The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis

We compute explicitly the action of the group of affine diffeomorphisms on the relative homology of two remarkable origamis discovered respectively by Forni (in genus 3) and Forni-Matheus (in genus 4). We show that, in both cases, the action on the non trivial part of the homology is through finite groups. In particular, the action on some 4-dimensional invariant subspace of the homology leaves invariant a root system of $D_4$ type. This provides as a by-product a new proof of (slightly stronger versions of) the results of Forni and Matheus: the non trivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the Teichmuller disks of these two origamis are equal to zero.


INTRODUCTION
Our main objective is the explicit description of the action on homology of the affine group of the square-tiled translation surfaces constructed by Forni [7] and Forni-Matheus [8], characterized by the total degeneracy of the so-called Kontsevich-Zorich cocycle. Before going to the statements of our results, let us briefly recall some basic material about these notions.  See Veech [16], [18], and the surveys of Yoccoz [20] and Zorich [23] for more details.
Once we get the existence of a natural invariant measure µ (1) κ for the Teichmüller flow, it is natural to ask whether µ (1) κ has finite mass and/or µ (1) κ is ergodic with respect to the Teichmüller dynamics. In this direction, we have the following result: Theorem 1.2 (Masur [14], Veech [16]). The total volume of µ (1) κ is finite and the Teichmüller flow G t = diag(e t , e −t ) is ergodic on each connected component of M(M, κ) with respect to µ (1) κ . Remark 1.3. Veech [18] showed that the strata are not always connected. More recently, Kontsevich and Zorich [12] gave a complete classification of the connected components of all strata of the moduli spaces of holomorphic 1-forms (see Lanneau [13] for the same result for quadratic differentials).

1.2.
The affine group of a translation surface. Let (M, ω) be a translation surface, i.e let ω be a non-zero holomorphic 1-form w.r.t. some complex structure on M . We denote as above by Σ the set of zeros of ω.
Definition 1.4. The affine group Aff(M, ω) of (M, ω) is the group of orientation preserving homeomorphisms of M which preserve Σ and are given by affine maps in the charts defined by local primitives of ω. In these charts, the differential of an affine map is an element of SL(2, R). We obtain in this way a homomorphism from Aff(M, ω) into SL(2, R). The automorphism group Aut(M, ω) of (M, ω) is the kernel of this homomorphism. Definition 1.5. The image of this homomorphism is called the Veech group of (M, ω) and denoted by SL(M, ω). It is a discrete subgroup of SL(2, R), equal to the stabilizer of (M, ω) for the action of SL(2, R) on M(M ).
One has thus an exact sequence For a nice account on affine and Veech groups see the survey of Hubert and Schmidt [10]. In genus g ≥ 2, the affine group Aff(M, ω) injects into Γ(M ) (and a fortiori into Γ(M, Σ)): for elements with non trivial image in SL(M, ω), this can be viewed from the period map (see Veech [19]); for elements in Aut(M, ω), this is a consequence of the Lefschetz fixed point theorem, as fixed points then have index 1.
Consider the natural surjective map: between the relative and absolute cohomology groups of (M, Σ). We denote by H 1 st (M, Σ, R) the subspace of H 1 (M, Σ, R) spanned by dx = (ω) and dy = (ω) and by H 1 st its image in H 1 (M, R). The subspace H 1 st (and therefore also H 1 st (M, Σ, R)) is 2-dimensional: this can be seen either as a standard fact from Hodge theory or more concretely from Veech's zippered rectangles construction (see [17], [20], [23]). We denote by H 1 (0) the orthogonal of H 1 st with respect to the exterior product in H 1 (M, R). We have The intersection form defines a non-degenerate pairing between the homology groups H 1 (M − Σ, R) and H 1 (M, Σ, R), and also between H 1 (M, R) and itself.
Let H One has These decompositions only depend on the image of the translation surface structure in Teichmüller space; they are constant along SL(2, R)-orbits, invariant under the action of Aff(M, ω), and covariant under the action of the mapping class groups. The same is true for the decompositions of the cohomology groups. In particular, the decomposition 1.3. Veech surfaces and square-tiled surfaces. Let (M, ω) be a translation surface.
The stabilizer of this SL(2, R)-orbit in Γ(M ) is exactly the affine group Aff(M, ω). We can thus view the reduced Kontsevich-Zorich cocycle G KZ,red t over this closed SL(2, R)orbit as the quotient of the trivial cocycle by the action of the affine group Aff(M, ω).
The two examples that we will consider belong to a special kind of Veech surfaces. Definition 1.9. (M, ω) is a square-tiled surface if the integral of ω over any path joining two zeros of ω belongs to Z + iZ. Equivalently, there exists a ramified covering π : M → R 2 /Z 2 unramified outside 0 ∈ R 2 /Z 2 such that ω = π * (dz). Every square-tiled surface is a Veech surface. One says that the square-tiled surface (M, ω) is primitive if the relative periods of ω span the Z-module Z + iZ. In this case the Veech group SL(M, ω) is a subgroup of SL(2, Z) of finite index (see [10], [23]).
In a square-tiled surface (M, ω), the squares are the connected components of the inverse image π −1 ((0, 1) 2 ), with π : M → R 2 /Z 2 as above. The set Sq(M, ω) of squares of (M, ω) is finite and equipped with two one-to-one self maps r (for right) and u (for up) which associate to a square the square to the right of it (resp. above it). The connectedness of the surface means that the group of permutations of Sq(M, ω) generated by r and u acts transitively on Sq(M, ω). Conversely, a finite set S, equipped with two one-to-one maps r and u such that the group of permutations generated by r and u acts transitively on S, defines a square-tiled surface. See [23].
For a square-tiled surface, it is easy to identify the factors H st 1 and H 1 in the decomposition of the homology groups, and to see that in this case they are defined over Q.
Let Σ ⊃ Σ be the inverse image of {0} under the ramified covering π. For each square i ∈ Sq(M, ω), let σ i ∈ H 1 (M, Σ , Z) be the homology class defined by a path in i from the bottom left corner to the bottom right corner; let ζ i be the homology class defined by a path in i from the bottom left corner to the upper left corner. Let σ (resp. ζ) be the sum over Sq(M, ω) of the σ i (resp. of the ζ i ). It is clear that both σ and ζ belong to H 1 (M, Z). Let σ (resp., ζ) be the class in H 1 (M − Σ, Z) obtained from σ (resp., ζ) by shifting each σ i (resp., ζ i ) slightly upwards (resp., to the right). (1) The subspace H 1 (M, Σ, R)) is the kernel of the homomorphism from H 1 (M, R) (resp., H 1 (M, Σ, R)) to H 1 (R 2 /Z 2 , R) (resp., H 1 (R 2 /Z 2 , {0}, R)) induced by the ramified covering π.
(2) One has Moreover, the action of the affine group on H st 1 is through the homomorphism from the affine group to SL(M, ω) ⊂ SL(2, R) and the standard action of SL(2, R) on Rσ ⊕ Rζ.
Proof. The first part is an immediate consequence of the definitions of H On the other hand, we have θ ∈ H We will denote by H and H st 1 , we omit the coefficients to keep the notation simple: H st 1 designates Qσ ⊕ Qζ or Rσ ⊕ Rζ (the context should remove the ambiguity); H (0) 1 is the kernel of the homomorphism between the first absolute homology groups of M and R 2 /Z 2 with real or rational coefficients.
1.4. Degenerate SL(2, R)-orbits. Veech has asked how "degenerate" the Lyapunov spectrum of G KZ t can be along a non-typical SL(2, R)-orbit, for instance along a closed orbit.
Similarly, Forni and Matheus' example (M 4 , ω (4) ) is the Riemann surface of genus 4 equipped with the Abelian differential ω (4) = zdz/w 3 . Remark 1.11. Actually, these formulas for (M 3 , ω (3) ) and (M 4 , ω (4) ) are the description of entire closed SL(2, R)-orbits. The corresponding square-tiled surfaces (in the sense of the previous definition) belonging to these orbits are obtained by appropriate choices of the points x µ . Also, we point out that these examples are particular cases of a more general family studied by I. Bouw and M. Möller [5].
13. An unpublished work of Martin Möller [15] indicates that such examples with totally degenerate KZ spectrum are very rare: they don't exist in genus g ≥ 6, Forni's example is the unique totally degenerate SL(2, R)-orbit in genus 3 and the Forni-Matheus example is the unique totally degenerate SL(2, R)-orbit in genus 4; also, the sole stratum in genus 5 possibly supporting a totally degenerate SL(2, R)-orbit is M (2,2,2,2) , although this isn't probably the case (namely, Möller pursued a computer program search and it seems that the possible exceptional case M (2,2,2,2) can be ruled out).
Remark 1.14. Forni's version of Kontsevich formula for the sum of the Lyapunov exponents reveals the following interesting feature of the Kontsevich-Zorich cocycle over a totally degenerate SL(2, R): it is isometric with respect to the Hodge norm on the cohomology H 1 (M, R) on the orthogonal complement of the subspace associated to the exponents ±1. For more details see [6] and [7]. Observe that this fact is far from trivial in general since the presence of zero Lyapunov exponents only indicates a subexponential (e.g., polynomial) divergence of the orbits (although in the specific case of the KZ cocycle, Forni manages to show that this "subexponential divergence" suffices to conclude there is no divergence at all).
1.5. Statement of the results. We start with (M 3 , ω (3) ). For this square-tiled surface, the Veech group is the full group SL(2, Z) and the automorphism group is the 8-element quaternion group Q := {±1, ±i, ±j, ±k}: see F. Herrlich and G. Schmithüsen [9] (and also Figure 1 below). We have tried to summarize the main conclusions of the computations of the next section.
(1) One has a decomposition (2) There exists a root system R of D 4 type spanning H 2 This stratum has two connected components distinguished by the parity of the spin structure (see [12]). In particular, one can ask about the connected component of Forni-Matheus's surface. In the Appendix A below, we'll use a square-tiled representation of this example to show that its spin structure is even. on H rel is sent isomorphically onto the principal congruence subgroup Γ(4) by the canonical morphism from Aff(M 3 , ω (3) ) to SL(2, Z).
Remark 1.16. Avila and Hubert communicated to the authors that they checked that the action of generators of Aff( was through matrices of finite order. One of the referees also brought our attention to the PhD thesis of Oliver Bauer [2], who does some computations on the action of the affine group similar to ours. We now consider (M 4 , ω (4) ). For this square-tiled surface, we will see that the Veech group is the full group SL(2, Z) and the automorphism group is the cyclic group Z/3.
(1) One has decompositions The subspace H τ is 2-dimensional and the action of Aff(M 4 , ω (4) ) on it is through a homomorphism to the cyclic group Z/6 (acting by rotations). Actually, in Section 3, we study a family of square-tiled surfaces parametrized by an odd integer q ≥ 3, the surface (M 4 , ω (4) ) corresponding to q = 3. Some of the computations are valid for all q ≥ 3, but the stronger statements only hold for q = 3.
In view of the observation following Definition 1.8 (in Subsection 1.3), an immediate consequence of the theorems is that, for the two square-tiled surfaces considered above, all Lyapunov exponents of the reduced Kontsevich-Zorich cocycle are equal to zero.
Conversely (and less trivially), Möller has shown ( [15]) that the action on H (0) 1 for a totally degenerate square-tiled surface has to be through a finite group.
Acknowledgements. This research has been supported by the following institutions: the Collège de France, French ANR (grants 0863 petits diviseurs et resonances en géométrie, EDP et dynamique 0864 Dynamique dans l'espace de Teichmüller). We thank the Collège de France, IMPA (Rio de Janeiro), the Max-Planck Institute für Mathematik in Bonn and the Mittag-Leffler Institute for their hospitality. We are also grateful to the referees for suggestions that greatly improved the presentation of the paper.

PROOF OF THEOREM 1.15
This section is organized as follows. In Subsection 2.1, we recall the description of (M 3 , ω (3) ) as a square-tiled surface. Also, the automorphism group is identified with the quaternion group. In order to understand the action of this group on the homology, we recall in Subsection 2. 2.1. The square-tiled surface (M 3 , ω (3) ). We follow here F. Herrlich and G. Schmithüsen [9]. The set Sq(M 3 , ω (3) ) is identified with the quaternion group Q = {±1, ±i, ±j, ±k}. We denote by sq(g) the square corresponding to g ∈ Q. The map r (for right) is sq(g) → sq(gi) and the map u (for up) is sq(g) → sq(gj). The automorphism group Aut(M 3 , ω (3) ) is then canonically identified with Q, the element h ∈ Q sending the square sq(g) on the square sq(hg). See Figure 1 below.
Here, it is shown the horizontal and vertical cylinder decompositions, and the right and top neighbors of each square, so that the (implicit) side identifications are easily deduced.
We will denote by Q the quotient of Q by its center {±1}. It is isomorphic to Z 2 × Z 2 . We denote by 1, i, j, k the images of ±1, ±i, ±j, ±k in Q.
For g ∈ Q, the lower left corners of sq(g) and sq(−g) correspond to the same point of Σ (3) . We will identify in this way Σ (3) with Q.
The map π : M 3 → R 2 /Z 2 factors as Here, the first map π 1 is a two fold covering ramified over the four points of order 2 in R 2 /(2Z) 2 , and may be viewed as the quotient map by the involution −1 ∈ Q of M 3 , the four squares of R 2 /(2Z) 2 being naturally labelled by Q.

2.2.
Irreducible representations of Q. The group Q has 5 distinct irreducible representations, 4 one-dimensional χ 1 , χ i , χ j , χ k and one 2-dimensional χ 2 . The character table is 1 −1 ±i ±j ±k In the regular representation of Q in Z(Q), the submodules associated to these representations are generated by . We consider the direct sum Z(Q) ⊕ Z(Q) of two copies of Z(Q). We denote by (σ g ) g∈Q the canonical basis of the first copy and by (ζ g ) g∈Q the canonical basis of the second copy. We define a homomorphism p from , Z) by sending σ g on the homology class defined by the lower side of sq(g) (oriented from left to right) and ζ g on the homology class defined by the left side of sq(g) (oriented from upwards) (see Figure 2 below). The homomorphism p is compatible with the actions of Q on Z(Q) ⊕ Z(Q) (by the regular representation) and The kernel of the homomorphism p is the submodule Ann of Z(Q) ⊕ Z(Q) generated by the elements (2.1) g := σ g + ζ gi − ζ g − σ gj . We have Q g = 0, hence Ann has rank 7. Observe that, for g, h ∈ Q we have h. g = hg .
We note that We have ∂(w i ) = 4(j + k − i − 1) and for g ∈ Q. Similar statements hold for w j and w k . Let H rel = Qw i ⊕ Qw j ⊕ Qw k be the subspace of H 1 (M 3 , Σ (3) , Q) spanned by w i , w j , w k . The formulas imply the following lemma: Lemma 2.1. The subspace H rel is invariant under the action of Aut(M 3 , ω (3) ) and complements Proof. The first part is clear. The second part follows from the fact that the images of w i , w j , w k under ∂ are linearly independent. Let From Subsection 1.3, we have H st 1 = Qσ ⊕ Qζ.
We set, for g ∈ Q, We have ∂( σ g ) = ∂( ζ g ) = 0. The subspace of H 1 (M 3 , Q) generated by the σ g , ζ g , g ∈ Q has rank 4 and it is the kernel H For g ∈ Q, we have and The subspace H (0) 1 is Q-invariant, being (in many ways) sum of two copies of χ 2 . One has, for g, h ∈ Q, h. σ g = σ hg , (2.10) At this stage, we have written the homology group H 1 (M 3 , Σ (3) , Q) as the direct sum Let f ∈ Aff(M 3 , ω (3) ) and g := −1; then, f −1 g −1 f g belongs to Aut(M 3 , ω (3) ) and fixes each point of Σ (3) , hence is equal to id or g; but the second possibility would imply that f −1 g −1 f = id. Thus, −1 belongs to the center Aff(M 3 , ω (3) ).
Finally, the sequence is exact because the intersection of Q with Aff (1) (M 3 , ω (3) ) is precisely Z. Remark 2.3. We will see below that the center of Aff(M 3 , ω (3) ) is cyclic of order 4.

2.5.
The action of S, T on H 1 (M 3 , Σ (3) , Z). One easily checks that From these formulas, we deduce and also Finally, we have ). We introduce also These four points are the vertices of a regular tetrahedron in H rel .

Lemma 2.4.
(1) The sequence is exact, where the morphism from Aff(M 3 , ω (3) ) to S 4 is through the action of Aff(M 3 , ω (3) ) on Σ (3) . Proof. We first prove that the homomorphism from Aff(M 3 , ω (3) ) to S 4 is onto. Indeed, S fixes 1 and j and exchanges i and k, while T fixes 1 and i and exchanges j and k; therefore, Aff (1) (M 3 , ω (3) ) acts by the full permutation group of i, j, k. As Q = Aut(M 3 , ω (3) ) acts transitively on Σ (3) , the claim follows. We now prove that the kernel of the morphism from Aff(M 3 , ω (3) ) to S 4 is Γ(2). Indeed, S 2 , T 2 and ( S T −1 S) 2 act trivially on Σ (3) . These elements generate Γ(2), which is therefore contained in the kernel. On the other hand, the index of Γ(2) in Aff(M 3 , ω (3) ) is 24 which is also the cardinality of S 4 . This proves the first part of the lemma.
We have already noted that H rel is invariant under S and T . As Aff(M 3 , ω (3) ) is generated by Q and these two elements, the first assertion of the second part follows. In fact, the formulas (2.2) and (2.16) show that S, T and Q preserve the set { w(1), w(i), w(j), w(k)} and act on this set as on Σ (3) . This completes the proof of the lemma.
It is worth mentioning that the group S 4 through which Aff(M 3 , ω (3) ) acts on H rel is the Weyl group W (A 3 ) of an A 3 root system. 1 . In order to get a nice description of this action, we briefly recall the main features of D l root systems (see [4] for proofs and further details). They are typically obtained from Lie algebras isomorphic to so(2l).
Let l ≥ 4 and V be a l-dimensional space equipped with a scalar product; let (ε m ) 1≤m≤l be an orthonormal basis of V .
The set R = {±ε m ± ε n , 1 ≤ m < n ≤ l} is a root system of type D l . For every α ∈ R, let s α be the orthogonal symmetry with respect to the hyperplane orthogonal to α. Then, s α (R) = R. The subgroup of O(V ) generated by the s α is the Weyl group W (R). It is isomorphic to the semi-direct product of the symmetric group S l (acting by permutation of the ε m ) by the group (Z/2) l−1 (acting by ε m → (±1) m · ε m with m (±1) m = 1).

The finite subgroup of O(V ) preserving R is the automorphism group of R and is denoted by A(R). It contains W (R).
A basis for R is a family {α 1 , . . . , α l } ⊂ R such that, for every α ∈ R, either α or −α is a combination of the α m with nonnegative integer coefficients. One can for instance take When l > 4, the quotient A(R)/W (R) is isomorphic to Z/2. When l = 4, with α 1 , . . . , α 4 as above, the quotient A(R)/W (R) is isomorphic to the permutation group S 3 of {1, 3, 4} in the following way: for every a ∈ A(R), there exists a unique w ∈ W (R) and a unique permutation τ of {1, 3, 4} such that a(α 2 ) = w(α 2 ) and a(α m ) = w(α τ (m) ) for m = 1, 3, 4.
From the description of the Weyl group and (2.10) it follows that Q acts on H The action of Aff(M 3 , ω (3) ) on H 1 is therefore given by a homomorphism Z : Aff(M 3 , ω (3) ) → A(R) for which we obtain in the sequel an explicit description. Let p be the canonical projection p : given by π = p • Z.
We have then This shows that the images of S, T by π are the transpositions (1, 4), (1,3) respectively. Therefore π is onto.
Remark 2.6. The element e iπ satisfies ( e iπ ) 2 = −1 ∈ Q. It is easily checked that it commutes with S, T and the elements of Q. Therefore, it belongs to the center of Aff(M 3 , ω (3) ). The center of Aff(M 3 , ω (3) ) is in fact equal to the cyclic subgroup of order 4 generated by e iπ . Indeed, the center of Aff(M 3 , ω (3) ) has to be contained in the inverse image in Aff(M 3 , ω (3) ) of the center {±Id} of SL(2, Z). This inverse image is the subgroup of order 16 (isomorphic to the subgroup Q below) whose center is generated by e iπ (see also [9]).
Then Γ(2)/Γ(4) is isomorphic to (Z/2) 3 , generated by the images of S 2 , T 2 and −Id. Let s ∈ W (R) be the element defined by Identify Q with its image in W (R).
Proof. The first part is a direct calculation; it shows that the elements ±1, ±i, ±j, ±k, ±s, ±is, ±js, ±ks form a subgroup of W (R); as s / ∈ Q, these elements must be distinct. This proves the second part of the lemma. The subgroup {±1} is contained in the center of Q; and the quotient Q/{±1} is abelian with every element of order ≤ 2, hence isomorphic to (Z/2) 3 . Moreover, the image of is, js, ks clearly generate this quotient. This proves the third part of the lemma.
The formulas for the actions of S 2 , T 2 and e iπ on the ε g , g ∈ Q, show that the images of these elements in W (R) are respectively equal to is, js, ks.
Proof. The first assertion follows from the fact that the images by Z of the generators S 2 , T 2 and e iπ of Γ(2) are the generators is, js, ks of Q. The second assertion follows from the exact sequence just before the statement of the lemma. The third assertion follows from the fact that the kernel of the morphism Γ(4) → Γ(4) is sent isomorphically by Z onto {±1}.
Remark 2.9. It is not difficult to determine explicitly the kernel of Z in Γ(2). One first observes, looking at the action of Γ(4) in the upper half plane, that Γ(4) is the free subgroup generated by  Proof. A direct inspection reveals that the intersection form on the elements σ g , g ∈ {1, i, j, k}, is given by Using the definition ε g := σ g − σ gj , we obtain the following intersection table: In particular, since any symplectic transformation sends symplectic planes into symplectic planes, we conclude that symplectic elements of W (R) either fix or exchange the subsets {±ε 1 , ±ε k } and {±ε j , ±ε i }. It follows that the subgroup of symplectic elements of W (R) has order 16, so it is equal to Q.
The proof of Theorem 1.15 is now complete.

PROOF OF THEOREM 1.17
The outline of this section is the following. In Subsection 3.1, we introduce a family of square-tiled surfaces parametrized by an odd integer q ≥ 3 such that the one featuring in Theorem 1.17 corresponds to q = 3. The Veech group is computed, as well as a representation of these surfaces as algebraic curves. We then compute in Subsection 3.2 the action of the automorphism group (cyclic of order q) on homology, and then in Subsection 3. 1 breaks into invariant subspaces H τ andH which are analyzed in Subsections 3.5, 3.6 respectively (for any odd q ≥ 3). The special case q = 3 (corresponding to a larger Veech group equal to SL(2, Z)) is further analyzed in Subsection 3.7.
In Figures 4 and 5 below, we illustrate the square-tiled surfaces corresponding to the cases q = 3 and q = 5.
The automorphism group Aut(M, ω) is Z/q, acting by translations of the i variable. It fixes each point of Σ.
When q is odd ≥ 5, i − 2 is not equal to i + 4 in Z/q and therefore every element of Aff(M, ω) must fix A 1,1 . On the other hand, one sees easily that the matrices  When q = 3, it is easy to see that S, T ∈ SL(M, ω). The Veech group in this case is equal to SL(2, Z).
In all cases, by considering generators of SL(M, ω), one checks that any element of Aff (1) (M, ω) actually fixes all the A i . It follows that Aut(M, ω) is contained in the center of Aff(M, ω), and that Aff(M, ω) is the direct product of Aut(M, ω) Z/q by SL(M, ω).
Although we do not need it in the following, let us give formulas associated to this cyclic covering.
We claim that M is isomorphic to the desingularization of the algebraic curve with the holomorphic form 2 dz w q , and the automorphism t given by Here, c is the elliptic integral (cf. [3], thm. 1.7). Indeed, let M * be the curve {w 2q = z q−2 (z 2 − 1)}, desingularized in order to have q distinct points at infinity given by z = exp 2πij q w 2 + O(1), j ∈ Z/q. Observe that (cω * ) 2 is the pull-back by the projection (z, w) → z of the quadratic differential with simple poles (dz) 2 z(z 2 −1) . The elliptic curve E = {y 2 = z(z 2 − 1)} is isomorphic to

Similarly, let
Then, M has q connected components indexed naturally by Z/q, with Each connected component M j contains one point at infinity; the formula One also sees that, for 0 ≤ s ≤ 1, j ∈ Z/q, This completes the justification of our claim that (M * , ω * ) is isomorphic to (M, ω). The q points at infinity on M * correspond to the A 0,0,j , the points (0, 1), (0, −1), (0, 0) of M * correspond to the points A 0,1 , A 1,0 , A 1,1 of Σ ⊂ M . The sides of the squares correspond to the locus on M * where the function z is real. See Figure 6 below. The claim about t is easily checked. When q = 3, one recovers the equation of Subsection 1.4 (except that z = 0 has been sent to infinity).
We have for i, g ∈ Z/q. From the formula for i , we have, for i ∈ Z/q, We define We have We observe that {τ i ,σ i ,ζ i ; i ∈ Z/q, i = 0} form a basis for H When q = 3, one has When q ≥ 5, one has  3.6. The subspaceH. We denote byH the (2q − 2)-dimensional subspace of H 1 (M, Q) spanned by theσ i ,ζ i , i ∈ Z/q. We have thus In this subsection, we discuss general considerations valid for all q ≥ 3. The special case q = 3 is treated in the next subsection.
Each of the (q − 1) 2-dimensional subspacesH(ρ) is thus invariant under the action of the affine group, and their direct sum is equal toH ⊗ C. Over R,H ⊗ R splits into the q−1 2 4-dimensional subspaces induced by theH(ρ) ⊕H(ρ −1 ).
Remark 3.2. The determinant of the restriction toH(ρ) of S 2 T 2 is equal to 1 and its trace is equal to 2(1 + ρ + ρ −1 ). When ρ = exp 2iπ q and q ≥ 5, the trace is > 2. In particular, the affine group does not act on H (0) 1 through a finite group. Therefore, by [15], the square-tiled surface is not totally degenerate for q ≥ 5.

3.7.
The subspaceH for q = 3. We assume in this subsection that q = 3. The formulas of Subsection 3.3 show that the set of 24 vectors inH: is invariant under the action of Aff(M, ω). More precisely, we have The set R is a root system of D 4 type. More precisely, let V be the set of nonzero vectors in (Z/3) 2 . Define a map ε : V →H by One has then The action of the affine group onH is therefore given by a homomorphism Z from Aff(M, ω) to the automorphism group A(R). This last group and the Weyl group W (R) ⊂ A(R) have been described in Subsection 2.7. We will now describe Z.
The proof of Theorem 1.17 is now complete.

APPENDIX A. PARITY OF THE SPIN STRUCTURE
After the work of Kontsevich and Zorich [12] on the complete classification of the connected components of the strata of the moduli space of Abelian differentials, we know that the stratum H(1, 1, 1, 1) of (M 3 , ω (3) ) is connected, but the stratum H(2, 2, 2) of (M 4 , ω (4) ) has exactly two connected components distinguished by a topological invariant called the parity of the spin structure. After recalling the definition of the parity of the spin structure , we will show that the parity of spin of (M 4 , ω (4) ) is even.
Let (M, ω) be a translation surface such that the zeros p 1 , . . . , p n of ω have even orders 2l 1 , . . . , 2l n . For an oriented smoothly immersed closed curve γ avoiding the zeros of ω, the index ind ω (γ) ∈ Z is the integer such that the total change of angle between the tangent vector of γ and the rightwards horizontal direction determined by ω is 2π ·ind ω (γ). Because the zeros of ω have even order, the parity of ind ω (γ) depends only on the class [γ] of γ in H 1 (M, Z) and will be denoted by ind ω ([γ]) ∈ Z/2.
We conclude that the parity of the spin of (M 4 , ω (4) ) is even.
Remark A.1. Another way to compute the parity of the spin structure could have been to determine the Rauzy class of an interval exchange map obtained as the first return map of a nearly vertical linear flow of (M, ω) on an appropriate transversal. See Appendix C of [22] for a detailed explanation of this method (including basic definitions and main technical steps). For sake of convenience of the reader, we compute below some combinatorial data associated to (M 4 , ω (4) ) needed to perform the arguments of [22]. Fix a small angle θ; we consider, as shown on Figure 9 below, the first return of the upwards vertical flow for e iθ ω (4) on a transverse interval L which is horizontal for e iθ ω (4) .
Here, the interval L starts at the singularity and ends at the black diamond dot. Also, we indicated the "top" partition of L (corresponding to the domain of the return map) into 10 intervals (named A, B, C, D, E, F , G, H, I and J) and their images by the return map (the "bottom" partition of L). The combinatorial data of the return map is thus We have seen that, for both (M 3 , Σ (3) , ω (3) ) and (M 4 , Σ (4) , ω (4) ), the absolute homology group H 1 (M, R) admits a Aff(M k , ω (k) )-invariant supplement H rel within the relative homology group H 1 (M, Σ, R). This appendix shows a simple example of a square-tiled surface of genus 2 with two simple zeros where this does not hold.