [61], there is a complete classification of SL(2, R)-invariant measures in genus 2 (i.e., C = H(2) or H(1, 1)). In particular, it follows that such measures are always algebraic (in genus 2). Furthermore, it was recently announced by A. Eskin and M. Mirzakhani [24] that the full conjecture is true.
In any case, the result obtained byÁvila and Gouëzel [4] is:  [75], [47] performed several numerical experiments leading them to conjecture that the Lyapunov spectra of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures µ C are simple, i.e., the multiplicity of each Lyapunov exponent λ µ C i , i = 1, . . . , 2g is 1: As we discussed in the previous section, the Kontsevich-Zorich cocycle G KZ t is symplectic, so that its Lyapunov exponents (with respect to any invariant ergodic probability µ) are symmetric with respect to the origin: λ µ 2g−i = −λ µ i+1 . Also, the top Lyapunov exponent 1 = λ µ 1 is always simple (i.e., λ µ 1 > λ µ 2 ). Therefore, the Kontsevich-Zorich conjecture is equivalent to In 2002, G. Forni [27] was able to show that λ µ C g > 0 via second variational formulas for the Hodge norm and certain formulas for the sum of the Lyapunov exponents of the KZ cocycle (inspired by M. Kontsevich's work). In Subsection 5.2 below, we'll illustrate some of G. Forni's techniques by showing the positivity of the second Lyapunov exponent λ µ C 2 of the KZ cocycle with respect to Masur-Veech measure µ C . While the fact λ µ C 2 > 0 is certainly a weaker statement than Forni's theorem λ µ C g > 0, it turns out that it is sufficient to some interesting applications to interval exchange transformations and vertical translation flows. Indeed, using a technical machinery of parameter exclusion strongly based on the fact that λ µ C 2 > 0, A.Ávila and G. Forni [3] were able to show that almost every i.e.t. (not corresponding to "rotations") and almost every vertical translation flow (on genus g ≥ 2 translation surfaces) are weakly mixing. Here, we say that an i.e.t. corresponds to a rotation if its combinatorial data π : {1, . . . , d} → {1, . . . , d} has the form In this case, one can see that the corresponding i.e.t. can be conjugated to a rotation of the circle, and hence it is never weak-mixing. Observe that, in general, weak-mixing property is the "best" dynamical property we can expect: indeed, as it was shown by A. Katok [44], interval exchange transformations and suspension flows over i.e..t's with a roof function of bounded variation (e.g., translation flows) are never mixing. We will come back to this point later in this section.
In 2007, A.Ávila and M. Viana [7] proved the full Kontsevich-Zorich conjecture by studying a discrete-time analog of Kontsevich-Zorich cocycle over the Rauzy-Veech induction. In few words, Avila and Viana showed that the symplectic monoid associated to Rauzy-Veech induction is pinching ("it contains matrices with simple spectrum") and twisting ("any subspace can be put into generic position by using some matrix of the monoid"), and they used the pinching and twisting properties to ensure simplicity of Lyapunov spectra. In a certain sense, these conditions (pinching and twisting) are analogues (for deterministic chaotic dynamical systems) of the strong irreducibility and proximality conditions (sometimes derived from a stronger Zariski density property) used by Y. Guivarch and A. Raugi [37], and I. Goldsheid and G. Margulis [36] to derive simplicity of Lyapunov exponents for random products of matrices.
Remark 41. More recently, G. Forni extended some techniques of his article [27] to prove in [29] a geometric criterion for the non-uniform hyperbolicity of KZ cocycle (i.e., λ µ g > 0) of "general" SL(2, R)-invariant ergodic probability measures µ (see Remark 58 below). As a matter of fact, this general recent criterion strictly includes Masur-Veech measures, but it doesn't allow to derive simplicity of the Lyapunov spectrum in general (see the appendix to [29] for more details). Also, it was recently shown by V. Delecroix and the second author [18] that there is no converse 12 to G.
Forni's criterion. Here, the arguments of V. Delecroix and the second author are based on a recent criterion for the simplicity of the Lyapunov exponents of KZ cocycle with respect to SL(2, R)invariant ergodic probabilities supported on the SL(2, R)-orbits of square-tiled surfaces due to M.
Möller, J.-C. Yoccoz and the second author [57]. For more comments on this, see Section D below.
As the reader can imagine, the Kontsevich-Zorich conjecture has applications to the study of For the case of vertical translation flows, we begin with a typical vertical translation flow φ ω t on a translation surface (M, ω) (so that it is uniquely ergodic) and we choose a typical point p (so that φ ω t is defined for every time t), e.g., as in Figure 11 above. For all T > 0 large enough, let us denote by γ T (x) ∈ H 1 (M, R) the homology class obtained by "closing" the piece of (vertical) For genus g = 1 translation surfaces (i.e., flat torii), this is very good and fairly complete result: indeed, it is not hard to see that the deviation of γ T (x) from the line E 1 := R · c spanned by the Schwartzman asymptotic cycle is bounded.
For genus g = 2 translation surfaces, the global scenario gets richer: by doing numerical experiments, what one sees is that the deviation of γ T (x) from the line E 1 has amplitude T λ2 with λ 2 < 1 around a certain line. In other words, the deviation of γ T (x) from the Schwartzman asymptotic cycle is not completely random: it occurs along an isotropic 2-dimensional plane containing E 1 . Again, in genus g = 2, this is a "complete" picture in the sense that numerical experiments indicate that the deviation of γ T (x) from E 2 is again bounded.
More generally, for arbitrary genus g, the numerical experiments indicate that existence of an asymptotic Lagrangian flag, i.e., a sequence of isotropic subspaces For instance, the reader can see below two pictures (Figures 13 and 14) extracted from A. Zorich's survey [74] and showing numerical experiments related to the deviation phenomenon or Zorich phenomenon discussed above in a genus 3 translation surface. There, we have a slightly different notation for the involved objects: c n denotes γ Tn (x) for a convenient choice of T n , the subspaces V i correspond to the subspaces E i , and the numbers ν i correspond to the numbers λ i .   We dedicate this subsection to give a sketch of proof of the following result: Theorem 42. Let C be a connected component of some stratum of H g and denote by µ C the corresponding Masur-Veech measure. Then, λ µ C 2 > 0.
As we already mentioned, this result is part of one of the main results of [27] showing that However, we'll not discuss the proof of the more general result λ µ C g > 0 because • one already finds several of the ideas used to show λ µ C g > 0 during the sketch of proof of λ µ C 2 > 0, and • by sticking to the study of λ µ C 2 > 0 we avoid the (rather technical) discussion of characterizing Oseledets unstable subspaces of KZ cocycle via basic currents.
In any event, we start the sketch of proof of Theorem 42 by recalling (from Subsection 3.5, Theorem 29) that the form B ω (α, β) := i 2 αβ ω ω, α, β ∈ H 1,0 (M ), is relevant in the study of first variation of the Hodge norm in view of the formula: is KZ cocycle invariant and it contributes with the ±1 = ±λ µ 1 Lyapunov exponents. Hence, since KZ cocycle preserves the symplectic intersection form, we get that the Lyapunov exponents λ µ i , 2 ≤ i ≤ g come from the restriction of KZ cocycle to H 1 (0) (M, R).
Hodge representation theorem (cf. Subsection 3.4), we obtain the following nice immediate consequence of this discussion: 13 Note that it depends real-analytically (in particular continuously) on ω.
Geometrically, B ω is essentially the second fundamental form (or Kodaira-Spencer map) of the holomorphic subbundle H 1,0 of the complex Hodge bundle H 1 C equipped with the Gauss-Manin connection. Roughly speaking, recall that the second fundamental form II ω : See, e.g., [33] for more discussion on this differential-geometrical interpretation 14 of B.
Next, by taking {ω 1 , . . . , ω g } a Hodge-orthonormal basis of C equipped with the Gauss-Manin connection (see [33] for more details), i.e., the matrix H also a differential-geometrical interpretation (similarly to B). In particular, this geometrical interpretation hints that H should naturally enter into second variation formulas for the Hodge norm 15 and, a fortiori, the eigenvalues of H should provide nice consequences to the study of Lyapunov exponents. In fact, as it was proposed by M. Kontsevich [47] and proved by G. Forni [27], one can relate the eigenvalue of H to Lyapunov exponents of KZ cocycle via the following formula: Theorem 44 (M. Kontsevich, G. Forni). Let µ be a SL(2, R)-invariant g t -ergodic probability on a connected component C of some stratum of H g . Then, one has the following formula for the sum of non-negative Lyapunov exponents of KZ cocycle with respect to µ: Remark 45. Since B ω (ω, ω) := 1, one can use the argument (Cauchy-Schwarz inequality) of the proof of Corollary 30 to see that Λ 1 (ω) ≡ 1 for all ω. In particular, since λ µ 1 = 1, one can rewrite 14 A word of caution: the second fundamental form Aω(c) considered in [33] differs from II ω(c) by a sign, i.e., Of course, this should be compared with the fact that B naturally enters into first variation formulas for the Hodge norm.
the formula above as Remark 46. Note that there is an important difference in the hypothesis of Theorem 29 and Theorem 44 is: in the former µ is any g t -invariant while in the latter µ is SL(2, R)-invariant! Before giving a sketch of proof of Theorem 44, we observe that from it (and Remark 45) one can immediately deduced the following "converse" to Corollary 43: Evidently, this corollary shows how one can prove Theorem 42: since Masur-Veech measures µ C are fully supported, it suffices to check that rank(B . In other words, by assuming Theorem 44, we just saw that: Now, before trying to use this corollary, let's give an outline of proof of Theorem 44.
Sketch of proof of Theorem 44.
where I k is a k-dimensional isotropic subspace of the real Hodge bundle H 1 R and {c 1 , . . . , c k } is any 16 Hodge-orthonormal basis of I k .
In the sequel, we will use the following three lemmas (see [27] or [33] for proofs and more details).
Lemma 49 (Lemma 5.2' of [27]). Let {c 1 , . . . , c k , c k+1 , . . . , c g } be any Hodge-orthonormal completion of {c 1 , . . . , c k } into basis of a Lagrangian subspace of H 1 (M, R). Then, Remark 50 (M. Kontsevich's fundamental remark). In the extremal case k = g, the right-hand side of the previous equality doesn't depend on the Lagrangian subspace I g : This fundamental observation of Maxim Kontsevich lies at the heart of the main formula of Theorem 44. 16 Of course, it is implicit here that the expression 2 It is not hard to see that the notion of Hodge norm . ω on vectors c ∈ H 1 (M, R) can be extended to any polyvector c 1 ∧ · · · ∧ c k coming from a (Hodge-orthonormal) basis {c 1 , . . . , c k } of an isotropic subspace I k . By slightly abusing of the notation, we will denote by c 1 ∧ · · · ∧ c k ω the Hodge norm of such a polyvector.
Note that the Hodge norm . ω depends only on the complex structure, so that . ω = . ω whenever ω = constant · ω. In particular, it makes sense to consider the Hodge norm . h over the Teichmüller disk h ∈ SO(2, R)\SL(2, R) · ω. For subsequent use, we denote by ∆ hyp the hyperbolic (leafwise) Laplacian on SO(2, R)\SL(2, R) · ω (here, we're taking advantage of the fact . Finally, in order to connect the previous two lemmas with Oseledets theorem (and Lyapunov exponents), one needs the following fact about hyperbolic geometry: at the origin 0 ∈ D and area P is Poincaré's area form on D.
Next, the idea to derive Theorem 44 from the previous three lemmas is the following. Denote by R θ = cos θ − sin θ sin θ cos θ , and, given ω ∈ H g , for SO(2, R)\SL(2, R) · ω h = g t R θ ω = (t, θ), let L(h) := c 1 ∧ · · · ∧ c k h . In plain terms, L is measuring how the (Hodge norm) size of the polyvector c 1 ∧ · · · ∧ c k changes along the Teichmüller disk of ω. In particular, as we're going to see in a moment, it is not surprising that L has "something to do" with Lyapunov exponents.
By Lemma 52, one has Then, by integrating with respect to the t-variable in the interval [0, T ] and by using Lemma 51 for the computation of ∆ hyp L, one deduces At this point, by taking an average with respect to µ and using the SL(2, R)-invariance of µ to get rid of the integration with respect to θ, we deduce that Now, we observe that: • by Oseledets theorem, for a "generic" isotropic subspace I k and µ-almost every ω, one has that 1 T L(g T (ω)) converges 17 to λ µ 1 + · · · + λ µ k as T → ∞, and • by Remark 50, for k = g, Φ g (ω, I g ) = Φ g (ω) = Λ 1 (ω) + · · · + Λ g (ω) is independent on I g .
So, for k = g, this discussion 18 allows to show that This completes the sketch of proof of Theorem 44.
Remark 53. Essentially the same argument above allows to derive formulas for partial sums of Lyapunov exponents. More precisely, given µ a SL(2, R)-invariant g t -ergodic probability with where E + k (ω) is the Oseledets subspace associated to the k top Lyapunov exponents. In general, this formula is harder to use than Theorem 44 because the right-hand side of the former implicitly assumes some a priori control of E + k (ω) while the right-hand side of the latter is independent of Lagrangian subspaces (as noticed by M. Kontsevich).
After obtaining Theorem 44, we're ready to use Corollary 47 to reduce the proof of Theorem 42 to the following theorem: Theorem 54. In any connected component C of a stratum of H g one can find some ω ∈ C with Roughly speaking, the basic idea (somehow recurrent in Teichmüller dynamics) to show this result is to look for ω near the boundary of C after passing to an appropriate compactification.
More precisely, one shows that, by considering the so-called Deligne-Mumford compactification there exists an open set U ⊂ C near some boundary point ω ∞ ∈ ∂C such that rank(B R ω | H 1 (0) (M,R) ) = 2g − 2 for any ω ∈ U "simply" because the "same" is true for ω ∞ . A complete formalization of this idea is out of the scope of these notes as it would lead us to a serious discussion of Deligne-Mumford compactification, some variational formulas of J. Fay and A. Yamada, etc. Instead, we offer below a very rough sketch of proof of Theorem 54 based on some "intuitive" properties of Deligne-Mumford compactification (while providing adequate references for the omitted details).
The first step towards finding the boundary point ω ∞ is to start with the notion of Abelian differentials with periodic Lagrangian horizontal foliation: 17 Recall that, by definition, the function t → L(gt(ω)) is measuring the growth (in Hodge norm) of the polyvector c 1 ∧ · · · ∧ c k along the Teichmüller orbit gt(ω). 18 Combined with an application of Lebesgue dominated convergence theorem and the fact that tanh(t)/area(Dt) → 1 as t → ∞. See [27] and [33] for more details.

Definition 55.
Let ω be an Abelian differential on a Riemann surface M . We say that the horizontal foliation F hor (ω) := { ω = constant} is periodic whenever all regular leaves of F hor (ω) are closed, i.e., the translation surface (M, ω) = ∪ i C i can be completely decomposed into maximal cylinders C i corresponding to some closed regular geodesics γ i in the horizontal direction.
The homological dimension of ω with periodic horizontal foliation is the dimension of the (isotropic) subspace of H 1 (M, R) generated by the waist curves γ i of the horizontal maximal cylinders C i decomposing (M, ω).
We say that ω has periodic Lagrangian horizontal foliation whenever its homological dimension is maximal (i.e., g).
In general, it is not hard to find Abelian differentials with periodic horizontal foliation: for instance, any square-tiled surface (see Example 15) verifies this property and the class of squaretiled surfaces 19 is dense on C.
Next, we claim that: Lemma 56. C contains Abelian differentials with periodic Lagrangian horizontal foliation.
Proof. Of course, the lemma follows once we can show that given ω ∈ C with homological dimension k < g, one can produce an Abelian differential ω with homological dimension k+1. In this direction, given such an ω, we can select a closed curve γ disjoint from (i.e., zero algebraic intersection with) the waist curves γ i of horizontal maximal cylinders C i of (M, ω) and γ = 0 in H 1 (M, R).
Then, let's denote by [df ] ∈ H 1 (M, Z) the Poincaré dual of γ given by taking a small tubular neighborhoods V ⊂ U of γ and taking a smooth function f on M − γ such that with respect to its orientation of γ (see the figure below) 19 As square-tiled surfaces (M, ω) are characterized by the rationality of their periods (i.e., R γ ω ∈ Q ⊕ iQ for any γ ∈ H 1 (M, Σ, Z)). See [38] for more details.
In this setting, since the waist curves γ i of maximal cylinders of C i of ω generate a k-dimensional is possible to check (see the proof of Lemma 4.4 of [27]) that the Abelian differential ω = ω + r[df ] has homological dimension k + 1 whenever r ∈ Q − {0} is sufficiently small.
This completes the proof of the lemma. Now, let's fix ω ∈ C with periodic Lagrangian horizontal foliation and let's try to use ω to reach some nice boundary point ω ∞ on the Deligne-Mumford compactification of C (whatever this means...). Intuitively, we note that horizontal maximal cylinders C i of ω and their waist curves γ i looks like this In particular, by applying Teichmüller flow g t = diag(e t , e −t ) and letting t → −∞, we start to pinching off the waist curves γ i . As it was observed by H. Masur (see Section 4 of [27] and references therein), by an appropriate scaling process on ω t = g t (ω), one can makes sense of a limiting object ω ∞ in the Deligne-Mumford compactification of C looking like this: Roughly speaking, this picture is intended to say that ω ∞ lives in a stable curve M ∞ , i.e., a Riemann surface with nodes at the punctures p i obtained after pinching γ i 's off, and it is a meromorphic quadratic differential with double poles (and strictly positive residues) at the punctures and the same zeroes of ω t .
If ω has homological dimension g, it is possible to check that ω ∞ lives in a sphere with 2g paired punctures and ω has strictly positive residues on each of them. In this situation, certain variational formulas 20 of J. Fay and A. Yamada allowing to show that as ω t approaches ω ∞ , one has Hodge-orthonormal basis of the dual of the (g-dimensional) subspace of H 1 (M, R) generated by the waist curves γ i 's of ω. In other words, up to orthogonal matrices, the matrix of the form B ωt approaches −Id g×g as t → −∞. Hence, rank(B R ωt ) := 2 · rank(B ωt ) = 2g as t → −∞, and, a fortiori, the rank of Thus, this completes the sketch of proof of Theorem 54.
Remark 57. Actually, the fact that B ωt "approaches" −Id g×g can be used to show that In a nutshell, the previous discussion around Theorem 54 can be resumed as follows: firstly, we searched (in C) some ω with periodic Lagrangian horizontal foliation; then, by using the Teichmüller flow orbit ω t = g t (ω) of ω and by letting t → −∞, we spotted an open region U of C (near a certain "boundary" point ω ∞ ) where the form B becomes an "almost" diagonal matrix with non-vanishing diagonal terms, so that the rank of B is maximal. Here, we "insist" that the inspiration for spotting U near the boundary of C comes from the fact that B is a sort of derivative of the so-called period matrix Π, and one knows since the works of J. Fay and A. Yamada that the period matrix Π (and therefore B) has nice asymptotic expansions near the boundary of C. The following picture is a résumé of the discussion of this paragraph: 20 Here, it is implicit the fundamental fact that B can be interpreted as the "derivative of the period matrix".
See Section 4 of [27] for more comments.  Forni [29] to give the following far-reaching criterion for the non-uniform hyperbolicity of KZ cocycle with respect to a SL(2, R)-invariant g t -ergodic probabilities µ (satisfying a certain local product structure property): if one can find ω in the support of µ with periodic Lagrangian horizontal foliation (i.e., there is some ω ∈ supp(µ) with homological dimension g), then λ µ g > 0.
5.3. Weak mixing property for i.e.t.'s and translation flows. The plan for this subsection is to vaguely sketch how the knowledge of the positivity of the second Lyapunov exponent λ µ C 2 of KZ cocycle with respect to Masur-Veech measures µ C on connected components C of strata was used by A. Avila and G. Forni [3] to show weak mixing property for i.e.t.'s and translation flows.
The basic references for the subsection are the original article [3] and the survey [30].
Recall that a dynamical system T : X → X preserving a probability µ is weak mixing whenever For the case of i.e.t.'s and translation flows, it is particularly interesting to consider the following spectral characterization of weak mixing. 21 I.e., lim Similarly, a (vertical) translation flow φ ω s on a translation surface (M, ω) represented by the suspension of an i.e.t. T : This spectral characterization of weak mixing allowed W. Veech to setup a criterion of weak mixing for i.e.t.'s and translation flows: roughly speaking, in the case of translation flows φ ω s , it says that if φ ω s is not weak mixing, say the equation has a non-constant measurable solution f for some t ∈ R, then, by considering the times when the Teichmüller orbit of the translation surface comes back near itself, i.e., the times t n such that Actually, this is a very crude approximation of Veech's criterion: the formal statement depends on the relationship between Teichmüller flow/KZ cocycle and Rauzy-Veech-Zorich algorithm, and we will not try to recall it here.
Instead, we will close this subsection by saying that the idea to deduce weak mixing for "almost every" i.e.t.'s and translation flows is to carefully analyze the KZ cocycle in order to prove that G KZ t "tends" to keep "typical" lines t · h ∈ R · h ⊂ H 1 (M, Σ, R) in homology sufficiently "far away" from the integral lattice H 1 (M, Σ, Z) when the second Lyapunov exponent λ µ C 2 (with respect to Masur-Veech measures µ C ) is positive. In other words, one of the (many) key ideas in [3] is to show that Equation 5.1 can be contradicted for "almost every" i.e.t.'s and translation flows when λ µ C 2 > 0, so that Veech's criterion implies weak mixing property for "almost every" i.e.t.'s and translation flows.

Veech's question
As we saw in the previous chapter, after the works of G. Forni [27], and A. Avila and M. Viana Therefore, it is natural to ask how much of discussion of the previous chapter still applies to other (Teichmüller and/or SL(2, R)) invariant measures. In this direction, after the completion of the work [27], W. Veech asked G. Forni whether the positivity of the gth top exponent (i.e., λ µ g > 0) and/or the "simplicity scenario" from the Kontsevich-Zorich conjecture can be extended to arbitrary SL(2, R)-invariant probability measures µ.
The reader maybe wondering why W. Veech doesn't also include all Teichmüller invariant probabilities in his question. As it turns out, there are at least two good reasons to do so: • the Teichmüller flow is non-uniformly hyperbolic with respect to Masur-Veech measure, and hence it has a lot of invariant measures and a complete study of the Lyapunov spectrum of all such measures seems a very hard task. More concretely, W. Veech [68] (see also Appendix B below) constructed (with the aid of the so-called Rauzy-Veech diagrams) a periodic orbit of the Teichmüller flow (i.e., a pseudo-Anosov element of the mapping class group) in the stratum H(2) of genus 2 Abelian differentials with a single double zero such 22 Cf. Example 13 of Section 2 that the Teichmüller invariant probability supported in this periodic orbit has a vanishing second Lyapunov exponent for the KZ cocycle. This is in sharp contrast with the fact that the second Lyapunov exponent of KZ cocycle w.r.t. the Masur-Veech measure µ H(2) of the (connected) stratum H(2) is non-zero (as it follows from the work of G. Forni [27], and Avila and Viana [7]) and it shows that the description of Lyapunov spectra of KZ cocycle with respect to arbitrary Teichmüller invariant measures can be difficult even at the level of periodic orbits (pseudo-Anosov elements).
• As we mentioned in the previous chapter, in analogy to Ratner's work on unipotent flows, it is conjectured that any SL(2, R)-invariant probability in the moduli space of Abelian differentials is "algebraic" (and a recent progress on this conjecture was announced by A. As the reader can check, p is not ramified at ∞, and, by Riemann-Hurwitz formula, M 3 (x 1 , . . . , x 4 ) has genus 3.
Note that ω EW is anti-invariant with respect to the action T * (by pull-back) of the automor- Therefore, the quadratic differential q EW = ω 2 EW is T * -invariant. Since T generates the Galois group of deck transformations of p, this means that q projects under p to a quadratic differential q 0 on C with 4 simple poles.
Because EW is a closed connected locus of real dimension 3 (as it is a copy of the stratum By combining this discussion with Lemma 62, we have that EW supports an unique SL(2, R)invariant probability µ EW . 26 In particular, the translation atlas of (M 3 , ω EW ) is obtained by pre-composing (half) translation charts of (C, q 0 ) with the covering map p. 27 This is true because any stratum of quadratic/Abelian differentials is closed and SL(2, R)-invariant. 28 See Theorem 1.7 of [11] for a proof of this identity. 29 We will see a concrete model of this translation surface in next section.
Theorem 63 (Forni [28]). The Lyapunov spectrum of KZ cocycle with respect to µ EW is totally degenerate in the sense that Proof. As the reader can check, the set Let's denote by λ(n) the T * -eigenvalue of θ n , i.e., T * (θ n ) = λ(n)θ n .
We can compute the (symmetric, complex-valued) form B ω EW on H 1,0 in this basis as follows.
Firstly, we recall that B ω EW (ω EW , ω EW ) = 1. Secondly, by using the automorphism T to perform a change of variables, we get that In particular, B ω EW (θ n , θ m ) = 0 whenever λ(n)λ(m) = 1. Since λ(n)λ(m) = 1 for (n, m) = (1, 1), we obtain that the matrix of B in the basis {θ 1 , θ 2 , θ 3 } is  [39] because it has marvelous algebro-geometrical properties (see [39]) in addition to its totally degenerate Lyapunov spectrum. In fact, the German term Eierlegende Wollmilchsau literally is "egg-laying wool-milksow" in English and it means "a tool for several purposes" (after "Wiktionary"). The picture below (found on internet) resumes the meaning of this German expression: Remark 64. In fact, by analyzing the argument above, one realizes that it was shown that B R where x 1 , . . . , x 4 ∈ C are four distinct points.
The map p : M 4 (x 1 , . . . , x 4 ) → C, p(x, y) = x, is a covering branched (precisely) at x 1 , . . . , x 4 , and the Galois group of its deck transformations is generated by the automorphism T (x, y) = (x, ε 6 y), ε 6 = exp(2πi/6) (and hence it is isomorphic to Z/6Z). We leave the verification of this lemma as an exercise to the reader.
Proof. The argument is similar to the proof of Theorem 63. One starts by noticing that the set is a basis of the space Again, let's denote by λ(n) the T * -eigenvalue of θ n , i.e., T * (θ n ) = λ(n)θ n , and let's use the automorphism T to perform a change of variables to compute B ω O : As before, B ω O (θ n , θ m ) = 0 whenever λ(n)λ(m) = 1, and λ(n)λ(m) = 1 for (n, m) = (1, 1).
Since B ω O (θ 1 , θ 1 ) = 1, we conclude that the matrix of B in the basis {θ 1 , θ 2 , θ 3 , θ 4 } is In resume, µ O is another example answering (negatively) Veech's question. This example was announced in [31] and it appeared later in [32].  The motivation for the nomenclature Teichmüller curve comes from the following facts: In this setting, M. Möller [59] proved that: Actually, M. Möller [59] showed that any candidate for Shimura-Teichmüller curve in genus g = 5 must satisfy several constraints (e.g., they must belong to specific strata, etc.). In particular, he conjectures that there are no Shimura-Teichmüller curves in genus 5.
In resume, we have a fairly satisfactory understanding of SL(2, R)-invariant probabilities with totally degenerate Lyapunov exponents coming from Veech surfaces. Next, let's pass to the study of the analogous question for more general classes of SL(2, R)-invariant probabilities. 1 area(ω) ω ∈ supp(µ)} is an affine suborbifold of H(κ) in the sense that it is described by affine subspaces of relative cohomology in local period coordinates (cf. Subsection 2.3 of Section 2 for the definitions); • the measure ν on H(κ) given by dν = da · dµ (where a(ω) = area(ω) is the total area function) is equivalent to the Lebesgue measure on the affine suborbifold R · supp(µ) (or equivalently, the Lebesgue measure of the affine subspaces representing R · supp(µ) in local period coordinates).
It was recently announced 31 by A. Eskin and M. Mirzakhani [20] that all SL(2, R)-invariant g t -ergodic probability are affine.
In any event, one of the main goals in [21] was the development of a formula for the sums of Lyapunov exponents of KZ cocycle with respect to affine measures. However, for technical reasons (related to a certain "integration by parts" argument), A. Eskin, M. Kontsevich and A. Zorich need a "regularity" condition. More precisely, we say that an affine µ on Here, C 2 (K, ε) is the set of (unit area) translation surfaces (M, ω) ∈ H (1) (κ) possessing two non- In plain terms, µ is regular if the probability of seeing non-parallel "very thin and high" cylinders in translation surfaces in the support of µ is "very small".
As a matter of fact, all known examples of affine measures are regular and it is conjectured in [21] that any affine measure is regular. Of course, the quantity N area (ω, L) depends a lot on the geometry of (M, ω) and the real number L > 0. However, W. Veech and Ya. Vorobets discovered that given any SL(2, R)-invariant g t -ergodic probability µ, the quantity doesn't depend on L > 0. In the literature, c(µ) is called the Siegel-Veech constant of µ.
Remark 72. Our choice of normalization of the quantity 1 L 2 N area (ω, L)dµ(ω) leading to the Siegel-Veech constant here is not the same of [21]. Indeed, what [21] call Siegel-Veech constant is 3c(µ)/π 2 in our notation. Of course, there is no conceptual different between these normalizations, but we prefer to take a different convention from [21] because c(µ) appears more "naturally" in the statement of Eskin-Kontsevich-Zorich formula. In this context, for each S j ∈ SL(2, Z)·(M 0 , ω 0 ), we write S j = C ij where C ij are the maximal horizontal cylinders of S j , and we denote the width and height of C ij by w ij and h ij . Here, we're using the fact that the group SL(2, Z) is generated by the matrices S = Then, the sum of the top g Lyapunov exponents of KZ cocycle with respect to µ is The proof of this fundamental theorem is long 34 and sophisticated, and hence a complete discussion is out of the scope of these notes. Instead, we offer only a very rough idea on how the argument goes on. Firstly, one uses the formula (Theorem 44) of M. Kontsevich and G. Forni for sums of Lyapunov exponents to think of λ µ 1 + · · · + λ µ g as a certain integral over the stratum H (1) (k 1 , . . . , k s ). Then, by studying the integral, one can apply an integration by parts ("Stokes") argument to express it as a main term and a boundary term. At this point, the so-called Riemann-Roch-Hirzebruch-Grothendieck theorem allows to compute the main term and the outcome (depending only on the stratum) is precisely 1 12 If the (strata of) moduli spaces of Abelian differentials were compact, there would be no contribution from the boundary term and the deduction of the formula would be complete. Of course, . . , k s ) is never compact, and the contribution of the boundary term is not negligible.
Here, the study of the geometry of translation surfaces near the boundary of the moduli spaces (and the regularity assumption on µ) plays a role to show that the boundary term is given by the Siegel-Veech constant c(µ) and this completes this crude sketch of the arguments in [21]. Proof. Since λ µ 1 = 1, it suffices to show that the right-hand side of Eskin-Kontsevich-Zorich formula is > 1 to get that λ µ 2 > 0, and this follows from the computation λ µ 1 + · · · + λ µ g = 1 12 based on the non-negativity of the Siegel-Veech constant c(µ) and the assumption g ≥ 7.
At this stage, we can resume this section as follows. We close this section with the following remarks.
Remark 77. A. Eskin, M. Kontsevich and A. Zorich also showed in [21] a version of the their formula for quadratic differentials, and they used it to compute Siegel-Veech constants of SL(2, R)-invariant g t -ergodic probabilities µ supported in the hyperelliptic connected components H hyp (2g − 2) and H hyp (g − 1, g − 1) of the strata H(2g − 2) and H(g − 1, g − 1). The outcome of their computation is the fact that Siegel-Veech constant of any such SL(2, R)-invariant g t -ergodic µ is and, hence (by Theorem 75), in this case, the sum of Lyapunov exponents is In particular, since the sole two strata H(2) and H(1, 1) in genus 2 are hyperelliptic connected components, one has that, for any SL(2, R)-invariant g t -ergodic µ, because λ µ 1 = 1. This fact was conjectured by M. Kontsevich and A. Zorich, and it was firstly demonstrated by M. Bainbridge [9] a few years before the article [21] was available.
Remark 78. In a very recent work, D. Aulicino [2] further studied the problem of classifying SL(2, R)-invariant measures with totally degenerate spectrum from the point of view of the Teichmüller disks 35 contained in the rank-one locus. More precisely, following [27] and [28], we define the rank -k locus of the moduli space H g of Abelian differentials of genus g is D g (k) := {ω ∈ H g : rank(B ω ) ≤ k}. Note that D g (1) ⊂ · · · ⊂ D g (g − 1). In the literature, the locus D g (g − 1) is sometimes called determinant locus (because D g (g − 1) = {ω ∈ H g : det B ω = 0}). Observe that these loci are naturally related to the study of Lyapunov exponents of KZ cocycle: for instance, by Theorem 44, any SL(2, R)-invariant probability µ with supp(µ) ⊂ D g (1) has totally degenerate spectrum. In his work [2], D. Aulicino showed that there are no Teichmüller disks SL(2, R) · (M, ω) contained in D g (1) for g = 2 or g ≥ 13, the Eierlegende Wollmilchsau and Ornithorynque are the sole Teichmüller disks contained in D 3 (1) and D 4 (1), and, furthermore, if there are no Teichmüller curves contained in D 5 (1), then there are no Teichmüller disks contained in D g (1) for g ≥ 5. It is worth to point out that Teichmüller disks are more general objects than regular affine measures, so that Proposition 76 doesn't allow to recover the results of D. Aulicino.

SL(2, Z).
See Remark 83 below for more details. By plugging these facts into Theorem 74, one can compute the Siegel-Veech constants of the measures µ EW and µ O , and then, by Theorem 75, one can calculate the sum of their Lyapunov exponents. By doing so, one finds: Since λ µ 1 = 1 for any g t -invariant ergodic µ, one concludes that λ µ EW In fact, such computer programs for Mathematica and SAGE were written by, e.g., A. Zorich and V. Delecroix, and it is likely that they will be publicly available soon. 36 A pair of permutations h, v ∈ S N gives rise to a square-tiled surface with N squares by taking N unit squares

Explicit computation of Kontsevich-Zorich cocycle over two totally degenerate examples
We saw in the previous section that examples of SL(2, R)-invariant g t -ergodic probabilities with totally degenerate spectrum are rare and it is likely that there are only two of them coming from two square-tiled surfaces, namely, the Eierlegende Wollmilchsau and the Ornithorynque.
In this section, we will investigate more closely the Kontsevich-Zorich cocycle over the SL(2, R)- It is possible to show that, in genus g ≥ 2, the affine group Aff(M, ω) injects in the modular group Γ g , and the stabilizer of the SL(2, R)-orbit of (M, ω) in Γ g is precisely Aff(M, ω) (see [70]).
In particular, since KZ cocycle is the quotient of the trivial cocycle over ramified only at 0 ∈ T 2 with ω = p * (dz).
Using the action of p on homology groups, we define H through Aff(M, ω), we see that (in the present context) the tautological Lyapunov exponents λ µ 1 = 1 and λ µ 2g = −1 of KZ cocycle (with respect to µ) come from the restriction of KZ cocycle to the 2-dimensional symplectic subspace H st 1 . Therefore, the interesting part λ µ 2 ≥ · · · ≥ λ µ g of the Lyapunov spectrum of KZ cocycle with respect to µ comes from its restriction to H (0) 1 . In particular, we "reduced" the study of KZ cocycle over the Eierlegende Wollmilchsau and Ornithorynque to the computation of the homological action of their affine diffeomorphisms on H (0) 1 . Evidently, it is convenient to get concrete models of these square-tiled surfaces because they allow to write down explicit basis of H (0) 1 , so that the action of affine diffeomorphisms can be encoded by concrete matrices.
Here, (a) ensures that the Riemann surfaces are connected, (b) ensures that they are cyclic covers of C branched at x 1 , . . . , x 4 but not at ∞, and (c) ensures that ω is a well-defined, holomorphic 1-form.
The family M N (a 1 , . . . , a 4 ) is called square-tiled cyclic cover. In the sequel, our discussion will follow closely [32].
Note that the square of the Abelian differential ω is the pull-back of the quadratic differential on C under the natural projection p(x, y) = x. By choosing (x 1 , . . . , x 4 ) = (−1, 0, 1, ∞), we know 37 that the flat structure associated to q 0 is given by two flat unit squares glued by their boundaries: Now, we can use this concrete description of the flat structure of q 0 to obtain a concrete model for (M N (a 1 , . . . , a 4 ), ω) as follows. We have 2 squares tilling the flat model of q 0 , a white and 37 This is better appreciated by noticing that the square of the Abelian differential dx/z on the elliptic curve ). generating the Galois group of the covering p, we number 2k the white square S 2k = T k (S 0 ) and 2k + 1 the black square S 2k+1 = T k (S 1 ). Here, we take k modulo N (so that one may always think 0 ≤ k < N ).
The endpoint of the lift of the path τ h (see the figure above) to M N (a 1 , . . . , a 4 ) is deduced from the starting point of the lifted path by applying T a1+a4 = T −a2−a3 (here item (b) above was used).
In this way, by moving to two squares to the right, we go from the square number j to the square number j + 2(a 1 + a 3 )(mod N ). In particular, by successively applying T a1+a3 we can deduce all horizontal cylinders of (M N (a 1 , . . . , a 4 ), ω).
Similarly, we can deduce neighbors in the vertical direction by using small (positively oriented) paths σ i encircling z i (see the picture above for σ 1 ). Indeed, since the extremal points of the lift of σ i to M N (a 1 , . . . , a 4 ) differ by T ai , by going around a corner (in the counterclockwise sense) of the square numbered j, we end up in the square numbered j + 2a i (mod N ).
These "local moves" obtained by lifting τ h and σ 1 , σ −1 1 , σ 2 , . . . are described in a nutshell in the following picture: Of course, the picture above makes clear that we can extract concrete square-tiled models for M N (a 1 , . . . , a 4 ).
For example, it is an instructive exercise to the reader to apply this procedure to Eierlegende Wollmilchsau M 4 (1, 1, 1, 1) and Ornithorynque M 6 (3, 1, 1, 1), and check that the pictures one get for them are the following: Remark 83. From Figure 15, we see that the Eierlegende Wollmilchsau can be decomposed into two maximal horizontal cylinders, both of height 1 and width 4. Similarly, from Figure 16, we see that the Ornithorynque can be decomposed into two maximal horizontal cylinders, both of height 1 and width 6. Moreover, by applying the matrices S = 1 0 1 1 and T = 1 1 0 1 to the two figures above, and by using adequate elements of the modular group to cut and paste the resulting objects, the reader can verify that S and T stabilize both the Eierlegende Wollmilchsau Once we dispose of these concrete models for Eierlegende Wollmilchsau and Ornithorynque (and more generally square-tiled cyclic covers), it is time use them to produce nice basis of their homology groups.
7.4. Eierlegende Wollmilchsau and the quaternion group. By carefully looking at Figure   15, the reader can verify that Eierlegende Wollmilchsau admits the following presentation: Here, the squares of Eierlegende Wollmilchsau are labelled via the elements of the quaternion group 38 Q = {±1, ±i, ±j, ±k}.
A great advantage of this presentation is the fact that one can easily deduce the neighbors of squares by right multiplication by ±i or ±j: indeed, given a square g ∈ Q, its neighbor to the right is the square g · i and its neighbor on the top is g · j. In this way, we can identify the group of automorphisms Aut(M EW , ω EW ) of Eierlegende Wollmilchsau (M EW , ω EW )with the quaternion group Q by associated to h ∈ Q the automorphism sending the square g to the square h · g deduced from g by left multiplication by h. 38 The multiplication rules are i 2 = j 2 = k 2 = −1 and ij = k.  In what follows, we'll follow closely [56] to compute the homological action of affine group of Eierlegende Wollmilchsau using the cycles σ g , ζ g ∈ H 1 (M EW , Σ EW , ω EW ) introduced in Subsection 7.2 above, i.e., Remark 84. Note that, from this picture, we have that σ g + ζ gi − σ gj − ζ g = 0 in homology. We'll systematically use this relation in the sequel.
However, before rushing to the study of the whole action of the affine group, let's first investigate the action of the group of automorphisms. The automorphism group of Eierlegende Wollmilchsau is isomorphic to Q. In particular, one can select a nice basis on the homology of Eierlegende Wollmilchsau using the representation theory of Q. More precisely, we know Q has 4 irreducible 1dimensional representations χ 1 , χ i , χ j , χ k and 1 irreducible 2-dimensional representation χ 2 . They can be seen inside the regular representation of Q in Z(Q) via the submodules generated by: Thus, we see that the character table of Q is 1 −1 ±i ±j ±k This motivates the introduction of the following relative cycles: Remark 85. Note that Indeed, we have that the cycle w i is relative because its boundary is ∂w i = 4(j + k − i − 1) and the action of an automorphism g ∈ Q is g · w i = χ i (g) · w i (see Figure 17). Also, one has similar formulas for w j and w k , so that the action of Q = Aut(M EW , ω EW ) on the subspace Observe that H rel is a relative subspace in the sense that it complements the absolute homology Next, we consider the following absolute cycles: Using the notations introduced in Subsection 7.2, one can check that H st 1 = Qσ ⊕ Qζ and H (0) 1 (M EW , Q) is spanned by the cycles σ g , ζ g . Here, we notice that, since Eierlegende Wollmilchsau has genus g = 3, i.e., the dimension of H 1 (M EW , Q) is 2g − 2 = 6, H st 1 has dimension 2 and one has that H 1 (M EW , Q) has dimension 4. For later use, we observe that σ −g = − σ g , ζ −g = − ζ g , ε −g = −ε g and Therefore, one can write H Finally, we observe that, for any g ∈ Q Aut(M EW , ω EW ), g · σ = σ, g · ζ = ζ and g · σ h = σ gh , g · ζ h = ζ gh , g · ε h = ε gh See Figure 17. Hence, the action of Q = Aut(M EW , ω EW ) on the absolute homology H 1 (M EW , Q) = is also fairly well-understood.
In resume, by looking at the representation theory of the (finite) group of automorphisms Q of Eierlegende Wollmilchsau, we selected a nice generating set of the relative cycles w i , w j , w j and the absolute cycles σ, ζ, σ g , ζ g such that the action of Q is easily computed.
After this first (preparatory) step of studying the homological action of Aut(M EW , ω EW ) Q, we are ready to face the homological action of Aff(M EW , ω EW ). A direct inspection of Figure 17 reveals that the actions of S and T on the cycles σ g and ζ g are From these formulas, one deduces that S and T act on H st 1 = Qσ ⊕ Qζ in the standard way (cf. Subsection 7.2) while they act on the relative part H rel = Qw i ⊕ Qw j ⊕ Qw k via the symmetry group of a tetrahedron : Last, but not least, we have S and T act on H (0) 1 (that is, the subspace containing the nontautological exponents of KZ cocycle) as: Here, our choice of computing S and T in terms of ε g was not arbitrary: indeed, a closer inspection of these formulas shows that S and T are acting on the 4-dimensional subspace H with the inner product such that {ε 1 , ε i , ε j , ε k } is orthonormal, one has that S and T act on H In particular, this discussion shows that Aff (1) (M EW , ω EW ) acts on H   [59] for more details. However, we avoided using this fact during this section to convince the reader that the homological action of the affine group is so concrete that one 39 The order of O(R) is order 1152, so that the affine group of Eierlegende Wollmilchsau acts via a index 12 can actually determine (with bare hands) the matrices involved in such action (at least if one is sufficiently patient). 7.6. The action of the affine diffeomorphisms of the Ornithorynque. As the reader can imagine, the calculations of the previous two subsections can be mimicked in the context of Ornithorynque. Evidently, the required modifications are somewhat straightforward, so let's content ourselves with a mere outline of the computations (referring to the original article [56] for details).
We start by considering a "better" presentation of Ornithorynque (M O , ω O ) (compare with Figure 16): The indication i = 0, 1, 2 (mod 3) means that we're considering three copies of the "basic" pattern, and we identify sides with the same "name" σ i , σ i , ζ i or ζ i by taking into account that the subindices i are thought modulo 3. The (three) black dots are regular points while the other "special" points indicated the (three) double zeroes of the Abelian differential ω O . Also, it is clear from this picture that the cycles σ i , σ i , ζ i and ζ i are relative, and they verify the relation The relation between the cycles σ i , σ i , ζ i and ζ i written above implies that This suggests the introduction of the following cycles in H (0) 1 : Actually, this is the sole relation satisfied by them because it is possible to show that Observe that this is coherent with the fact that H Therefore, Aff Finally, a careful inspection of the formulas above shows that the action of Aff (1) H preserves the root system

Cyclic covers
In last section we studied combinatorial models of Eierlegende Wollmilchsau and Ornithorynque by taking advantage of the fact that they belong to the class of square-tiled cyclic covers. Then, we used these combinatorial models to "put our hands" on the KZ cocycle over their SL(2, R)-orbits via the homological action of the group of affine diffeomorphisms.
In this section, we'll be "less concrete but more conceptual" in order to systematically treat the Lyapunov spectrum of KZ cocycle over square-tiled cyclic covers in an unified way. From this framework we will prove that all zero Lyapunov exponents in the class of square-tiled cyclic covers have a nice geometrical explanation: they come from the annihilator of the second fundamental form B ω . A striking consequence of this fact is the continuous (actually, real-analytic) dependence of the neutral Oseledets subspace on the base point. However, by the end of this section, we will see that this beautiful scenario is not true in general: indeed, we'll construct other (not square-tiled) cyclic covers leading to a merely measurable neutral Oseledets subspace.
8.1. Hodge theory and the Lyapunov exponents of square-tiled cyclic covers. Consider a square-tiled cyclic cover where gcd (N, a 1 , . . . , a 4 ) = 1, 2b j = a j − 1, N is even and 0 < a j < N are odd. Cf. Subsection 7.3 for more comments.
The arguments in the previous two sections readily show that the locus of ( M N (a 1 , . . . , a 4 ), ω) is the SL(2, R)-orbit of a square-tiled surface. 40 Thus, it makes sense to discuss the Lyapunov exponents of KZ cocycle with respect to the unique SL(2, R)-invariant probability supported on the locus of ( M N (a 1 , . . . , a 4 ), ω).
For "linear algebra reasons", it is better to work with complex version of the KZ cocycle on the complex Hodge bundle: indeed, as we shall see in a moment, it is easier to diagonalize by blocks the complex KZ cocycle, while we keep the same Lyapunov exponents of usual (real) KZ cocycle.
We can diagonalize by blocks the complex KZ cocycle over (M N (a 1 , . . . , a 4 ), ω) using the automorphism T (x, y) = (x, εy), ε = exp(2πi/N ). More precisely, since T N = Id, one can write where H 1 (ε j ) is the eigenspace of the cohomological action T * of the automorphism T .
We affirm that these blocks are invariant under the complex KZ cocycle. Indeed, since the automorphism T acts by pre-composition with the translation charts of (M N (a 1 , . . . , a 4 ), ω) and SL(2, R) acts by post-composition with translation charts, it follows that T * commutes with the complex KZ cocycle and, a fortiori, the eigenspaces of T * serve to diagonalize by blocks the complex KZ cocycle.
Next, we recall that the complex KZ cocycle preserves the Hodge form (α, β) = i 2 α ∧ β, a positive definite form on H 1,0 and negative definite form on H 0,1 (cf. Subsection 3.4). Since , and, thus, the restriction of the complex KZ cocycle to H 1 (ε j ) acts via some elements of the group U (p j , q j ) of complex matrices preserving a non-degenerate (pseudo-)Hermitian form of signature In the setting of square-tiled cyclic covers, these signatures are easy to compute in terms of N , In particular, p j , q j ∈ {0, 1, 2} and p j + q j ∈ {0, 1, 2}.
Proof. A sketch of proof of this result goes as follows. Since H 0,1 (ε j ) = H 1,0 (ε N −j ), we have that q j = p N −j and, therefore, it suffices to compute p j .
Note that this completes the sketch of proof of lemma: indeed, since 0 < a n < N and a 1 +· · ·+a 4 is a multiple of N , we have that 1 ≤ 4 n=1 a n j N ≤ 3 because a 1 + · · · + a 4 ∈ {N, 2N, 3N }. So, our discussion above covers all cases.
From the point of view of Lyapunov exponents, this "separation" is natural because the groups U (p j , 0) or U (0, q j ) are compact while the group U (1, 1) SL(2, R) is not compact. More concretely, an immediate consequence of the compactness of the groups U (p j , 0) or U (0, q j ) is: . See, e.g., [33] for more details. So, we can reobtain that this corollary by showing that B ω vanishes on H 1,0 (ε j ) ⊕ H 1,0 (ε N −j ) when j ∈ N .
Then, one realizes that this is true since j ∈ N implies that H 1,0 (ε j ) ⊕ H 1,0 (ε N −j ) = H 1,0 (ε j ) or H 1,0 (ε N −j ), and the restriction of B ω to H 1,0 (ε k ) vanishes for every k = N/2 because of the following computation 41 : for any α, β ∈ H 1,0 (ε k ), Actually, one can further play with the form B ω to show a "converse" to this corollary, that is, the Lyapunov exponents of the restriction of the complex KZ cocycle to H 1 (ε j ) are non-zero whenever j ∈ P. In fact, if j ∈ P, the restriction of KZ cocycle to H 1 (ε j ) has Lyapunov exponents ±λ (j) (because it acts via matrices in U (1, 1) SL(2, R)). Moreover, the restriction of KZ cocycle to H 1 (ε j ) is conjugated to the restriction of KZ cocycle to H 1 (ε N −j ), so that λ (j) = λ (N −j) . Therefore, by the discussion of the previous paragraph, we can deduce that λ (j) = λ (N −j) is nonzero by showing that the restriction of B ω to H 1,0 (ε j ) ⊕ H 1,0 (ε N −j ) is not degenerate. Here, this last fact is true because H 1,0 (ε j ) := C · α j := C · α j (b 1 (j), . . . , b 4 (j)), b k (j) := [a k j/N ], for j ∈ P (cf. Lemma 88), so that Here, we used that [a k j/N ] + [a k (N − j)/N ] = a k − 1 for j ∈ P.
In other words, we just showed that Corollary 90. If j ∈ P, the Lyapunov exponents ±λ (j) of the restriction of the complex KZ cocycle to H 1 (ε j ) are non-zero.
At this point, we can say (in view of Corollaries 89 and 90) that the Lyapunov spectrum of KZ cocycle over square-tiled cyclic covers is qualitatively well-known: it can be diagonalized by blocks by restriction to H 1 (ε j ) and zero Lyapunov exponents come precisely from blocks H 1 (ε j ) with j ∈ N . However, in some applications 42 , it is important to determine quantitatively individual exponents of KZ cocycle. In the case of cyclic covers, A. Eskin, M. Kontsevich and A. Zorich [22] determined the value of λ (j) for j ∈ P. Roughly speaking, they start the computation λ (j) = λ (N −j) from the fact (already mentioned) that 2λ (j) = λ (j) + λ (N −j) coincides with the average of the . Then, they use the fact that H 1,0 (ε j ) has complex dimension 1 for j ∈ P to reduce the calculation of the aforementioned average to the computation of the orbifold degree of the line bundle H 1,0 (ε j ). After this, the calculation of the orbifold degree of H 1,0 (ε j ) can be performed explicitly by noticing that H 1,0 (ε j ) has a global section H 1,0 (ε j ) = C · α j (b 1 (j), . . . , b 4 (j)) =: C · α j over the SL(2, R)-orbit Since the orbifold degree is expressed as a certain integral depending on α j , a sort of integration by parts argument can be used to rewrite the orbifold degree in terms of the "behavior at infinity" of α j , that is, the behavior of α j when x i approaches x j for some i = j. This led them to the following result: for any j ∈ P. 42 For instance, the precise knowledge of Lyapunov exponents of KZ cocycle for a certain SL(2, R)-invariant gt-ergodic probability supported in H 5 recently permitted V. Delecroix, P. Hubert and S. Lelièvre [17] to confirm a conjecture of the physicists J. Hardy and J. Weber that the so-called Ehrenfest wind-tree model of Lorenz gases has abnormal diffusion for typical choices of parameters.
Coming back to the qualitative analysis of KZ cocycle (and its Lyapunov spectrum) over squaretiled cyclic covers, we observe that our discussion so far shows that the neutral Oseledets bundle E c (associated to zero Lyapunov exponents) of KZ cocycle coincides with the annihilator Ann(B R ) in the case of square-tiled cyclic covers. In other words, the zero Lyapunov exponents of KZ cocycle have a nice geometrical explanation in the case of square-tiled cyclic covers: they come precisely from the annihilator of second fundamental form B R ! In particular, since B ω depends continuously (real-analytically) on ω, we conclude that the neutral Oseledets bundle E c of KZ cocycle is where θ is the positive multiple of (x − x 1 )dx/z ∈ H(2) with unit area.
Because any Riemann surface of genus 2 is hyperelliptic, it is not hard to see that the family Here, the six Weierstrass points are located over x 1 , . . . , x 6 . In other words, any genus 2 Riemann surface is biholomorphic to some M 2 (x 1 , . . . , x 6 ). Also, the zero of an Abelian differential θ in M 2 (x 1 , . . . , x 6 ) must be located at one of the Weierstrass points. Thus, by renumbering the points x 1 , . . . , x 6 (in order to place the zero over x 1 ), we can write a θ ∈ H(2) in M 2 (x 1 , . . . , x 6 ) as a multiple of (x − x 1 )dx/z. Alternatively, one can show that Just to get a "feeling" on how the flat structure of translation surfaces in Z look like, we notice the following facts. It is not hard to check that the flat structure associated to (M 2 (x 1 , . . . , x 6 ), θ)) is described by the following octagon (whose opposite parallel sides are identified): is located at the "symmetry center" of the octagon. See the picture above for an indication of the relative positions of x 1 (marked by a black dot) and x 2 , . . . , x 6 (marked by crosses). In this way, we obtain a concrete description of H(2). Now, since Z is defined by Abelian differentials (M 10 (x 1 , . . . , x 6 ), ω) given by certain triple (ramified) covers of the Abelian differentials (M 2 (x 1 , . . . , x 6 ), θ)) ∈ H(2), one can check that the flat structure associated to (M 10 (x 1 , . . . , x 6 ), ω) is described by the following picture: Here, we glue the half-sides determined by the vertices (black dots) and the crosses of these five pentagons in a cyclic way, so that every time we positively cross the side of a pentagon indexed by j, we move to the corresponding side on the pentagon indexed j + 1 (mod 5). For instance, in the In what follows, we will study the Lyapunov spectrum of KZ cocycle with respect to µ Z . By the reasons explained in the previous subsection, we consider the complex KZ cocycle over Z.
Denoting by T (x, y) = (x, εy), ε = exp(2πi/6), the automorphism of order 6 of M 10 (x 1 , . . . , x 6 ) generating the Galois group of the covering p : M 10 (x 1 , . . . , x 6 ) → C, p(x, y) = x, we can write where H 1 (ε j ) is the eigenspace of the eigenvalue ε j of the cohomological action T * of T . Again, the fact that T is an automorphism implies that the complex KZ cocycle G KZ,C t preserves each H 1 (ε j ), that is, we can use these eigenspaces to diagonalize by blocks the complex KZ cocycle.
is the complex conjugate of G KZ,C t | H 1 (ε 4 ) , it suffices to study the latter cocycle.
Proof. Since the cocycle C t := G KZ,C t | H 1 (ε 4 ) preserves the Hodge intersection form (., .), we have Let v, w be two vectors in some Oseledets subspaces associated to Lyapunov exponents λ, µ with λ + µ = 0. By definition of Lyapunov exponents, one has that Therefore, we conclude that (v, w) = 0 whenever v, w belong to Oseledets subspaces associated to Lyapunov exponents λ, µ with λ + µ = 0. In particular, denoting by E u , resp. E s , the unstable, resp. stable, Oseledets subspace associated to the positive, resp. negative, Lyapunov exponents of has dimension min{p, q} at most.
Proof. By taking adequate coordinates, we may assume that (., .) is the standard Hermitian form of signature (p, q) in C p+q : (z, w) = z 1 w 1 + · · · + z p w p − z p+1 w p+1 − z p+q w p+q By symmetry, we can assume that p ≤ q, i.e., min{p, q} = p.
Suppose that V is a vector space of dimension ≥ p + 1 inside the light-cone C. Let's select Because w (1) , . . . , w (p+1) is a collection of p + 1 vectors in C p , we can find a non-trivial collection of coefficients (a 1 , . . . , a p+1 ) ∈ C p+1 − {0} with p+1 j=1 a j w (j) = 0 Since w (j) were built from the p first coordinates of v (j) , we deduce that This shows that dimV ≤ p whenever V ⊂ C, as desired.
By applying this lemma in the context of the cocycle G KZ,C t | H 1 (ε 4 ) , we get that E u and E s have dimension 1 at most because they are in the light-cone of a pseudo-Hermitian form of signature We refer the reader to [33] and [34] where this is discussed in details.
After this discussion, we understand completely the Lyapunov spectrum of KZ cocycle with respect to the "Masur-Veech" measure µ Z of the locus Z: Proposition 96. The non-negative part of the Lyapunov spectrum of KZ cocycle with respect to µ Z is {1 > 4/9 = 4/9 > 1/3 > 0 = 0 = 0 = 0 = 0 = 0} Now, we pass to the study of the neutral Oseledets subspace of KZ cocycle over Z, or, more Here, it is worth to notice that the neutral Oseledets subspace of G KZ,C t | H 1 (ε 4 ) and the intersection of the annihilator of B ω with H 1,0 (ε 2 ) ⊕ H 1,0 (ε 4 ) have the same rank (namely, 2). In particular, it is natural to ask whether these subspaces coincides, or equivalently, the neutral Oseledets subspace of G KZ,C t | H 1 (ε 4 ) has a nice geometrical explanation.
This was shown not to be true in [34] along the following lines. Since the neutral Oseledets subspace is g t -invariant and the annihilator of B ω is continuous and SO(2, R)-invariant, the coincidence of these subspaces would imply that the neutral Oseledets subspace of G KZ,C t | H 1 (ε 4 ) is a (rank 2) continuous SL(2, R)-invariant subbundle of H 1 (ε 4 ). This property imposes severe restrictions on the behavior G KZ,C t | H 1 (ε 4 ) : for instance, by considering two periodic (i.e., pseudo-Anosov) orbits of the Teichmüller flow in the same SL(2, R) associated to two Abelian differentials on the same Riemann surface, we get that the matrices A and B representing G KZ,C t | H 1 (ε 4 ) along these periodic orbits must share a common subspace of dimension 2, and this last property can be contradicted by explicitly computing with some periodic orbits. Unfortunately, while this idea is very simple, the calculations needed to implement it are somewhat long and we will not try to reproduce them here. Instead, we refer to Appendix A of [34] where the calculation is largely detailed (and illustrated with several pictures).
Remark 97. During an exposition of this topic by the second author, Y. Guivarch asked whether G KZ,C t | H 1 (ε 4 ) still acts isometrically on its neutral subspace. This question is very natural and interesting because now that one can't use variational formulas involving B ω to deduce this property (as we did in the case of square-tiled cyclic covers. As it turns out, the answer to Y. Guivarch's question is positive by the following argument: one has that the neutral Oseledets subspace E c is outside the light-cone C because the stable Oseledets subspace E s has dimension 1, and so, if E c ∩ C were non-trivial, we would get a subspace (E c ∩ C) ⊕ E s ⊂ C of dimension at least 2 inside the light-cone C of an Hermitian form of signature (3, 1), a contradiction with Lemma 95 above. In other words, the light-cone is a geometric mechanism of production of neutral Oseledets subbundles with isometric behavior genuinely different from the (also geometric) method of using the annihilator of the second fundamental form of Gauss-Manin connection of the Hodge bundle.
Of course, the fact that the neutral Oseledets subspace doesn't coincide with the annihilator of the second fundamental form B ω is not the "end of the road": indeed, by carefully inspecting the arguments of the previous paragraph one notices that it leaves open the possibility that the neutral Oseledets subspace maybe continuous despite the fact that it is not the annihilator of B ω .
Heuristically, one strategy to "prove" that the neutral Oseledets subspace is not very smooth goes as follows: as we knoe, the Lyapunov exponents of the Teichmüller flow can be deduced from the ones of the KZ cocycle by shifting them by ±1; in this way, the smallest non-negative Lyapunov 45 Here we "excluded" the part of the neutral Oseledets bundle coming from the blocks H 1 (ε) and H 1 (ε 5 ) because the complex KZ cocycle acts via U (0, 4) and U (4, 0), and it is not hard to show from this that the corresponding part of the neutral Oseledets bundle is "geometrically explained" by the annihilation of the form Bω on these blocks. exponent of the Teichmüller flow is 5/9 = 1−4/9; therefore, the generic points tend to be separated by Teichmüller flow by ≥ e 5t/9 after time t ∈ R; on the other hand, the largest Lyapunov exponent on the fiber H 1 (ε 4 ) is 4/9, so that the angle between the neutral Oseledets bundle over two generic points grows by ≤ e 4t/9 after time t ∈ R; hence, in general, one can't expect the neutral Oseledets bundle to be better than α = (4/9)/(5/9) = 4/5 Hölder continuous.
Of course, there are several details missing in this heuristic, and currently we don't know how to render it into a formal argument. However, in a recent work still in progress [6], A. Avila, J.-C.
Yoccoz and the second author proved (among other things) that the neutral Oseledets subspace is not continuous at all (and hence only measurable by Oseledets theorem). In the sequel, we provide a brief sketch of this proof of the non-continuity of E c .
As we mentioned a few times in this text, the Teichmüller flow and the Kontsevich-Zorich cocycle over (connected components of) strata can be efficiently coded by means of the so-called Rauzy-Veech induction. Roughly speaking, given a (connected component of a) stratrum C of Abelian differentials of genus g ≥ 1, the Rauzy-Veech induction associates the following objects: a finite oriented graph G(C), a finite collection of simplices ("Rauzy-Veech boxes") and a finite number of copies of a Euclidean space C 2g over each vertex of G(C), and, for each arrow of G(C), a (expanding) projective map between (parts of) the simplices over the vertices connected by this arrow, and a matrix between the copies of C 2g over the vertices connected by this arrow. We strongly recommend J.-C. Yoccoz's survey [72] for more details on the Rauzy-Veech induction.
In this language, the simplices (Rauzy-Veech boxes) over the vertices of this graph represent admissible paramaters determining translations surfaces (Abelian differentials on Riemann surfaces M ) in C, the (expanding) projective map between (parts of) the simplices (associated to vertices connected by a given arrow) correspond to the action of the Teichmüller flow on the parameter space (after running this flow for an adequate amount of time), and the matrices (attached to the arrows) on C 2g are the action of the Kontsevich-Zorich cocycle on the first cohomology group Among the main properties of the Rauzy-Veech induction, we can highlight the fact that it permits to "simulate" almost every (with respect to Masur-Veech measure) orbit of Teichmüller flow on on C in the sense that these trajectories correspond to (certain) infinite paths on the graph G(C). In order words, the Rauzy-Veech induction allows to code the Teichmüller flow as a subshift of a Markov shift on countably many symbols (as one can use loops on G(C) based on an arbitrarily fixed vertex as basic symbols / letters of the alphabet of our Markov subshift). Moreover, the KZ cocycle over these trajectories of Teichmüller flow can be computed by simply multiplying the matrices attached to the arrows one sees while following the corresponding infinite path on G(C).
Equivalently, we can think the KZ cocycle as a monoid of (countably many) matrices (as we can only multiply the matrices precisely when our oriented arrows can be concatened, but in principle we don't dispose of the inverses of our matrices because we don't have the right to "revert" the orientation of the arrows).
In the particular case of H(2), the associated graph G(H(2)) is depicted below: Figure 18. Schematic representation of the Rauzy diagram associated to H(2). The letters near the arrows are not important here, only the 7 vertices (black dots) and the arrows between them. Now, we observe that Z was defined by taking certain triple covers of Abelian differentials of H(2), so that it is also possible to code the Teichmüller flow and KZ cocycle on Z by the same graph and the same simplices over its vertices, but by changing the matrices attached to the arrows: in the case of H(2), these matrices acted on C 4 , but in the case of Z they act on C 20 and they contain the matrices of the case of H(2) as a block.
At this stage, one can prove non-continuity of the neutral Oseledets subspace E c of G KZ,C t | H 1 (ε 4 ) as follows.
Firstly, one computes the restriction of KZ cocycle (or rather the matrices of the monoid) to E c on certain "elementary" loops and one checks that they have finite order. In particular, every time we can get the inverses of the matrices associated to these elementary loops by simply repeating these loops an appropriate number of times (namely, the order of the matrix minus 1). On the other hand, since these elementary loops are set up so that any infinite path (coding a Teichmüller flow orbit) is a concatenation of elementary loops, one conclude that the action (on E c ⊂ C 20 ) of our monoid of matrices is through a group! In particular, given any loop γ (not necessarily an elementary one), we can find another loop δ such that the matrix attached to δ (i.e., the matrix obtained by multiplying the matrices attached to the arrows forming δ "in the order they show up" with respect to their natural orientation of δ) is exactly the inverse of the matrix attached to γ.
Secondly, by computing with a pair of "sufficiently random" loops γ A and γ B , it is not hard to see that we can choose such that their attached matrices A and B have distinct and/or transverse central eigenspaces E c A and E c B (associated to eigenvalues of modulus 1). In this way, the periodic orbits (pseudo-Anosov orbits) of the Teichmüller flow coded by the infinite paths . . . γ A γ A γ A . . . and . . . γ B γ B γ B . . . obtained by infinite concatenation of the loops γ A and γ B have distinct and/or transverse neutral Oseldets bundle, but this is no contradiction to continuity since the base points of these periodic orbits are not very close. However, we can use γ A and γ B to produce a contradiction as follows. Let k 1 a large integer. Since our monoid acts by a group, we can find a loop γ C,k such that the matrix attached to it is A −k . It follows that the matrix attached to the loop γ A,B,k : i.e., B. Therefore, the infinite paths . . . γ A γ A γ A . . . and . . . γ A,B,k γ A,B,k γ A,B,k . . . produce periodic orbits whose neutral Oseledets bundle still are E 0 A and E 0 B (and hence, distinct and/or transverse), but this time their basepoints are arbitrarily close (as k → ∞) because the first k "symbols" (loops) of the paths coding them are equal (to γ A ).
Remark 98. Actually, this argument is part of more general considerations in [6] on certain cyclic covers obtained by taking 2n copies of a regular polygon with m sides, and cyclically gluing the sides of these polygons in such a way that their middle points become ramification points: indeed, Z corresponds to the case n = 3 and m = 5 of this construction.
Remark 99. It is interesting to notice that the real version of Kontsevich-Zorich cocycle over Z on is an irreducible symplectic cocycle with non-continuous neutral Oseledets bundle. In principle, this irreducibility at the real level makes it difficult to see the presence of zero exponents, so that the passage to its complex version (where we can decompose it as a sum of two complex conjugated monodromy representations by matrices in U (1, 3) and U (3, 1)) reveals a "hidden truth" not immediately detectable from the real point of view (thus confirming the famous quotation of J. Hadamard: "the shortest route between two truths in the real domain passes through the complex domain"). We believe this example has some independent interest because, to the best of our knowledge, most examples of symplectic cocycles and/or diffeomorphisms exhibiting some zero Lyapunov exponents usually have smooth neutral Oseldets bundle due to some sort of "invariance principle" (see this article of A. Avila and M. Viana [8] for some illustrations of this).
We state below two "optimistic guesses" on the features of the KZ cocycle over the support of general SL(2, R)-invariant probabilities mostly based on our experience so far with cyclic covers.
Notice that we call these "optimistic guesses" instead of "conjectures" because we think they're shared (to some extent) by others working with Lyapunov exponents of KZ cocycle (and so it would be unfair to state them as "our" conjectures).
Optimistic Guess 1. Let µ be a SL(2, R)-invariant probability in some connected component of a stratrum of Abelian differentials and denote by L its support. Then, there exists a finite (ramified) cover L such that (the lift of ) the Hodge bundle H 1 C over L can be decomposed into a direct sum of continuous where W 1 , . . . , W m , U 1 , . . . , U n are distinct SL(2, R)-irreducible representations admiting Hodge are complex vector spaces (taking into account the multiplicities of the irreducible factors W i , U j ), and L is the tautological bundle L = L 1,0 ⊕ L 0,1 , L 1,0 = Cω, L 0,1 = Cω, ω ∈ L. Moreover, this decomposition is unique and it can't be further refined after passing to any further finite cover.
Optimistic Guess 2. In the setting of Optimistic Guess 1, denote by Then, the Lyapunov spectrum of the KZ cocycle on W i is simple, i.e., and the Lyapunov spectrum of the KZ cocycle on U j is "as simple as possible", i.e., Remark 101. This "guess" is based on the general philosophy (supported by works as the ones of A. Raugi and Y. Guivarch [37], and I. Goldscheid and G. Margulis [36]) that, after reducing our cocycle to irreducible pieces, if the cocycle restricted to such a piece is "sufficiently generic" inside a certain Lie group of matrices G, then the Lyapunov spectrum on this piece should look like the "Lyapunov spectrum" (i.e., collection of the logarithms of the norms of eigenvalues) of the "generic" matrix of G. For instance, since a generic matrix inside the group U (p, q) has spectrum λ 1 > · · · > λ r > 0 = · · · = 0 r > −λ r > · · · > −λ 1 where r = min{p, q}, the above guess essentially claims that, once one reduces the KZ cocycle to irreducible pieces, its Lyapunov spectrum on each piece must be as generic as possible.
Remark 102. Notice that the previous guess doesn't make any attempt to compare Lyapunov exponents within distinct irreducible factors: indeed, in general non-isomorphic representations may lead to the same exponent by "pure chance" (as it happens in the case of certain genus 5 Abelian differentials associated to the "wind-tree model", cf. [17]).
We close this section by mentioning that in Appendix D below it is presented some recent results on both non-simplicity and simplicity of Lyapunov spectrum of KZ cocycle in the context of square-tiled surfaces. claiming that there are square-tiled surfaces such that the representation ρ S associated to its SL(2, R)-orbit S has irreducible factors in the complementary series.
Observe that the natural identification between SL(2, R)-orbits S of square-tiled surfaces and the unit cotangent bundle of Γ\H where Γ is the corresponding Veech group permit to think of ρ S as the regular unitary SL(2, R)-representation ρ Γ on the space L 2 0 (Γ\H, ν Γ ) of zero-mean L 2 -functions with respect to the natural measure ν Γ on Γ\H.
In view of Ellenberg and McReynolds theorem, it suffices to find a finite-index subgroup Γ ⊂ Γ (2) such that ρ Γ has complementary series. As we promised, this will be achieved by a cyclic covering procedure. Firstly, we fix a congruence subgroup Γ(m) such that the corresponding modular curve Γ(m)\H has genus g ≥ 1, e.g., Γ(6). Next, we fix a homotopically non-trivial closed geodesic β of Γ(m)\H after the compactification of its cusps and we perform a cyclic covering of Γ(m)\H (i.e., we choose a subgroup Γ ⊂ Γ(m)) of high degree N such that a lift β N of β satisfies (β N ) = N · (β).  ) .
Indeed, suppose that there exists some ε 0 > 0 such that σ(Γ) < −ε 0 for every N . By Ratner's theorem 109, it follows that for any |t| ≥ 1. On the other hand, since the distance between the centers of U and V is ∼ N/2, the support of u is disjoint from the image of the support of v under the geodesic flow a(t N ) for a time t N ∼ N/2. Thus, u, ρ Γ (a(t N ))v = u · v • a(t N ) = 0, and, a fortiori, Putting these two estimates together and using the facts that f L 2 ≤ u L 2 ≤ 1 and g L 2 ≤ v L 2 ≤ 1, we derive the inequality In particular, ε 0 ≤ C(ε 0 ) · ln N N , a contradiction for a sufficiently large N . Of course, the naive strategy is to make the arguments of the previous two subsections as explicit as possible. By trying to do so, we notice that there are essentially two places where one needs to pay attention: • firstly, in Ellenberg-McReynolds theorem, given a group Γ, we need to know explicitly a square-tiled surface with Veech group Γ; • secondly, for the explicit construction of a group Γ with complementary series, we need explicit constants in Ratner's theorem 109: indeed, in terms of the notation of the previous subsection, by taking ε 0 = 1, we need to know the constant C(ε 0 ) = C(1) in order to determine a value of N (and hence Γ) violating the inequality 1 ≤ C(1) ln N N (imposed by an eventual absence of complementary series).
This reduces the problem of construction of explicit square-tiled surfaces with complementary series to find some square-tiled surface with Veech group Γ 6 (6). At this point, one can improve again over the naive strategy above by following J. Ellenberg and D. McReynolds methods only partly. More precisely, since λ 1 ( Γ\H) ≤ λ 1 (Γ\H) when Γ ⊂ Γ (i.e., Γ\H covers Γ\H), it suffices to construct a square-tiled surface with Veech group Γ ⊂ Γ 6 (6) and, for this purpose, we don't have to follow [20] until the end: by doing so, we "save" a few "covering steps" needed when one wants the Veech group to be exactly Γ 6 (6). In other words, by "stopping" the arguments in [20] earlier, we get "only" a square-tiled surface M with Veech group Γ ⊂ Γ 6 (6) but we "reduce" the total number of squares of M because we don't insist into taking further "coverings steps" to get Veech group equal to Γ 6 (6).
In fact, this "mildly improved" strategy was pursued on the article [55] by Gabriela Schmithüsen and the second author were it is shown that: ) with 576 squares, genus 147, and Veech group Γ ⊂ Γ 6 (6). In particular, the Teichmüller curve SL(2, R) · (M, ω) has complementary series.
We close this section by referring the reader to [55] for the explicit pair of permutations h, v quoted above and a complete proof of this result.
The Lie algebra sl(2, R) of SL(2, R) (i.e., the tangent space of SL(2, R) at the identity) is the set of all 2 × 2 matrices with zero trace. Given a C 1 -vector v of ρ and X ∈ sl(2, R), the Lie derivative Exercise 107. Show that L X v, w = − v, L X w for any C 1 -vectors v, w ∈ H of ρ and X ∈ sl(2, R). 49 Cf. Appendix C for more comments on this construction. The Casimir operator Ω ρ is Ω ρ := (L 2 V + L 2 Q − L 2 W )/4 on the dense subspace of C 2 -vectors of ρ. It is known that Ω ρ v, w = v, Ω ρ w for any C 2 -vectors v, w ∈ H, the closure of Ω ρ is self-adjoint, Ω ρ commutes with L X on C 3 -vectors for any X ∈ sl(2, R) and Ω ρ commutes with ρ(g) for any g ∈ SL(2, R).
Furthermore, when the representation ρ is irreducible, Ω ρ is a scalar multiple of the identity operator, i.e., Ω ρ v = λ(ρ)v for some λ(ρ) ∈ R and for any C 2 -vector v ∈ H of ρ.
In general, as we're going to see below, the spectrum σ(Ω ρ ) of the Casimir operator Ω ρ is a fundamental object.

A.2.
Some examples of SL(2, R) unitary representations. Given a dynamical system consisting of a SL(2, R) action (on a certain space X) preserving some probability measure (µ), we have a naturally associated unitary SL(2, R) representation on the Hilbert space L 2 (X, µ) of L 2 functions of the probability space (X, µ). More concretely, we'll be interested in the following two examples.
Hyperbolic surfaces of finite volume. It is well-known that SL(2, R) is naturally identified with the unit cotangent bundle of the upper half-plane H. Indeed, the quotient SL(2, R)/SO(2, R) on Riemann surfaces of genus g ≥ 1.
As we saw in the last Section 1.4 of Section 1, the case of Q 1 is particularly clear: it is wellknown that Q 1 is isomorphic to the unit cotangent bundle SL(2, Z)\SL(2, R) of the modular curve. In this nice situation, the SL(2, R) action has a natural absolutely continuous (w.r.t. Haar measure) invariant probability µ (1) , so that we have a natural unitary SL(2, R) representation on L 2 (Q 1 , µ (1) ).
After the works of H. Masur and W. Veech, we know that the general case has some similarities with the genus 1 situation, in the sense that connected components C of strata H κ of H g come equipped with a natural Masur-Veech measure invariant probability µ C . In particular, we get also an unitary SL(2, R) representation on L 2 (C, µ C ). More generally, there are plenty of SL(2, R)invariant probabilities µ on C (e.g., coming from square-tiled surfaces) and evidently all of them lead can be used to produce unitary SL(2, R) representations (on L 2 (C, µ)). We remember the reader that the subset C(Γ) detects the presence of complementary series in the decomposition of ρ Γ into irreducible representations. Also, since Γ is a lattice, it is possible to show that C(Γ) is finite and, a fortiori, β(Γ) < 0. Because β(Γ) essentially measures the distance between zero and the first eigenvalue of ∆ S on H Γ , it is natural to call β(Γ) the spectral gap.
Theorem 109. For any f, g ∈ H Γ and |t| ≥ 1, we have sup(σ(∆ S ) ∩ (−∞, −1/4)) < −1/4 and −1/4 is not an eigenvalue of the Casimir operator Ω ρΓ ; Observe that, generally speaking, the results of Avila, Gouezel and Yoccoz say that ρ µ doesn't contain all possible irreducible representations of the complementary series, but it is doesn't give any hint about quantitative estimates of the "spectral gap", i.e., how small ε > 0 can be in general.
In fact, at the present moment, it seems that the only situation where one can say something more precise is the case of the moduli space H 1 : 50 Recall that, roughly speaking, a SL(2, R)-invariant probability is algebraic whenever its support is an affine suborbifold (i.e., a suborbifold locally described, in periodic coordinates, by affine subspaces) such that, in period coordinates, µ is absolutely continuous with respect to Lebesgue measure and its density is locally constant.
Theorem 112 (Selberg/Ratner). The representation ρ H1 has no irreducible factor in the complementary series and it holds | v, ρ H1 w | ≤ C · t · e −t .
In fact, using the notation of Ratner's theorem, Selberg proved that C(SL(2, Z)) = ∅. Since we already saw that H 1 = SL(2, Z)\SL(2, R), the first part of the theorem is a direct consequence of Selberg's result, while the second part is a direct consequence of Ratner's result.
In view of the previous theorem, it is natural to make the following conjecture: For our current task, starting from the vertex at the center of the Rauzy we take the following Here, using the language of [72], we're coding arrows by the associated winning letter. The fact that the four letters A, B, C, D appear in the construction of γ means that it is ∞-complete, so that γ represents a periodic orbit of Teichmüller flow.
A direct calculation (with the formulas presented in [72]) shows that the KZ cocycle over γ is represented by a matrix B γ with characteristic polynomial By performing the substitution y = x + 1/x, we obtain y 2 − 7y + 9 whose discriminant is we see that B γ has a pair of complex conjugated eigenvalues of modulus 1, i.e., γ gives a pseudo-Anosov in H(2) such that KZ cocycle over γ has vanishing second Lyapunov exponent. In the case of volume of strata of moduli spaces of Abelian differentials, the strategy is "similar": • firstly, one realizes that the role of integral points is played by square-tiled surfaces, so that the volume of the "ball" H (R) (κ) of translation surfaces in the stratum H(κ) with total area at most R is reasonably approximated by the number N κ (R) of square-tiled surfaces in the stratum H(κ) composed of R unit squares at most; • secondly, one computes the asymptotics N κ (R) ∼ c(κ) · R 2g+s−1 (recall that the stratum H(κ) has complex dimension 2g + s − 1 when κ = (k 1 , . . . , k s ), 2g − 2 = s j=1 k j ); • finally, by homogeneity, one deduces that λ (1) κ (H (1) (κ)) = (4g + 2s − 2) · c(κ). Evidently, the most difficult step here is the calculation of c(κ). In rough terms, the main point is that one can reduce the computation of c(κ) to a combinatorial problem about permutations and the representation theory of the symmetric group helps us in this task. However, the implementation of this idea is a hard task and it is out of the scope of these notes to present the arguments of A. Eskin and A. Okounkov. In particular, we will content ourselves to reduce the calculation of c(κ) to a combinatorial problem and then we will simply state some of the main results of [25].
Finally, we will conclude this appendix by showing how the action of SL(2, Z) on square-tiled surfaces translates in terms of combinatorics of permutations, so that it will be "clear" that the SL(2, Z)-action on square-tiled surfaces with a "low" number of squares can be calculated with the aid of computer programs.
The computation of c(κ) essentially amounts to count the number of square-tiled surfaces with N squares inside a given stratum H(κ). Combinatorially speaking, a square-tiled surface with N squares can be coded numbering its squares from 1 to N and then considering a pair of permutations h, v ∈ S N such that • h(i) is the number of the square to the right of the square i; • v(i) is the number of the square on the top of the square i. 3 Logically, the codification by a pair of permutations is not unique because we can always renumber the squares without changing the square-tiled surface: In combinatorial terms, the operation of renumbering corresponds to perform a simultaneous conjugation of h and v, i.e., we replace the pair of permutations (h, v) by (φhφ −1 , φvφ −1 ). Because 51 In what follows, we will represent permutations by their cycles. By solving this combinatorial problem, A. Eskin and A. Okounkov [25] proved that Theorem 113. The number c(κ) and, a fortiori, the volume λ (1) κ (H (1) (κ)) is a rational multiple of π 2g . Moreover, the generating function is quasi-modular: indeed, it is a polynomial in the Eisenstein series G 2 (q), G 4 (q) and G 6 (q).
Remark 114. The (very) attentive reader may recall that strata are not connected in general and they may have 3 connected components at most distinguished by hyperellipticity and parity of spin structure. As it turns out, the volume of individual connected components can be translated into a combinatorial problem of counting certain equivalence classes of permutations, but the new counting problem becomes slightly harder because parity of spin structure is not completely easy to read off from pairs of permutations: they have to do with the so-called theta characteristics.
Nevertheless, this computation was successfully performed by A. Eskin, A. Okounkov and R.
Pandharipande [26] to determine explicit formulas for volumes of connected components of strata.
Closing this appendix, let's translate the action of SL(2, Z) in terms of pairs of permutations.  In this subsection we'll follow closely [58]. Consider a square-tiled surface M represented as a pair of permutations (h, v) ∈ S N ×S N (see the previous appendix). The subgroup G of S N generated by h and v is called monodromy group. Note that the stabilizers of the squares of M form a conjugacy class of subgroups of G whose intersection is trivial. Conversely, given a finite group G generated by two elements h and v, and a subgroup H of G whose intersection with its conjugated is trivial (i.e., H doesn't contain non-trivial normal subgroups of G), we recover an origami whose squares are labelled by the elements of H\G such that Hgh is the neighbor to the right of Hg ∈ H\G and Hgv is the neighbor on the top of Hg ∈ H\G. For the sake of this subsection, we'll think of a square-tiled surface M as the data of G, H, h, v as above.
As we explained in Section 7, the study of the non-tautological Lyapunov exponents of KZ   where c = [h, v] is the commutator of h and v, n(g) > 0 is the smallest integer such that gc n(g) g −1 ∈ N , and Fix α (n) is the subspace fixed by χ α (n).
Remark 117. An interesting consequence of this formula is the fact that the multiplicity α depend on h and v only by means of its commutator c = [h, v].
This formula is one of the ingredients towards the following result: Corollary 118. The multiplicity α is never equal to 1, i.e., either α = 0 or α > 1.
Once we understand the decomposition of the Aut(M ) N/H-module H • if a is real, a is even and Sp(W a ) is isomorphic to the symplectic group Symp( a , R); • if a is complex, there are integers p a , q a with a = p a +q a such that Sp(W a ) is isomorphic to the group U C (p a , q a ) of matrices with complex coefficients preserving a pseudo-Hermitian form of signature (p a , q a ); • if a is quaternionic, there are integers p a , q a with a = p a + q a such that Sp(W a ) is isomorphic to the group U H (p a , q a ) of matrices with quaternionic coefficients preserving a pseudo-Hermitian form of signature (p a , q a ).
From this discussion, we already can get derive some consequences for the Lyapunov exponents of KZ cocycle: indeed, the fact that Aff * * (M ) acts on complex and quaternionic isotypical components W a via the groups U C (p a , q a ) and U H (p a , q a ) can be used to ensure 52 the presence of |p a − q a | zero Lyapunov exponents (at least). See e.g. [58] and/or Appendix A of [34] for more details. Moreover, by looking at the definitions it is not hard to show that Oseledets subspaces W a (θ, x) associated to a Lyapunov exponent θ of the restriction of KZ cocycle (or, equivalently Aff * * (M )) to a isotypical component W a V a a at a point x in the SL(2, R)-orbit of M are Aut(M )-invariant. Therefore, these Oseledets subspaces W a (θ, x) is a R(Aut(M ))-module obtained as a finite sum of copies of V a , and, a fortiori, the multiplicity of the Lyapunov exponent θ (i.e., the dimension of W a (θ, x)) is a multiple of dim R (V a ).
In a nutshell, we can resume our discussion so far as follows: starting with a square-tiled surface M with a non-trivial group of automorphism Aut(M ) (in the sense that Aut(M ) has a rich representation theory), usually one finds: • several vanishing Lyapunov exponents, mostly coming from complex and/or quaternionic isotypical components, and • high multiplicity, i.e., non-simplicity, of general Lyapunov exponents.
The bounded distortion assumption says that, in some sense, µ is "not very far" from a Bernoulli measure. As an exercise, the reader can check that bounded distortion implies that µ is f -ergodic.
From now on, we will assume that µ has bounded distortion and we think of (f, µ) as our base dynamical system. Next, we discuss some assumptions concerning the class of cocycles we want to investigate over this base dynamics.
Definition 120. We say that a cocycle A : Σ → Sp(2d, R) is • locally constant if A(x) = A x0 , where A ∈ Sp(2d, R) for ∈ Λ, and x = (x 0 , . . . ) ∈ Σ; Remark 121. Following the work [7] of Avila and Viana, we'll focus here in the case A ∈ Symp(d, R), d even, because we want to apply their criterion to a symplectic cocycle closely related to the Kontsevich-Zorich cocycle. However, it is not hard to see that Avila-Viana simplicity criterion below can extended to the groups U C (p, q) and U H (p, q), and this particularly useful because KZ cocycle may act via these groups in some examples (as we already saw above). For more details on this, see [57].
The ergodicity of µ (coming from the bounded distortion property) and the integrability of the cocycle A allow us to apply the Oseledets theorem to deduce the existence of Lyapunov exponents We denote by G(k) the Grassmanian of • isotropic k-planes if 1 ≤ k ≤ d, and • coisotropic k-planes if d ≤ k < 2d.
At this point, we are ready to introduced the main assumptions on our cocycle A: Definition 122. We say that the cocycle A is • pinching if there exists * ∈ Ω such that the spectrum of the matrix A * is simple.
• twisting if for each k there exists (k) ∈ Ω such that Remark 123. Later on, we will refer to the matrices B with the same property as A (k) above as (k-)twisting with respect to (the pinching matrix) A := A * .
In this language, one has the following version of the simplicity criterion of A. Avila and M.

Viana [7]:
Theorem 124. Let A be a locally constant log-integrable cocycle over a base dynamics (f, µ) consisting of a countable shift f and a f -invariant probability measure µ with bounded distortion.
Suppose that the cocycle A is pinching and twisting. Then, the Lyapunov spectrum of A is simple.
In the sequel, we wish to apply this result to produce a simplicity criterion for the KZ cocycle over SL(2, R)-orbits of square-tiled surfaces. For this sake, we will need to briefly discuss how to This hints that it is a nice idea to start our discussion by reviewing how the geodesic flow on the modular surface is coded by continued fraction algorithm.
Below, we illustrate irrational lattices of top and bottom types: Using this proposition, we can describe the Teichmüller geodesic flow g t = e t 0 0 e −t on the space SL(2, R)/SL(2, Z) of normalized lattices as follows. Let L 0 be a normalized irrational lattice, and let (v 1 , v 2 ) be the basis of L 0 given by the proposition above, i.e., the top, resp. bottom, condition. Then, we see that the basis (g t v 1 , g t v 2 ) of L t := g t L 0 satisfies the top, resp. bottom condition for all t < t * , where λ 1 e t * = 1 in the top case, resp. λ 2 e t * = 1 in the bottom case.
However, at time t * , the basis {v * 1 = g t * v 1 , v * 2 = g t * v 2 } of L 0 ceases to fit the requirements of the proposition above, but we can remedy this problem by changing the basis: for instance, if the basis {v 1 , v 2 } of the initial lattice L 0 has top type, then it is not hard to check that where a = λ 2 /λ 1 is a basis of L t * of bottom type. This is illustrated in the picture below: Here, we observe that the quantity α := λ 1 /λ 2 ∈ (0, 1) giving the ratios of the first coordinates of the vectors g t v 1 , g t v 2 forming a top type basis of L t for any 0 ≤ t < t * is related to the integer a by the formula a = 1/α Also, the new quantity α giving the ratio of the first coordinates of the vectors v 1 , v 2 forming a bottom type basis of L t * is related to α by the formula where G is the so-called Gauss map. In this way, we find the classical relationship between the geodesic flow on the modular surface SL(2, R)/SL(2, Z) and the continued fraction algorithm.
At this stage, we're ready to code the Teichmüller flow over the unit tangent bundle of the Notice that this graph has finitely many vertices but countably many arrows. Using this graph, For sake of concreteness, let's assume that c = t (top case). Recalling the notations introduced after the proof of Proposition 126, we notice that the lattice L t * associated to m t * has a basis of where h 1 = h * h 0 and In other words, starting from the vertex (M i , t) associated to the initial point m 0 , after running the geodesic flow for a time t * , we end up with the vertex (M j , b) where M j = h * M i . Equivalently, the piece of trajectory from m 0 to g t * m 0 is coded by the arrow Evidently, we can iterate this procedure (by replacing L 0 by L t * ) in order to code the entire orbit g t m 0 by a succession of arrows. However, this coding has the "inconvenient" (with respect to the setting of Avila-Viana simplicity criterion) that it is not associated to a complete shift but only a subshift (as we do not have the right to concatenate two arrows γ and γ unless the endpoint of γ coincides with the start of γ ).
Fortunately, this little difficulty is easy to overcome: in order to get a coding by a complete shift, it suffices to fix a vertex p * ∈ Vert(Γ(M )) and consider exclusively concatenations of loops based at p * . Of course, we pay a price here: since there may be some orbits of g t whose coding is not a concatenation of loops based on p * , we're throwing away some orbits in this new way of coding. But, it is not hard to see that the (unique, Haar) SL(2, R)-invariant probability µ on SL(2, R)/SL(M ) gives zero weight to the orbits that we're throwing away, so that this new coding still captures most orbits of g t (from the point of view of µ). In any case, this allows to code g t by a complete shift whose (countable) alphabet is constituted of (minimal) loops based at p * .
Once we know how to code our flow g t by a complete shift, the next natural step (in view of Avila-Viana criterion) is the verification of the bounded distortion condition of the invariant measure induced by µ on the complete shift.
As we saw above, the coding of the geodesic flow (and modulo the stable manifolds, that is, the given by (p, α) → (p , G(α)) where G(α) = {1/α} = α is the Gauss map and p γa,p → p with a = 1/α . In this language, µ becomes (up to normalization) the Gauss measure dt/(1 + t) on each copy {p} × (0, 1), p ∈ Vert(Γ(M )), of the unit interval (0, 1). Now, for sake of concreteness, let us fix p * a vertex of top type. Given γ a loop based on p * , i.e., a word on the letters of the alphabet of the coding leading to a complete shift, we denote by I(γ) ⊂ (0, 1) the interval corresponding to γ, that is, the interval I(γ) consisting of α ∈ (0, 1) such that the concatenation of loops (based at p * ) coding the orbit of (p * , α) starts by the word γ.
In this setting, the measure induced by µ on the complete shift is easy to express: by definition, the measure of the cylinder Σ(γ) corresponding to concatenations of loops (based at p * ) starting by γ is the Gauss measure of the interval I(γ) up to normalization. Because the Gauss measure is equivalent to the Lebesgue measure (as its density 1/(1 + t) satisfies 1/2 ≤ 1/(1 + t) ≤ 1 in (0, 1)), we conclude that the measure of Σ(γ) is equal to |I(γ)| := Lebesgue measure of I(γ) up to a multiplicative constant.
In particular, it follows that the bounded distortion condition for the measure induced by µ on the complete shift is equivalent to the existence of a constant C > 0 such that for every γ = γ 0 γ 1 .
In resume, this reduces the bounded distortion condition to the problem of understanding the interval I(γ). Here, by the usual properties of the continued fraction algorithm, it is not hard to show that I(γ) is a Farey interval being t-reduced, i.e., 0 < p ≤ p, q < q.
Consequently, from this description, we recover the classical fact that Given γ = γ 0 γ 1 , and denoting by p 0 p 0 q 0 q 0 , resp. p 1 p 1 q 1 q 1 , resp. p p q q the matrices associated to γ 0 , resp. γ 1 , resp. γ, it is not hard to check that so that q = q 0 p 1 + q 0 q 1 . Because these matrices are t-reduced, we have that Therefore, in view of (D.1) and (D.2), the bounded distortion condition follows.
Once we know that the basis dynamics (Teichmüller geodesic flow  can be obtained in this way? In this direction, we recall the following definition (already encountered in the previous section): Observe that the product of two t-reduced (resp. b-reduced) matrices is also t-reduced (resp. b-reduced), i.e., these conditions are stable by products.
The following statement is the answer to the question above: Yoccoz and the second author [57] showed that pinching and twisting conditions (and, a fortiori, simplicity of Lyapunov spectrum) can be obtained from certain Galois theory conditions in the context of square-tiled surfaces: Theorem 129. Let M be a square-tiled surface. Suppose that there are two affine diffeomorphisms ϕ A and ϕ B whose linear parts Dϕ A and Dϕ B are either both t-reduced or both b-reduced, and assume that the action of ϕ A and ϕ B on the complementary subspace H i) The eigenvalues of A are real.
ii) The splitting field of the characteristic polynomial P of A has degree 2 g−1 (g − 1)!, i.e., the Galois group is the biggest possible.
iii) A and B 2 don't share a common proper invariant subspace.
Then the Lyapunov spectrum of M is simple.
Finally, the condition iii) above can be verified by checking that i) and ii) hold, and the disjointness of the splitting fields of A and B (see Remark 130 below).
In what follows, we'll give a sketch of proof of this theorem. We begin by noticing that the matrix A verifies the pinching condition (cf. Theorem 124 and Definition 122): indeed, since the Galois group G of P is the largest possible, we have that P is irreducible, and thus its roots are simple. By the assumption (i), all roots λ i , λ −1 i , 1 ≤ i ≤ d, of P are real, so that the pinching condition is violated by A precisely when there are i = j such that λ i = −λ ± j . However, this is impossible because G is the largest possible: for instance, since i = j, we have an element of G fixing λ i and exchanging λ j and λ −1 j ; applying this element to the relation λ i = −λ ± j , we would get that λ i = −λ j and λ i = −λ −1 j , so that λ j = ±1 a contradiction with the fact that P is irreducible.
Remark 130. Concerning the applications of this theorem to the case of origamis, we observe that item (iii) is satisfied whenever the splitting fields Q(P B ) and Q(P ) of the characteristic polynomials of B and A are disjoint as extensions of Q, i.e., Q(P B ) ∩ Q(P ) = Q. Indeed, if E ⊂ R 2d is invariant by A and B 2 , one has that • E is generated by eigenvectors of A (as A is pinching, i.e., A has simple spectrum), so that E is defined over Q(P ), and • E is invariant by B 2 , so that E is also defined over Q(P B ).
Since Q(P ) and Q(P B ) are disjoint, it follows that E is defined over Q. But this is impossible as A doesn't have rational invariant subspaces (by (i) and (ii)).
Once we know that the matrix A satisfies the pinching condition, the proof of Theorem 129 is reduced to checking the twisting condition with respect to A (Remark 123). Keeping this goal in mind, we introduced the following notations.
We denote by R the set of roots of the polynomial P (so that # R = 2d), for each λ ∈ R, we put p(λ) = λ + λ −1 , and we define R = p( R) (so that #R = d).
Given 1 ≤ k ≤ d, let R k , resp. R k be the set whose elements are subsets λ of R, resp. R with k elements, and let R k be the set whose elements are subsets λ of R with k elements such that p| λ is injective. In other words, R k consist of those λ ∈ R k such that if λ ∈ λ, then λ −1 / ∈ λ.
Next, we make a choice of basis of R 2d as follows. For each λ ∈ R, we select an eigenvector v λ of A associated to λ, i.e., Av λ = λv λ . In particular, v λ is defined over Q(λ) ⊂ Q(P )). Then, we assume that the choices of v λ 's are coherent with the action of the Galois group G, i.e., v gλ = gv λ (and thus A(v gλ ) = (gλ)v gλ ) for each g ∈ G. In this way, for each λ ∈ R k , we can associated a multivector v λ = v λ1 ∧ · · · ∧ v λ k ∈ k R 2d (using the natural order of the elements of λ = {λ 1 < · · · < λ k }).
From our assumptions (i) and (ii) on A, we have that: • v λ , λ ∈ R k , is a basis of k R 2d • the subspace generated by v λ1 , . . . , v λ k is isotropic if and only if λ = {λ 1 , . . . , λ k } ∈ R k Also, by an elementary (linear algebra) computation, it is not hard to check that a matrix C is twisting with respect to A if and only if λ,λ are the coefficients of the matrix k C in the basis v λ .
In order to organize our discussions, we observe that the condition (D.3) can be used to define an oriented graph Γ k (C) as follows. Its vertices Vert(Γ k (C)) are Vert(Γ k (C)) = R k , and we have an arrow from λ 0 to λ 1 if and only if C Of course, the verification of the completeness of Γ k (C) is not simple in general, and hence it could be interesting to look for more soft properties of Γ k (C) ensuring completeness of Γ k (D) for some matrix D constructed as a product of powers of C and A. Here, we take our inspiration from Dynamical Systems and we introduce the following classical notion: Definition 131. Γ k (C) is mixing if there exists m ≥ 1 such that for all λ 0 , λ 1 ∈ R k we can find an oriented path in Γ k (C) of length m going from λ 0 to λ 1 .
Here, we note that it is important in this definition that we can connect two arbitrary vertices by a path of length exactly m (and not of length ≤ m). For instance, the figure below shows a connected graph that is not mixing because all paths connecting A to B have odd length while all paths connecting A to C have even length.
As the reader can guess by now, mixing is a soft property ensuring completeness of a "related" graph. This is the content of the following proposition: Proof. To alleviate the notation, let's put D = D(n). By definition, Our goal is to prove that D so that the resulting expression vanishes. The idea is to show that can be chosen suitably to avoid such cancelations, and the heart of this argument is the observation that, for γ = γ , the linear forms L γ and L γ are distinct. Indeed, given λ ∈ R k and λ ∈ R k , λ = λ, we claim that the following coefficients of L γ and L γ differ: But, since λ ∈ R k , we have that if λ ∈ λ then λ −1 / ∈ λ. In particular, by taking an element λ(0) ∈ λ − λ , and by considering an element g of the Galois group G with g(λ(0)) = λ(0) −1 and g(λ) = λ otherwise, one would get on one hand that but, on the other hand, Finally, we complete the proof by noticing that if / ∈ V 1 ∪ · · · ∪ V t , then for n → ∞ sufficiently large because the coefficients L γ ( ) are mutually distinct.
At this point, the proof of Theorem 129 goes along the following lines: • Step 0: We will show that the graphs Γ k (C) are always non-trivial, i.e., there is at least one arrow starting at each of its vertices.
• Step 1: Starting from A and B as above, we will show that Γ 1 (B) is mixing and hence, by Proposition 132, there exists C twisting 1-dimensional (isotropic) A-invariant subspaces.
• By Step 1, the treatment of the case d = 1 is complete, so that we have to consider d ≥ 2.
Unfortunately, there is no "unified" argument to deal with all cases and we are obliged to separate the case d = 2 from d ≥ 3.
• Step 2: In the case d ≥ 3, we will show that Γ k (C) (with C as in Step 1) is mixing for all 1 ≤ k < d. Hence, by Proposition 132, we can find D twisting k-dimensional isotropic A-invariant subspaces for all 1 ≤ k < d. Then, we will prove that Γ d (D) is mixing and, by Proposition 132, we have E twisting with respect to A, so that this completes the argument in this case.
• Step 3: In the special case d = 2, we will show that either Γ 2 (C) or a closely related graph Γ * 2 (C) are mixing and we will see that this is sufficient to construct D twisting 2-dimensional isotropic A-invariant subspaces.
In the sequel, the following easy remarks will be repeatedly used: Remark 133. If C ∈ Sp(2d, Z), then the graph Γ k (C) is invariant under the action of Galois group G on the set R k × R k (parametrizing all possible arrows of Γ k (C)). In particular, since the Galois group G is the largest possible, whenever an arrow λ → λ belongs to Γ k (C), the inverse arrow λ → λ also belongs to Γ k (C). Consequently, Γ k (C) always contains loop of even length.
Remark 134. A connected graph Γ is not mixing if and only if there exists an integer m ≥ 2 such that the lengths of all of its loops are multiples of m.
Lemma 135. Let C ∈ Sp(2d, R). Then, each λ ∈ R k is the start of at least one arrow of Γ k (C).
Remark 136. Notice that we allow symplectic matrices with real (not necessarily integer) coefficients in this lemma. However, the fact that C is symplectic is important here and the analogous lemma for general invertible (i.e., GL) matrices is false.
Proof. For k = 1, since every 1-dimensional subspace is isotropic, R 1 = R and the lemma follows in this case from the fact that C is invertible. So, let's assume that k ≥ 2 (and, in particular, R k is a proper subset of R k ). Since C is invertible, for each λ ∈ R k , there exists λ ∈ R k with C (k) λ,λ = 0 Of course, one may have a priori that λ ∈ R k − R k , i.e., #p(λ) < k, and, in this case, our task is to "convert" λ into some λ ∈ R k with C (k) λ,λ = 0.
Step 1: Γ 1 (B) is mixing. For d = 1, the set R 1 consists of exactly one pair = {λ, λ −1 }, so that the possible Galois invariant graphs are: In the first case, by definition, we have that B(Rv λ ) = Rv λ (and B(Rv λ −1 ) = Rv λ −1 ), so that B and A share a common subspace, a contradiction with our hypothesis in Theorem 129.
In the second case, by definition, we have that B(Rv λ ) = Rv λ −1 and B(Rv λ −1 ) = Rv λ , so that B 2 (Rv λ ) = Rv λ and thus B 2 and A share a common subspace, a contradiction with our assumptions in Theorem 129.
Finally, in the third case, we have that the graph Γ 1 (B)  In this case, it is not hard to see that the addition of any extra arrow allows to build up loops of lenght 3, so that, by Remarks 133 and 134, the argument is complete.
Therefore, in any event, we proved that Γ 1 (B) is mixing.
Step 2: For d ≥ 3, Γ k (C) is mixing for 2 ≤ k < d, and Γ d (D) is mixing. Given C ∈ Sp(2d, Z) twisting 1-dimensional A-invariant subspaces, we wish to prove that Γ k (C) is mixing for all 2 ≤ k < d whenever d ≥ 3. Since Γ k (C) is invariant under the Galois group G (see Remark 133), we start by considering the orbits of the action of G on R k × R k .

Proposition 137. The orbits of the action of
We leave the proof of this proposition as an exercise to the reader. This proposition says that the orbits of the action G on R k × R k are naturalized parametrized by In particular, since Γ k (C) is G-invariant, we can write Γ k (C) = Γ k ( J) for some J := J(C) ⊂ I, where Γ k (J) is the graph whose vertices are R k and whose arrows are λ,λ = 0, and hence we can find w 1 , . . . , w k such that span{w 1 , . . . , w k } = span{v λ1 , . . . , v λ k } and In other words, as we also did in Step 0, we can use w 1 , . . . , w k to "convert" the minor of C associated to λ, λ into the identity.
From the claim above we deduce that e.g. C(v λ1 ) is a linear combination of v λ i , i = 1, . . . , k, a contradiction with the fact that C twists 1-dimensional A-invariant subspaces. In other words, we proved that Γ k (C) is mixing for each 2 ≤ k < d whenever C twists 1-dimensional A-invariant subspaces.
By Proposition 132, it follows that we can construct a matrix D twisting k-dimensional isotropic A-invariant subspaces for 1 ≤ k < d, and we wish to show that Γ d (D) is mixing. In this direction, we consider the orbits of the action of the Galois group G on R d × R d . By Proposition 137, the orbits are O e , = {(λ, λ ) ∈ R k × R k : #(λ ∩ λ ) = , #(p(λ) ∩ p(λ )) = } with ≤ k, ≥ 2k − d and k = d. In particular, = d in this case, and the orbits are parametrized by the set It is possible to show (again by the arguments with "minors" we saw above) that if D is ktwisting with respect to A, then J contains two consecutive integers say , + 1.
Thus, the connectedness of Γ d (J) follows.
Next, we show that Γ d (J) is mixing. Since Γ d (J) is invariant under the Galois group, it contains loops of length 2 (see Remark 133). By Remark 134, it suffices to construct some loop of odd length in Γ d (J). We fix some arrow λ → λ ∈ O( ) of Γ d (J). By the construction "λ 0 → λ → λ 1 " when #(λ 0 ∩ λ 1 ) = d − 1 performed in the proof of the connectedness of Γ d (J), we can connect λ to λ by a path of length 2 in Γ d (J). In this way, we have a loop (based on λ 0 ) in Γ d (J) of length 2 + 1.
By definition, ∧ 2 A(v * ) = v * , so that 1 is an eigenvalue of ∧ 2 A| K . In principle, this poses a problem to apply Proposition 132 (to deduce 2-twisting properties of C from Γ * 2 (C) is mixing), but, as it turns out, the fact that the eigenvalue 1 of ∧ 2 A| K is simple can be exploited to rework the proof of Proposition 132 to check that Γ * 2 (C) is mixing implies the existence of adequate products D of powers of C and A satisfying the 2-twisting condition (i.e., D twists 2-dimensional A-invariant isotropic subspaces). Therefore, it "remains" to show that either Γ 2 (C) or Γ * 2 (C) is mixing to complete this step. We write Γ 2 (C) = Γ 2 (J) with J ⊂ {0, 1, 2}.
• If J contains 2 two consecutive integers, then one can check that the arguments of the end of the previous sections work and Γ 2 (C) is mixing. For sake of concreteness, we will consider the cases J = {2} and J = {1} (leaving the treatment of their "symmetric" as an exercise). We will show that the case J = {2} is impossible while the case J = {1} implies that Γ * 2 (C) is mixing. We begin by J = {2}. This implies that we have an arrow λ → λ with λ = {λ 1 , λ 2 }. Hence, we can find w 1 , w 2 with span{w 1 , w 2 } = span{v λ1 , v λ2 } and Since J = {2}, the arrows λ → {λ −1 1 , λ 2 }, λ → {λ 1 , λ −1 2 }, and λ → {λ −1 1 , λ −1 2 } do not belong Γ 2 (J). Thus, C * 11 = C * 22 = 0 = C * 12 C * 21 . On the other hand, because C is symplectic, ω 1 C * 21 − ω 2 C * 12 = 0 (with ω 1 , ω 2 = 0). It follows that C * ij = 0 for all 1 ≤ i, j ≤ 2, that is, C preserves the A-subspace spanned by v λ1 and v λ2 , a contradiction with the fact that C is 1-twisting with respect to A. Now we consider the case J = {1} and we wish to show that Γ * 2 (C). We claim that, in this situation, it suffices to construct arrows from the vertex v * to R 2 and vice-versa. Notice that the action of the Galois group can't be used to revert arrows of Γ * 2 (C) involving the vertex v * , so that the two previous statements are "independent". Assuming the claim holds, we can use the Galois action to see that once Γ * 2 (C) contains some arrows from v * and some arrows to v * , it contains all such arrows. In other words, if the claim is true, we have the following situation: Thus, we have loops of length 2 (in R 2 ), and also loops of length 3 (based on v * ), so that Γ * 2 (C) is mixing. In particular, our task is reduced to show the claim above.
The fact that there are arrows from R 2 to v * follows from the same kind of arguments involving "minors" (i.e., selecting w 1 , w 2 as above, etc.) and we will not repeat it here.
Instead, we will focus on showing that there are arrows from v * to R 2 . The proof is by contradiction: otherwise, one has ∧ 2 C(v * ) ∈ Rv * . Then, we invoke the following elementary lemma (whose proof is a straightforward computation): Since v * := i(span(v λ1 , v λ −1 1 )), from this lemma we obtain that ∧ 2 C(v * ) ∈ Rv * implies a contradiction with the fact that C is 1-twisting with respect to A.
This completes the sketch of proof of Theorem 129. Now, we will conclude this appendix with some applications of this theorem to the first two "minimal" strata H(2) and H(4).
D.3. Some applications of Theorem 129. In the sequel we will need the following lemma about the "minimal" strata: Lemma 140. A translation surface in the stratum H(2g − 2)has no non-trivial automorphisms.
Proof. Let (M, ω) ∈ H(2g − 2) and denote by p M ∈ M the unique zero of the Abelian differential ω. Any automorphism φ of the translation structure (M, ω) satisfies φ(p M ) = p M . Suppose that there exists φ a non-trivial automorphism of (M, ω). We have that φ has finite order, say φ κ = id where κ ≥ 2 is the order of φ. Since φ fixes p M and φ is non-trivial, p M is the sole fixed point of φ. Hence, the quotient N of M by the cyclic group φ Z/κZ can be viewed as a normal covering π : (M, p M ) → (N, p N ) of degree κ such that π is not ramified outside p N := π(p M ), π −1 (p N ) = {p M } and π is ramified of index κ at p N . Now let us fix * ∈ N − {p N } a base point, so that the covering π is given by a homomorphism h : π 1 (N − {p N }, * ) → Z/κZ. In this language, we see that a loop γ ∈ π 1 (N − {p N }, * ) around p N based at * is a product of commutators in π 1 (N − {p N }, * ), so that h(γ) = 1 (as Z/κZ), a contradiction with the fact that π is ramified of order κ ≥ 2 at p N .
Remark 141. In a certain sense this lemma implies that origamis of minimal strata H(2g − 2) are not concerned by the discussions of Subsection D.1: indeed, there the idea was to use representation theory of the automorphism group of the origami to detect multiple (i.e., non-simple) and/or zero exponents; in particular, we needed rich groups of automorphisms and hence, from this point of view, origamis without non-trivial automorphisms are "uninteristing".
Next, we recall that M. Kontsevich and A. Zorich classified all connected components of strata H(k 1 −1, . . . , k s −1), and, as an outcome of this classification, any stratum has at most 3 connected components. We will come back to this latter when discussing the stratum H(4). For now, let us mention that the two strata H(2) and H(1, 1) of genus 2 translation structures are connected.
Once we fix a connected component C of a stratum, we can ask about the classification of SL(2, Z)-orbits of primitive 53 square-tiled surfaces in C. In this direction, it is important to dispose of invariants to distinguish SL(2, Z)-orbits. As it turns out, the monodromy group of a square-tiled surface (cf. Subsection D.1) is such an invariant, and the following result (from the PhD thesis) of D. Zmiaikou [73] shows that this invariant can take only two values when the number of squares of the origami is sufficiently large: Theorem 142 (D. Zmiaikou). Given an stratum H(k 1 , . . . , k s ), there exists an integer N 0 = N 0 (k 1 , . . . , k s ) such that any primitive origami of H(k 1 , . . . , k s ) with N ≥ N 0 squares has monodromy group isomorphic to A N or S N .
Remark 143. The integer N 0 = N 0 (k 1 , . . . , k s ) has explicit upper bounds (as it was shown by D. Zmiaikou), but we did not include it in the previous statement because it is believed that the current upper bounds are not sharp.
Remark 144. In order to aleviate the notation, we will refer to square-tiled surfaces as origamis in what follows.
After this (brief) general discussion, let's specialize to the case of genus 2 origamis. D.3.1. Classification of SL(2, Z)-orbits of square-tiled surfaces in H(2). Denote by N ≥ 3 the number of squares of a origami in H(2). By the results of P. Hubert and S. Lelièvre [42], and C.
McMullen [62], it is possible to show that the SL(2, Z)-orbits of origamis are organized as follows • if N ≥ 4 is even, then there is exactly 1 SL(2, Z)-orbit and the monodromy group is S N .
• if N ≥ 5 is odd, then there are exactly 2 SL(2, Z)-orbits distinguished by the monodromy group being A N or S N .
• if N = 3, there is exactly 1 SL(2, Z)-orbit and the monodromy group is S 3 .
Concerning the Lyapunov exponents of the KZ cocycle, recall that M. Bainbridge [9], and A.
Of course, they say much more as the explicit value of λ 2 is given, while the simplicity in H(2) amounts only to say that λ 2 > 0(> −λ 2 ). But, as it turns out, the arguments employed by Bainbridge and EKZ are sophisticated (involving the Deligne-Mumford compactification of the moduli space of curves, etc.) and long (both of them has ≥ 100 pages). So, partly motivated by this, we will discuss in this section the application of the simplicity criterion of Theorem 129 to the case of square-tiled surfaces in H(2): evidently, this gives only the fact that λ 2 > 0 but it has the advantage of relying on the elementary methods developed in previous posts.
We begin by selecting the following L-shaped origami L(m, n): 53 By definition, this means that the square-tiled surface is not obtained by taking a covering of an intermediate square-tiled tiled surface.
In terms of the parameters m, n ≥ 2, the total number of squares of L(m, n) is N = n + m − 1 ≥ 3. Notice that the horizontal permutation r associated L(m, n) is a m-cycle while the vertical permutation u is a n-cycle. Thus, by our discussion so far above, one has that: • if m + n is odd, the monodromy group is S N , • if m and n are odd, the monodromy group is A N , and • if m and n are even, the monodromy group is S N .
As we already know, given an origami p : M → T 2 , we have a decomposition In the particular case of L(m, n), in terms of the homology cycles σ 1 , σ 2 , ζ 1 , ζ 2 showed in the previous picture, we can construct the following basis of H Since L(m, n) ∈ H(2) has no automorphisms (see Lemma 140), we can also think of S and T as elements of the affine group of L(m, n).
As the reader can check in the figures, the action of S on the homology cycles σ 1 , σ 2 , ζ 1 , ζ 2 is • S(σ i ) = σ i for i = 1, 2; • S(ζ 1 ) = ζ 1 + σ 1 + (n − 1)mσ 2 • S(ζ 2 ) = ζ 2 + σ 1 Therefore, S(σ) = σ and S(ζ) = ζ + (n − 1)σ. Actually, the same computation with S replaced by any power S a , a ∈ N, gives the "same" result S a (σ) = σ, S a (ζ) = ζ + a(1 − n)σ By the natural symmetry between the horizontal and vertical directions in L(m, n), there is no need to compute twice to get the corresponding formulas for T and/or T b : By combining S and T , we can find pinching elements acting on H is not a square, A has two real irrational eigenvalues, and hence A is a pinching matrix whose eigenspaces are defined over Q( √ t A ).
Thus, we can apply the simplicity criterion as soon as one has a twisting matrix B with respect to A. We take B := S 2 T 2 By applying the previous formulas for the powers S a and T b of S and T (with a = b = 2), we find that the trace of B is t B = 2 + 4(m − 1)(n − 1) so that t 2 B − 4 = 16(m − 1)(n − 1)[(m − 1)(n − 1) + 1] is also not a square. Hence, B is also a pinching matrix whose eigenspaces are defined over Q( √ t B ). Furthermore, these formulas for t A and t B above show that the quadratic fields Q( √ t A ) and Q( √ t B ) are disjoint in the sense that So, B is twisting with respect to A, and therefore 1 = λ 1 > λ 2 > 0.
Remark 145. For the "other" stratum of genus 2, namely H(1, 1), the results of Bainbridge and EKZ show that λ 2 = 1/2. In principle, the (weaker) fact that λ 2 > 0 in this situation can be derived for particular origamis in H(1, 1) along the lines sketched above for H(2), but it is hard to treat such origamis in a systematic way because currently there is only a conjectural classification of SL(2, Z)-orbits.
This closes our (preliminary) discussions of the application of the simplicity criterion in the (well-established) case of H(2). Now, we pass to the case of the stratum H(4). Remark 146. The sole zero of M ∈ H(2g − 2) is fixed under the involution φ, so that 0 ≥ 1.
Remark 147. For i > 0, i ≡ N (mod 2): for instance, the involution permute the squares forming M and, e.g. 2 is the number of squares fixed by φ; since φ is an involution, the φ-orbits of squares have size 1 or 2, so that 2 ≡ N (mod 2); finally, since SL(2, Z) acts transitively on the set of half-integer points of T 2 , we can "replace" 2 by i , i > 0, to get the same conclusion for all i > 0.
The invariant [ 0 , { 1 , 2 , 3 }] was introduced by E. Kani [43], and P. Hubert and S. Lelievre assisted computations as we prefer to postpone them to the forthcoming article, and we will end this series with this "incomplete" argument as it already contains the main ideas and it is not as technical as the complete argument.
Recall that if the roots of P A are all real, and the Galois group is G is the largest possible, i.e., G S 2 × (Z/2Z) 2 , then the matrix A is pinching.
Following the usual trick to determine the roots of P A , we pose y = x + 1/x, so that we get an equation Q A (y) = y 2 + ay + (b − 2) with a, b ∈ Z. The roots of Q A are y ± = −a ± √ ∆ 1 2 where ∆ 1 := a 2 − 4(b − 2). Then, we recover the roots of P A by solving the equation For later use, we denote by K A , resp. K B , the splitting field of P A , resp. P B .
We leave the following proposition as an exercise in Galois theory for the reader: Proposition 150. It holds: (a) If ∆ 1 is a square (of an integer number), then Q A and P A are not irreducible.
In the sequel, we will always assume that ∆ 1 is not a square.   (e) If (∆ 1 ,) ∆ 2 and ∆ 1 ∆ 2 are not squares, and ∆ 1 ∆ 2 > 0, then the roots of P A are real and the Galois group of P A is the largest possible.
This proposition establishes a sufficient criterion for A, B ∈ Sp(4, Z) to fit the pinching and twisting conditions. Indeed, we can use item (e) to get pinching matrices A, and we can apply item (d) to produce twisting matrices B with respect to A as follows: recall that (cf. Remark 130) the twisting condition is true if the splitting fields K A and K B are "disjoint" (i.e., K A ∩ K B = Q), and, by item (d), this disjointness can be checked by computing the quantities ∆ 1 , ∆ 2 , ∆ 1 ∆ 2 associated to A, and ∆ 1 , ∆ 2 , ∆ 1 ∆ 2 associated to B, and verifying that they generated distinct quadratic fields. Now, we consider the following specific family M = M n , n ≥ 5, of origamis in H(4) odd : Here, the total number of squares is N = n + 4 and −id doesn't belong to the Veech group (i.e., this origami is "generic" in H(4) odd ). The horizontal permutation is the product of a n-cycle with a 2-cycle (so its parity equals the parity of n), and the vertical permutation is the product of a 4-cycle with a 2-cycle (so that its parity is even). In particular, here the monodromy group is S N if N is odd, and A N if N is even.
In terms of the homology cycles σ 1 , σ 2 , σ 3 , ζ 1 , ζ 2 , ζ 3 showed in the figure above, we can select the following basis of H 1 (M, Q): Notice that when n is even we can replace 2n by n in the definition of S, but we prefer to consider S directly (to avoid performing to similar discussions depending on the parity of n).
By computing with S and T , and using that y ± are the solutions of y 2 − ay + (b − 2) = 0, one finds a = −7n + 6, b = 8n 2 − 2n − 14 and ∆ 1 = 17n 2 − 76n + 100 By Proposition 150, we wish to know e.g. how frequently ∆ 1 is a square. Evidently, one can maybe produce some ad hoc method here, but since we want to verify simplicity conditions in a systematic way, it would be nice to this type of question in "general".
Here, the idea is very simple: the fact that ∆ 1 = 17n 2 − 76n + 100 is a square is equivalent to get integer and/or rational solutions to z 2 = 17t 2 − 76t + 100, that is, we need to understand integer/rational points in this curve.
This hints the following solution to our problem: if we can replace A = ST by more complicated products of (powers of) S and T chosen more or less by random, it happens that ∆ 1 becomes a polynomial P (n) of degree ≥ 5 without square factors. In this situation, the problem of knowing whether ∆ 1 = P (n) is a square of an integer number becomes the problem of finding integer/rational points in the curve z 2 = P (n).
Since this a non-singular curve (as P (n) has no square factors) of genus g ≥ 2 (as deg(P ) ≥ 5), we know by Faltings' theorem (previously known as Mordell's conjecture) that the number of rational solutions is finite.
Remark 151. In the case we get polynomials P (n) of degree 3 or 4 after removing square factors of P (n), we still can apply (and do [apply in [57]]) Siegel's theorem saying that the genus 1 curve z 2 = P (n) has finite many integer points, but we preferred to mention the case of higher degree polynomials because it is the "generic situation" of the argument (in some sense).
In other words, by the end of the day, we have that ∆ 1 = P (n) is not the square of an integer number for all but finitely many values of n.
Of course, at this point, the general idea to get the pinching and twisting conditions (and hence simplicity) for the origamis M = M n for all but finitely many n ∈ N is clear: one produces "complicated" products of S, T (and also a third auxiliary parabolic matrix U ) leading to elements A, B ∈ Sp(4, Z) such that the quantities ∆ i = ∆ i (n), i = 1, 2, ∆ 3 = ∆ 1 ∆ 2 (n) associated to A, ∆ i = ∆ i (n), ∆ 3 = ∆ 1 ∆ 2 (n) associated to B (and also the "mixed products" ∆ i ∆ j for i, j = 1, 2, 3) are polynomials of the variable n of high degree (and without square factors if possible), so that these quantities don't take square of integers as their values for all but finitely many n ∈ N.
Unfortunately, it seems that there is no "systematic way" of choosing these "complicated"