Square-tiled cyclic covers

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichm\"uller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichm\"uller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in \cite{Forni:Matheus}) of a Teichm\"uller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example found previously by Forni in genus three also corresponds to a square-tiled cyclic cover \cite{ForniSurvey}). We present several new examples of Teichm\"uller curves in strata of holomorphic and meromorphic quadratic differentials with maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichm\"uller curves with maximally degenerate spectrum. We prove that this is indeed the case within the class of square-tiled cyclic covers.


INTRODUCTION
The Kontsevich-Zorich cocycle is a dynamical system on the total space of the Hodge bundle over the moduli space of Abelian or quadratic differentials. It is a continuous-time cocycle in the standard sense: it is a flow which acts linearly on the fibers of the bundle. Its projection to the moduli space is given by the Teichmüller geodesic flow. Since the Kontsevich-Zorich cocycle is closely related to the tangent cocycle of the Teichmüller flow, its Lyapunov structure determines that of the Teichmüller flow and has implications for its dynamics. The Kontsevich-Zorich spectrum also plays a crucial role in applications of Teichmüller theory to the dynamics of translation flows and interval-exchange transformations-in particular, to results on the deviation of ergodic averages (see [38,39,40,22,14]), on the existence and nature of the limit distributions (see [7,8]) and on the weak mixing property (see [2]).
M. Kontsevich and A. Zorich conjectured that the Lyapunov spectrum of the cocycle is simple (in particular, that all the exponents are nonzero) for SL(2, R)invariant canonical absolutely continuous measures on all connected components of strata of the moduli spaces of Abelian (and quadratic) holomorphic differentials. G. Forni [14] proved that for the canonical measures on strata of Abelian differentials, the exponents are all nonzero. A. Avila and M. Viana [3] later completed the proof of the Kontsevich-Zorich conjecture in this case. Recently, G. Forni developed his approach to give a general criterion for the nonvanishing property of the Kontsevich-Zorich exponents for SL(2, R)-invariant measures on the moduli space of Abelian differentials [16]. Based on this criterion (and on a standard construction of an orienting double-cover), R. Treviño [33] proved the nonvanishing of the exponents for all canonical measures on strata of quadratic differentials. The full Kontsevich-Zorich conjecture is still open for strata of nonorientable quadratic differentials.
The aforementioned results lead to the natural questions as to whether or not it is possible for the Kontsevich-Zorich cocycle to have zero exponents with respect to other invariant measures. It is well known to experts (cf. [34]) that it is possible to construct invariant measures for the Teichmüller geodesic flow (for instance, supported on periodic orbits) with maximally degenerate Kontsevich-Zorich spectra, that is, with all exponents equal to zero with the exception of the "trivial" ones. Answering a question of W. Veech, G. Forni found in [15] the first example of a SL(2, R)-invariant measure with a maximally degenerate spectrum. The example is given by the measure supported on the SL(2, R)-orbit of a genus three square-tiled cyclic cover; that is, a branched cover of the fourpunctured Riemann sphere endowed with a quadratic differential with four simple poles at the punctures. Any cover of this type is "parallelogram-tiled" in the sense that it is also a branched cover of the torus with a single branching point. The flat surface in Forni's example, found independently by F. Herrlich, M. Möller and G. Schmithuesen, is very peculiar, very symmetric, and has so many remarkable properties that it has been aptly named Eierlegende Wollmilchsau [20]. Later G. Forni and C. Matheus announced in the preprint [17] a second example of the same kind in genus four (see also [27]). The present article has, in fact, grown out of that announcement.
M. Möller conjectured in [30] that these two examples are the only Teichmüller curves with maximally degenerate Kontsevich-Zorich spectra. He was able to prove his conjecture up to a few strata in genus five where the arithmetic conditions he derives to rule out maximally degenerate spectra could not be verified [30]. Möller result naturally led to the more general conjecture on whether the Teichmüller curves of the Eierlegende Wollmilchsau and the Forni-Matheus curve indeed give the only examples of SL(2, R)-invariant measures (or even of SL(2, R)-orbits) with maximally degenerate Kontsevich-Zorich spectra on strata of Abelian differentials. For sufficiently high genus the conjecture is proved in [11] for SL(2, R)-invariant suborbifolds in the moduli space of holomorphic Abelian and quadratic differentials, as a corollary of the key formula for the sum of the exponents. Other results in this direction for moduli spaces of holomorphic (Abelian or quadratic) differentials in all genera have been announced by A. Avila and M. Möller and, independently, by D. Aulicino. Similar conjectures for strata of meromorphic quadratic differentials are at the moment wide open, to the authors' best knowledge.
In this paper, we systematically investigate these questions within the class of all square-tiled cyclic covers. We remark that the idea of the construction of a square-tiled cyclic cover already appeared in [15] (and in [17]) but only in a particular case. Here we generalize the construction and derive the main topological, geometric, and combinatorial properties of the resulting translation and half-translation surfaces. We then classify all the examples of Teichmüller curves derived from square-tiled cyclic covers with maximally degenerate spectra in strata of Abelian holomorphic differentials and of quadratic holomorphic and meromorphic differentials. The main tool in our investigation of the spectrum of Lyapunov exponents is the formula from [11] for the sum of the nonnegative Lyapunov exponents (that is, for the exponent of the determinant bundle). The formula takes a particularly simple, explicit form in the case of square-tiled cyclic covers. In fact, our paper can be considered as a companion to the paper by A. Eskin, M. Kontsevich and A. Zorich [12] in which the authors derive a completely explicit formula for each individual Lyapunov exponent of a square-tiled cyclic cover. In a related paper D. Chen [9] computes the Lyapunov exponent of the determinant bundle for square-tiled cyclic covers by different methods and relates it to the slope of the corresponding Teichmüller curve.
Finally, we remark that the Eierlegende Wollmilchsau and the Forni-Matheus example were originally found in [15] and [17], respectively, by a completely different method based on the analysis of the action of the cyclic group of deck transformations on the second fundamental form of the Hodge bundle (related to the Kontsevich-Zorich spectrum by the variational formulas of [14,15]). This symmetry approach has led us to conduct a systematic investigation of the spectrum of the Kontsevich-Zorich cocycle on equivariant subbundles of the Hodge bundle, which will appear in forthcoming paper [18].
Additional bibliographic remarks. Cyclic covers were already studied by I. Bouw and M. Möller in [6] in a similar context, but with respect to completely different (not square-tiled) flat structures. The papers [5] of I. Bouw and [28] of C. McMullen investigate more general cyclic covers but without any relation to flat metrics. The paper [37] of A. Wright studies general square-tiled Abelian (versus cyclic) covers.
1.1. Reader's guide. Cyclic covers are defined in Section 2.1. In Section 2.2 we introduce a square-tiled flat structure on any appropriate cyclic cover. A reader interested in the main results can then choose to pass directly to Section 3.2. In Section 2.3 we characterize square-tiled cyclic covers defined by holomorphic one-forms. We determine the corresponding ambient strata of holomorphic 1-forms (of quadratic differentials in the general situation). In Section 2.4 we describe in detail how to explicitly construct the square-tiled surface (M N (a 1 , a 2 , a 3 , a 4 ), p * q 0 ), and in Section 2.6 we characterize its Veech group and the corresponding arithmetic Teichmüller curve. In Section 2.5 we describe the automorphism group of a cyclic cover. In Section 3.1 we recall a general formula from [11] for the sum of positive Lyapunov exponents of the Hodge bundle over an arithmetic Teichmüller curve. From this formula we derive in Section 3.2 an explicit expression for the sum of exponents in the case of an arbitrary square-tiled cyclic cover. We apply these results in Section 3.3 to determine square-tiled cyclic covers giving rise to arithmetic Teichmüller curves with a maximally degenerate Kontsevich-Zorich spectrum.
In Appendix A we present an analytic computation of the spin-structure (different from the original computation in [27]), which allows us to determine the connected component of the ambient stratum corresponding to the exceptional square-tiled cyclic cover in genus four. Appendix B is provided for the sake of completeness: it is a technical exercise related to the proof of one of the main Theorems (namely Theorem 18).

SQUARE-TILED FLAT STRUCTURE ON A CYCLIC COVER
2.1. Cyclic covers. Consider an integer N such that N > 1 and a 4-tuple of integers (a 1 , . . . , a 4 ) satisfying the following conditions: Let z 1 , z 2 , z 3 , z 4 ∈ C be four distinct points. Conditions (1) imply that, possibly after a desingularization, a Riemann surface M N (a 1 , a 2 , a 3 , a 4 ) defined by equation is closed, connected and nonsingular. By construction, M N (a 1 , a 2 , a 3 , a 4 ) is a ramified cover over the Riemann sphere P 1 (C) branched over the points z 1 , . . . , z 4 . By puncturing the ramification points we obtain a regular N -fold cover over P 1 (C) {z 1 , z 2 , z 3 , z 4 }.

REMARK 1.
It is easy to see that quadruples (a 1 , a 2 , a 3 , a 4 ) and (ã 1 ,ã 2 ,ã 3 ,ã 4 ) with a i =ã i (mod N ) for i = 1, 2, 3, 4, define isomorphic cyclic covers, which explains the first condition in formula (1). The condition on gcd in (1) is a necessary and sufficient condition of connectedness of the resulting cyclic cover. The third condition in formula (1) implies that there is no branching at infinity.
A group of deck transformations of this cover is the cyclic group Z/N Z with a generator T : M → M given by where ζ is a primitive N th root of unity, ζ N = 1. Throughout this paper we will use the term cyclic cover when referring to a Riemann surface M N (a 1 , . . . , a 4 ), with parameters N , a 1 , . . . , a 4 satisfying relations (1).

2.2.
Square-tiled surface associated to a cyclic cover. Any meromorphic quadratic differential q(z)(d z) 2 with at most simple poles on a Riemann surface defines a flat metric g (z) = |q(z)| with conical singularities at zeroes and poles of q. Let us consider a meromorphic quadratic differential q 0 on P 1 (C) of the form It has simple poles at z 1 , z 2 , z 3 , z 4 and no other zeroes or poles. The quadratic differential q 0 defines a flat metric on a sphere obtained by the following construction. Consider an appropriate flat cylinder. On each boundary component of the cylinder mark a pair of opposite points and glue the resulting pairs of cords by isometries. The four marked points become conical points of the flat metric. For a convenient choice of parameters c 0 , z 1 , . . . , z 4 the resulting flat sphere can be obtained by identifying the boundaries of two copies of a unit square. Metrically, we get a square pillow with four corners corresponding to the four poles of q 0 , see Figure 1. Now consider some cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) and the canonical projection p : M N (a 1 , a 2 , a 3 , a 4 ) → P 1 (C). Consider an induced quadratic differential q = p * q 0 on M N (a 1 , a 2 , a 3 , a 4 ) and the corresponding flat metric. By construction, the resulting flat surface is tiled with unit squares. In other words, we get a square-tiled surface, see [10,41] (also called an origami, [25,32]; also called an arithmetic translation surface, see [19]).
In this paper we mostly focus on square-tiled cyclic covers, which are pairs where the meromorphic quadratic differential q 0 on the underlying P 1 (C) defines a "unit square pillow with vertical and horizontal sides" as in Figure 1.
2.3. Singularity pattern of a square-tiled cyclic cover. The Riemann surface M N (a 1 , a 2 , a 3 , a 4 ) has gcd(N , a i ) ramification points over each branching point z i ∈ P 1 (C), where i = 1, 2, 3, 4, on the base sphere. Each ramification point has degree N / gcd(N , a i ). The flat metric has four conical points z = z i , i = 1, . . . , 4, on the base sphere with a cone angle π at each conical point. Hence, the induced flat metric on M N (a 1 , a 2 , a 3 , a 4 ) has gcd(N , a i ) conical points over z i ; each conical point has cone angle N / gcd(N , a i ) π. If one of the cone angles is an odd multiple of π, then the flat metric has nontrivial holonomy; in other words, the quadratic differential q = p * q 0 is not a global square of a holomorphic 1-form. Note, however, that although the condition that all cone angles be even multiples of π is necessary, it is not a sufficient condition for triviality of the holonomy of the flat metric.
Denote by H (m 1 , . . . , m n ) the stratum of Abelian differentials with zeroes of degrees m 1 , . . . , m n and by Q(d 1 , . . . , d n ) the stratum of meromorphic quadratic differentials with singularities of degrees d 1 , . . . , d n . Here m i ∈ N, for i = 1, . . . , n, and n i =1 m i = 2g −2. We do not allow poles of orders higher than one for meromorphic quadratic differentials, so d i ∈ {−1, 1, 2, 3, . . . }, for i = 1, . . . , n, and the sum of degrees of singularities satisfies the equality n i =1 d i = 4g − 4. LEMMA 2. If N is even and all a i , i = 1, 2, 3, 4, are odd, the quadratic differential q = p * q 0 is a global square of a holomorphic 1-form ω on M N (a 1 , a 2 , a 3 , a 4 ), where ω belongs to the stratum .
The associated flat metric on such a square-tiled cyclic cover has a trivial linear holonomy. If N is odd, or if N is even but at least one of a i , i = 1, 2, 3, 4, is also even, the quadratic differential q = p * q 0 is not a global square of a holomorphic 1-form on M N (a 1 , a 2 , a 3 , a 4 ). In this case q belongs to the stratum , and the flat metric on such a square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) has nontrivial linear holonomy.

REMARK 3.
In the case, when gcd(N , a i ) = N 2 , the corresponding "conical points" have cone angles 2π and so are actually regular points of the metric. Depending on the context we either consider such points as marked points or simply ignore them. REMARK 4. Lemma 2 implies, in particular, that the quadratic differential q = p * q 0 is holomorphic if and only if inequalities a i ≤ N are strict for all i = 1, 2, 3, 4. If a i = N for at least one index i , then the quadratic differential q = p * q 0 is meromorphic; that is, it has simple poles.
Proof of the Lemma. Let σ i be a small contour around z i in the positive direction on the sphere (see Figure 1). The paths σ i , i = 1, 2, 3 generate the fundamental group of the sphere punctured at the four ramification points.
Since the cone angle at each cone singularity of the underlying "flat sphere" is π (whether it is glued from squares or not), the parallel transport along each loop σ i brings a tangent vector v to − v. Let z ∈ P 1 (C) be a point of the loop σ i , and let (w, z) be one of its preimages in the cover M N (a 1 , a 2 , a 3 , a 4 ). By lifting the loop σ i to a path on the cover which starts at the point (w, z), we land at the point (ζ a i w, z), where ζ is the primitive N th root of unity. Thus we get the following representation of the fundamental group of the punctured sphere in the cyclic group Z/N Z of deck transformations (3) and the holonomy group Z/2Z of the flat metric on the sphere: (7) Deck : σ i → a i ∈ Z/N Z Hol : Since the metric on M N (a 1 , a 2 , a 3 , a 4 ) is induced from the metric on the sphere, the holonomy representation hol of the fundamental group of the covering surface M N (a 1 , a 2 , a 3 , a 4 ) factors through the one of the sphere, i.e., hol = Hol •p * . Let us suppose that N is odd. Let α i = N / gcd(N , a i ). Then Hence σ α i i can be lifted to a closed path on M N (a 1 , a 2 , a 3 , a 4 ). On the other hand, Hol(σ . Thus, the flat metric on the cover M N (a 1 , a 2 , a 3 , a 4 ) has nontrivial holonomy, and our quadratic differential q is not a global square of a holomorphic 1-form.
Let us suppose that N is even, but some a i is also even. Since gcd(N , a 1 , . . . , a 4 ) = 1 , at least one of a j is odd. Since a 1 +· · ·+a 4 is divisible by N , this sum is even, and hence we have exactly two even entries. By relabeling the ramification points on the sphere, if necessary, we may assume that a 1 , a 3 are odd while a 2 , a 4 are even. Let us consider a path For the induced deck transformation we get Hence, a lift of ρ is closed on M N (a 1 , a 2 , a 3 , a 4 ). On the other hand, this path acts as an element Hol(ρ) = (a 2 + a 3 ) − a 1 − a 1 = 1 (mod 2) in the holonomy group Z/2Z. Hence, the holonomy of the metric along this closed path on the cover M N (a 1 , a 2 , a 3 , a 4 ) is nontrivial, and our quadratic differential q is not a global square of a holomorphic 1-form. Finally, suppose that N is even and all a i , i = 1, 2, 3, 4 are odd. Any element of the fundamental group of the underlying four-punctured sphere can be represented by a product Let k i be the algebraic number of entries of the "letter" σ i in the word as above, where σ −1 i is counted with the sign minus, and i = 1, 2, 3, 4. The loop τ as above can be lifted to a closed loop on the cover if and only if Deck(τ) = 0, that is, if and only if k 1 a 1 +· · ·+k 4 a 4 = 0 (mod N ). In this case Hol(τ) = k 1 +· · ·+k 4 . Since all a i are odd we have k 1 + · · · + k 4 = k 1 a 1 + · · · + k 4 a 4 (mod 2). Finally, since N is even and k 1 a 1 + · · · + k 4 a 4 = 0 (mod N ), we also have k 1 a 1 + · · · + k 4 a 4 = 0 (mod 2). Thus, in this case, the flat metric on M N (a 1 , a 2 , a 3 , a 4 ) has trivial linear holonomy.
Recall that a flat metric associated to a meromorphic quadratic differential with at most simple poles has cone angle (d + 2)π at a zero of order d (we consider a simple pole as a "zero of degree −1"). A flat metric associated to a holomorphic 1-form has cone angle 2(k + 1)π at a zero of degree k. We have already evaluated the number of ramification points and their ramification degrees which define cone angles at all conical singularities, and hence the degrees of the corresponding quadratic (Abelian) differential. This completes the proof of the Lemma.

LEMMA 5.
In the case when the quadratic differential q = p * q 0 determining a square-tiled flat structure is the global square of a holomorphic 1-form, that is, q = ω 2 , the form ω is anti-invariant with respect to the action of a generator of the group of deck transformations, Proof. By construction, the quadratic differential q = p * q 0 is invariant under the action of deck transformations on M N (a 1 , a 2 , a 3 , a 4 ). Hence, when N is even, all a i , i = 1, 2, 3, 4 are odd, and q = p * q 0 = ω 2 , the holomorphic 1-form ω is either invariant or anti-invariant under the action of a generator of the group of deck transformations (3). Invariance of ω under T * would mean that ω can be pushed forward to P 1 (C), which would imply in turn that q 0 is a global square of a holomorphic 1-form. This is not true. Hence, ω is anti-invariant.
By the Riemann-Hurwitz formula, the genus g of M N (a 1 , a 2 , a 3 , a 4 ) satisfies and hence, The same result can be obtained by summing up the degrees of zeroes, which gives 2g − 2 for a holomorphic 1-form in (5), and 4g − 4 for a quadratic differential in (6).

2.4.
Combinatorics of a square-tiled cyclic cover. It is convenient to define a square-tiled surface corresponding to a holomorphic 1-form by a pair of permutations on the set of all squares, π h and π v , indicating for each square its neighbor to the right and its neighbor on top respectively. Let us evaluate these permutations for a square-tiled surface defined by a holomorphic 1-form ω on M N (a 1 , a 2 , a 3 , a 4 ) (and in addition to the conditions in formula (1) we assume that N is even and all a i are odd).
We start with an appropriate enumeration of the squares. By construction, our "square-tiled pillow" (P 1 (C), q 0 ) in the base of the cover is tiled with two unit squares. Since the above cover has degree N , our squaretiled cyclic cover gets tiled with 2N squares.
Assume that the branch points z 1 , z 2 , z 3 , z 4 are associated to the corners of the pillow as in Figure 1. We associate the letters A, B,C , D to the four corners respectively. We also associate the corresponding letters to the corners of each square on the surface M N (a 1 , a 2 , a 3 , a 4 ).
Let us color one of the faces of our pillow white, and the other black. Let us lift this coloring to M N (a 1 , a 2 , a 3 , a 4 ). Choose some white square, and associate the number 0 to it. Take a black square adjacent to the side [C D] of the first one and associate the number 1 to it. Acting by deck transformations we associate to a white square T k (S 0 ) the number 2k, and to a black square T k (S 1 ) the number 2k + 1. As usual, k is taken modulo N , so we may assume that 0 ≤ k < N .
Flat sphere glued from two squares Let us consider a small loop σ i encircling z i in a positive direction; we assume that σ i does not have other ramification points inside an encircled domain, see Figure 1. Let us then consider a lift of σ i to M N (a 1 , a 2 , a 3 , a 4 ). The end-point of the lifted path is the image of the action of T a i on the starting point of the lifted path. Hence, starting at a square number j and "going around a corner" on M N (a 1 , a 2 , a 3 , a 4 ) in the positive (counterclockwise) direction we get to a square number j + 2a j (mod 2N ) (see Figure 2).
Let us consider a horizontal path τ h as in Figure 1 and a lift of τ h to the surface M N (a 1 , a 2 , a 3 , a 4 ). The end-point of the resulting path is the image of the action of T a 1 +a 4 = T −(a 2 +a 3 ) on the starting point of the lifted path. Hence, "moving two squares to the right" on M N (a 1 , a 2 , a 3 , a 4 ) we move from a square number j to a square number j + 2(a 1 + a 4 ) (mod 2N ) if vertices B and C are at the bottom of the squares, and to a square number j − 2(a 1 + a 4 ) (mod 2N ) if vertices B and C are on top of the squares (see Figure 3).
Local moves on a square-tiled cyclic cover Using these rules, it is easy to determine the permutations π h and π v . Start with two neighboring squares numbered by 0 and 1. By iterating the operation τ h , we can determine all the squares to the right of 0 and 1 till we close up and get a cylinder. Recall that we associate letters A, B,C , D to the corners of the squares. By applying appropriate operations σ i we find the squares located atop those which are already constructed. By applying appropriate operations σ −1 i we find a direct neighbor to the right for every square which does not belong to one of the previously constructed horizontal cylinders. Having two horizontally adjacent squares, we apply iteratively the operation τ h to obtain all 2N / gcd(N , a 1 + a 4 ) squares in the corresponding cylinder (row), etc. Figure 3 presents a construction of the enumeration for the squaretiling of M 6 (1, 1, 1, 3), where the exponents {1, 1, 1, 3} are represented by vertices {A, B,C , D} respectively. Note that by moving two squares to the right in the first row (say, 0 −→ 8) we apply τ h , while by moving two squares to the right in the second row (say, 10 −→ 2) we apply τ −1 h . In this example the permutations π h and π v have the following decompositions into cycles π h = (0, 1, 8, 9, 4, 5)(11, 10, 3, 2, 7, 6) 6,9,2,5,10) When N is odd, or when N is even but at least one of a i is also even, the quadratic differential q = p * q 0 on M N (a 1 , a 2 , a 3 , a 4 ) is not a global square of a holomorphic 1-form. The holonomy of the flat structure defined by q is no longer trivial: a parallel transport of a tangent vector v along certain closed paths brings it to −v. In particular the notions of "up-down" or "left-right" are no longer globally defined. However, the notions of horizontal direction and of vertical direction are still globally well-defined. Our flat surface is square-tiled and its combinatorial geometry can still be encoded by a pair of permutations π h and π v . With this definition the remaining part of the construction literally coincides with the one for an Abelian differential, which was described above. Figure 4 illustrates a square-tiling of M 4 (1, 3, 2, 2). The flat metric has nontrivial linear holonomy; the corresponding quadratic differential belongs to the stratum Q(2, 2). In this example the permutations π h and π v are  π v = (0 , 5)(1, 4)(2, 7)(3, 6) REMARK 8. Note that when pairing sides of boundary squares of an abstract square-tiled surface one has to respect the orientation of the surface.

EXAMPLE 7.
We proceed below with an elementary lemma which will, however, be important later. LEMMA 9. Consider a decomposition M N (a 1 , a 2 , a 3 , a 4 ) = cyl i of a square-tiled cyclic cover into cylinders cyl i filled by closed horizontal trajectories. For every cylinder cyl i we denote by w i its width (the length of each closed horizontal trajectory) and by h i its height (the length of each vertical segment).
Assuming that the branch points z 1 , z 2 , z 3 , z 4 are numbered as indicated in Figure 1, the widths of the corresponding cylinders and the sum of the heights of all cylinders are given by the formulas: , for all i , and i h i = gcd(N , a 1 + a 4 ) .
Proof. Clearly, the operation τ h = T (a 1 +a 4 ) has order N gcd(N ,a 1 +a 4 ) . Hence, the length of each horizontal trajectory is equal to 2N gcd(N ,a 1 +a 4 ) which, in turn, is equal to the width of any cylinder. Since the area of the surface is 2N , the total height of all cylinders is equal to gcd(N , a 1 + a 4 ).

REMARK 10.
It is irrelevant whether or not the quadratic differential q = p * q 0 defining the square-tiled flat structure in Lemma 9 is a global square of a holomorphic 1-form.
2.5. Symmetries of cyclic covers. We continue with a description of the isomorphisms of cyclic covers (certainly known to all who worked with them). Note that in the Lemma below we do not use any flat structure.
if and only if for some primitive element k ∈ Z/N Z one has In particular, Proof. Any isomorphism g of cyclic covers as above induces an isomorphism g * of their groups of deck transformations such that, Since by construction Deck(σ i ) =ã i and Deck(σ i ) = a i in Z/N Z, for i = 1, . . . , 4, and any automorphism of a cyclic group Z/N Z is given by the multiplication by a primitive element k ∈ Z/N Z, the above relation yields formula (10). On the other hand, when condition (10) is satisfied, one can find integers m 1 , . . . , m 4 such thatã i = ka i + m i N , where 0 <ã i ≤ N , and, hence, we have an obvious isomorphism Consider the particular case when {a 1 , . . . , a 4 } and {ã 1 , . . . ,ã 4 } coincide as unordered sets (possibly with multiplicities).

DEFINITION 12.
A permutation π in S 4 is called a symmetry of a cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) if there exists an integer k such that k · a i (mod N ) = a π(i ) for i = 1, 2, 3, 4 .
2.6. The Veech group of a square-tiled cyclic cover. The groups SL(2, R) and PSL(2, R) act naturally on any stratum of holomorphic 1-forms and, respectively, meromorphic quadratic differentials with at most simple poles. The Veech group Γ(S) of a flat surface S is the stabilizer of the corresponding point of the stratum under this action. In this section we study the Veech groups of squaretiled cyclic covers. In particular, we prove the following statement. THEOREM 13. The Veech group Γ(S) of any square-tiled cyclic cover S contains the group Γ(2) (respectively Γ(2)/(± Id)) as a subgroup. The Veech group Γ(S) has one of the indices 1, 2, 3 or 6 in SL(2, Z) (respectively in PSL(2, Z)).

REMARK 14.
A proper subgroup of SL(2, Z) containing Γ(2) is said to be a congruence subgroup of level two. It is not hard to check that the conjugation class of a congruence subgroup of level two is uniquely determined by its index. An example of a congruence subgroup of SL(2, Z) of level two with index 3 is In the literature, the congruence subgroups of level two of SL(2, Z) are called theta groups (sometimes denoted Θ).
Theorem 13 will be derived from Lemma 15 and from a more precise Theorem 18.
Sometimes it is convenient to "mark" (in other words "label" or "give names to") the zeroes (and the simple poles) of the corresponding holomorphic 1form (respectively, quadratic differential). In this way, one gets the strata of marked (in other words "labeled", "named"), flat surfaces. The action of the group SL(2, R) (respectively PSL(2, R)) on the strata of marked flat surfaces, and the Veech group Γ(S marked ) of a marked flat surface S marked are defined analogously.
Let the flat surface S be a "unit square pillow", that is, let S be P 1 (C) endowed with the quadratic differential (4), where the parameters are chosen in such way that the flat surface S is glued from two unit squares, and their sides are vertical and horizontal. The Veech group Γ(S) of such a flat surface S coincides with PSL(2, Z). We will need the following elementary Lemma for the marked version of the latter surface. In particular, the Veech group Γ(S marked ) of the "unit square pillow with marked corners" S marked coincides with the kernel Γ(2)/(± Id) of this homomorphism.
Proof. Let the labels of the corners of the pillow be A, B,C , D, say, as in Figure  1. An element g of PSL(2, Z) acts on the flat surface (P 1 (C), q 0 ), giving a new flat surface isomorphic to the original one. It can still be obtained by gluing two squares, but the labels A, B,C , D have moved around. Using an appropri- is not canonical, since it depends on the choice of the fixed label. However, the conjugacy class of the image of any element is defined canonically. Note that the "pillow symmetries" are diffeomorphisms of our flat sphere with differential ± Id in flat coordinates. Thus, all "unit square pillows with marked corners" related by "pillow symmetries" define one and the same point of the stratum Q marked (−1, −1, −1, −1).
It is easy to see that under the convention that PSL(2, Z) keeps one of the labels fixed, the elements 1 2 0 1 and 1 0 2 1 of PSL(2, Z) fix all the labels, and, thus, belong to the Veech group of S marked . It is a well-known fact that the above two elements generate the kernel Γ(2)/(± Id) of the homomorphism of formula (12), and that the group PSL(2, Z/2Z) = SL(2, Z/2Z) is isomorphic to S 3 . It is also easy to check that under this identification the elements of S 3 act by the corresponding permutations of the three "free" labels; in particular, the action of S 3 on the six distinct "unit square pillows with marked corners" is free.
Let S be a cyclic cover of a type M N (a 1 , . . . , a 4 ) endowed with the flat structure induced from the flat structure (4) on P 1 (C). As always, we assume that when N is even and all a i are odd, the flat structure on S is defined by the holomorphic 1-form ω, such that ω 2 = p * q 0 ; otherwise it is defined by the quadratic differential q = p * q 0 .
It is easy to see, that for any g in SL(2, R) (respectively, for any g in PSL(2, R)), the imageS := g S is also represented by a cyclic cover. Moreover, any affine diffeomorphism A g : S →S of the corresponding flat surfaces intertwines the action of the group of the deck transformations on S andS, that is, A g • T = T • A g . Similarly, for any g ∈ SL(2, Z) (respectively, for any g ∈ PSL(2, Z)) the image of a square-tiled cyclic cover under the action of g is again a square-tiled cyclic cover.
In the rest of this section (and, basically, in the remaining part of the paper) we consider only square-tiled cyclic covers, in particular, to avoid cumbersome notations, we denote by M N (a 1 , a 2 , a 3 , a 4 ) a cyclic cover endowed with the flat structure induced from the "square pillow" as in Figure 1. Under this convention, the order of the entries a 1 , . . . , a 4 matters in the definition of a square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ). However, the square-tiled cyclic covers M N (a 1 , a 2 , a 3 , a 4 ), M N (a 2 , a 1 , a 4 , a 3 ), M N (a 4 , a 3 , a 2 , a 1 ), and M N (a 3 , a 4 , a 1 , a 2 ) , related by "pillow symmetries" (see formula (13)) define the same flat surface.
The fact that the square-tiled cyclic cover M N (a 3 , a 4 , a 1 , a 2 ) defines the same flat surface as M N (a 1 , a 2 , a 3 , a 4 ) implies, in particular, that in the case when the flat structure on a square-tiled cyclic cover is defined by a holomorphic 1-form, the element −1 0 0 −1 of SL(2, Z) belongs to the Veech group of the corresponding flat surface. Hence, the action of SL(2, Z) on square-tiled cyclic covers factors through the action of PSL(2, Z). We shall sometimes consider the latter action without specifying it explicitly. It is easy to see, that if an element g ∈ PSL(2, Z) permutes the marking of the initial "unit square pillow with marked corners" by a permutation π ∈ S 4 , then the square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) is mapped by g to the squaretiled cyclic cover M N (a π(1) , a π(2) , a π(3) , a π (4) ). Since the elements of the subgroup K of S 4 correspond to isomorphic square-tiled cyclic covers, we conclude that the action of SL(2, Z) (respectively of PSL(2, Z)) on square-tiled cyclic covers factors through the action on "unit square pillows with marked corners". By combining the latter observation with Lemma 15 we conclude that the Veech group of M N (a 1 , a 2 , a 3 , a 4 ) contains the group Γ(2) (respectively the group Γ(2)/(± Id)), that it has index at most 6 in SL(2, Z) (respectively, in PSL(2, Z)), and that it is determined by its index up to conjugation.
To complete the proof of Theorem 13 it remains to prove that all indices 1, 2, 3, 6 are realized.
In order to describe this Veech group more precisely we need the following remark. Letf : M N (a 1 , a 2 , a 3 , a 4 ) → M N (ã 1 ,ã 2 ,ã 3 ,ã 4 ) be an isomorphism of square-tiled cyclic covers, that is, a diffeomorphism with differential equal to Id (respectively ± Id) in flat coordinates. It is not hard to see thatf is part of a commutative diagram where p is the canonical projection and f : P 1 (C) → P 1 (C) is an automorphism of the underlying flat sphere. We have seen that the only automorphisms of the "square pillow" (P 1 (C), q 0 ) are the "pillow symmetries" (see Figure 5). It follows from formula (13) that all "pillow symmetries", that is, all elements of the Klein group K are involutions. Let s be the the "pillow symmetry" corresponding to the automorphism f : P 1 (C) → P 1 (C) or, equivalently, to its inverse. Let it act on the canonical labeling of the corners of the pillow by a permutation ∈ K.
Let us consider the case when {a 1 , . . . , a 4 } and {ã 1 , . . . ,ã 4 } coincide as unordered sets (possibly with multiplicities). Since all symmetries of cyclic covers such as those in formula (15) are described by Definition 12 and Lemma 11, our considerations imply the following statement.

LEMMA 16.
Consider π ∈ S 4 . The square-tiled cyclic covers M N (a 1 , a 2 , a 3 , a 4 ) and M N (a π(1) , a π(2) , a π(3) , a π(4) ) are isomorphic (that is define the same point of the corresponding stratum) if and only if there exists a symmetry π of the cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) such that the permutation π · π −1 belongs to the Klein subgroup K defined in formula (13).
By passing to the quotient S 3 S 4 /K we get the following immediate Corollary of the Lemma above.

COROLLARY 17.
For any square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) the index of its Veech group in SL(2, Z) (in PSL(2, Z) when the flat structure is defined by a quadratic differential) coincides with the index of the image of the subgroup of symmetries of M N (a 1 , a 2 , a 3 , a 4 ) in S 3 under the projection of formula (14).
In order to complete the proof of Theorem 13 it is sufficient to prove that all indices 1, 2, 3, 6 are realized. We prefer to prove a strengthened version of Theorem 13.

THEOREM 18. The index of the Veech group of a square-tiled cyclic cover is described by the following list.
• If for some triple of pairwise distinct indices i , j , k ∈ {1, 2, 3, 4} one has a i = a j = a k , the index of the Veech group of the corresponding square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) is 1.

If all a i are pairwise distinct, then
• If M N (a 1 , . . . , a 4 ) does not have nontrivial symmetries, or if any nontrivial symmetry decomposes into two cycles of length 2, the index of the Veech group is 6. • If M N (a 1 , . . . , a 4 ) has a symmetry represented by a cycle of length 4 or by a single cycle of length 2, the index of the Veech group is 3. • If M N (a 1 , . . . , a 4 ) has a symmetry represented by a cycle of length 3, the index of the Veech group is 2.
All the symmetries listed above are realized.
Proof. If for some triple of pairwise distinct indices i , j , k ∈ {1, 2, 3, 4} the numbers a i = a j = a k coincide, it is clear that the index of the Veech group of the square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) is 1, say, as for M 4 (1, 1, 1, 1) or for M 6 (3, 1, 1, 1). Let us suppose that there is no such triple of indices, but there is a pair of indices i = j , where i , j ∈ {1, 2, 3, 4}, such that a i = a j , say, as for M 10 (1,1,3,5) or for M 6 (1, 1, 5, 5). Then the transposition in the symmetric group S 4 which interchanges the two labels corresponding to a i and a j fixes the square-tiled cyclic cover M N (a 1 , . . . , a 4 ). The image of a transposition under the projection (14) is again a transposition. Hence, the index of the Veech group in this case is either 3 or 1.
Let us show that index 1 is excluded. Let us suppose that the index of the Veech group is 1, that is, for all permutations π ∈ S 4 the flat surfaces represented by square-tiled cyclic covers M N (a π(1) , a π(2) , a π(3) , a π(4) ) are isomorphic. Thus, without loss of generality we may assume that a 2 = a 1 , and a 3 = a 1 , a 4 = a 1 . Then Deck(σ 1 σ −1 2 ) = a 1 − a 2 = 0 . This property can be formulated in a form invariant under "pillow symmetries", namely: for at least one of the two vertical saddle connections of the "square pillow" (P 1 (C), q 0 ) the loop encircling first one of the corresponding singularities in the positive direction and then the other singularity in the negative direction lifts to a closed loop on M N (a 1 , a 2 , a 3 , a 4 ). Clearly this property is not valid for the surface M N (a 1 , a 3 , a 2 , a 4 ) and we get a contradiction.
Consider now the remaining case when all a i are pairwise distinct. If any nontrivial symmetry decomposes into two cycles of length 2, all the symmetries are reduced to "pillow symmetries" and by Corollary 17 the index of the Veech group of M N (a 1 , . . . , a 4 ) is 6. This situation realizes, for example, for M 8 (1, 3, 5, 7).
Let us suppose that M N (a 1 , . . . , a 4 ) has a symmetry represented by a cycle of length 4. As an example, consider M 10 (1, 3, 9, 7) and a symmetry corresponding to the multiplication by k = 3. The image of such symmetry under projection (14) is a transposition. By Corollary 17 this implies that the index of the Veech group of M N (a 1 , . . . , a 4 ) is either 3 or 1. It is left as an exercise to verify that index 1 is excluded (see Appendix B). M N (a 1 , . . . , a 4 ) has a symmetry represented by a cycle of length 2. As an example, consider M 40 (1,9,5,25) and a symmetry induced by multiplication by 9. A transposition is mapped by the projection (14) to a transposition. Hence, the index of the Veech group in this case is again either 3 or 1. It is left as an exercise to verify that index 1 is excluded (see Appendix B).

Let us suppose now that
Finally, let us suppose that M N (a 1 , . . . , a 4 ) has a symmetry represented by a cycle of length 3. As an example, consider M 14 (1,9,11,7) and a symmetry induced by multiplication by k = 9. A cycle of length 3 is mapped by the projection (14) to a cycle of length 3. Hence, the index of the Veech group is either 2 or 1. If it were 1, one of the symmetries would be an odd permutation, i.e., a single cycle of length 2 or 4. We have proved that the presence of such a symmetry excludes index 1. Theorem 18 and, thus, Theorem 13 are proved.
To complete this section we note that the SL(2, R)-orbit of any square-tiled surface (respectively, the PSL(2, R)-orbit in the case when the flat structure is represented by a quadratic differential) inside the ambient moduli space of Abelian or quadratic differentials is closed. Its projection to the moduli space of curves is a Riemann surface with cusps, often called an arithmetic Teichmüller curve, see [35,19]. Any arithmetic Teichmüller curve is a finite cover of the modular curve. Theorem 13 shows that for a square-tiled cyclic cover, the corresponding arithmetic Teichmüller curve is small: it is a 1, 2, 3, or 6-fold cover of the modular curve.

SUM OF LYAPUNOV EXPONENTS
3.1. Sum of the Lyapunov exponents for a square-tiled surface. Let us consider a Teichmüller curve C . Each point x of C is represented by a Riemann surface S x . We can consider the cohomology space of H 1 (S x , R) as a fiber of a vector bundle H 1 over C , called the Hodge bundle. Similarly one defines the bundles H 1,0 and H 1 C . Note that each fiber is endowed with a natural integer lattice H 1 (S x , Z), which enables us to identify the fibers at nearby points x 1 , x 2 . Hence, the bundle H 1 is endowed with a natural flat connection called the Gauss-Manin connection.
The Teichmüller curve C is endowed with a natural hyperbolic metric associated to the complex structure of C ; the total area of C with respect to this metric is finite. Consider a geodesic flow in this metric, and consider the monodromy of the Gauss-Manin connection in H 1 with respect to the geodesic flow on C . We get a 2g -dimensional symplectic cocycle. The geodesic flow on C is ergodic with respect to the natural finite Lebesgue measure. Let us denote by λ 1 ≥ λ 2 ≥ · · · ≥ λ 2g the Lyapunov exponents of the corresponding cocycle. Since the cocycle is symplectic, its Lyapunov spectrum is symmetric in the sense that λ k = −λ 2g −k+1 for all k = 1, . . . , 2g .
Note that C is isometrically immersed (usually embedded) into the corresponding moduli space of curves with respect to the hyperbolic metric on the Teichmüller curve C and Teichmüller metric in the moduli space. The cocycle described above is a particular case of a more general cocycle related to the Teichmüller geodesic flow on the moduli space (sometimes called the Kontsevich-Zorich cocycle).
We need two results from [11] concerning the sum λ 1 +· · ·+λ g of all nonnegative Lyapunov exponents of the Hodge bundle H 1 along the geodesic flow on an arithmetic Teichmüller curve. ([11]). The sum of nonnegative Lyapunov exponents of the Hodge bundle H 1 along the geodesic flow on an arithmetic Teichmüller curve in a stratum H (m 1 , . . . , m n ), where m 1 +· · ·+m n = 2g −2, satisfies the following relation:

THEOREM 19
where S 0 is a square-tiled surface representing the corresponding arithmetic Teichmüller curve. When an arithmetic Teichmüller curve belongs to a stratum Q(d 1 , . . . , d n ) of quadratic differentials, one can consider the same vector bundle H 1 as above and define the cocycle and the Lyapunov exponents exactly in the same way as for holomorphic 1-forms. By reasons which will become clear below, it is convenient to denote these Lyapunov exponents by λ + 1 ≥ · · · ≥ λ + 2g . Since the cocycle is symplectic, we again have the symmetry λ + k = −λ + 2g −k+1 for all k = 1, . . . , 2g . However, for quadratic differentials one has λ + 1 < 1 for any invariant suborbifold (in fact, for any invariant probability measure, see [14]).
Following M. Kontsevich [22], in the case of quadratic differentials one can define one more vector bundle, H 1 − , over Q(d 1 , . . . , d n ). Consider a pair (Riemann surface S, quadratic differential q) representing a point of a stratum of quadratic differentials Q(d 1 , . . . , d n ). By assumption, q is not a global square of a 1-form. There exists a canonical (possibly ramified) double cover p 2 :Ŝ → S such that p * 2 q = ω 2 , where ω is a holomorphic 1-form onŜ. Following [22] it will be convenient to introduce the following notation. Letĝ be the genus of the coverŜ. By effective genus we call the positive integer (17) g eff :=ĝ − g .
The cohomology space H 1 (Ŝ, R) splits into a direct sum H 1 (Ŝ, + and on H 1 − along Teichmüller geodesic flow defines two symplectic cocycles. Following the notations established above, we denote the Lyapunov exponents of the cocycle acting on H 1 + by λ + 1 ≥ · · · ≥ λ + 2g and the ones of the cocycle acting on H 1 − by λ − 1 ≥ · · · ≥ λ − 2g eff . As always for symplectic cocycles we have the symmetries λ + k = −λ + 2g −k+1 and λ − k = −λ − 2g eff −k+1 . It follows from the analogous result for the case of Abelian differentials that one always has λ − 1 = 1. Unlike H 1 + , the vector bundle H 1 − on a stratum of quadratic differentials is not induced from a vector bundle on an underlying moduli space of curves. However, it can be descended to Q(d 1 , . . . , d n )/C * , where C * is identified with the subgroup of GL(2, R) acting on Q(d 1 , . . . , d n ) by multiplying a quadratic differential by a nonzero constant. In particular, for any Veech surface (S, q) in Q(d 1 , . . . , d n ) the bundle H 1 − can be descended from the PSL(2, R)-orbit O of (S, q) in Q(d 1 , . . . , d n ) to the corresponding Teichmüller curve C = O /C * . THEOREM 21 ([11]). The sums of all nonnegative Lyapunov exponents of the bundles H 1 + and H 1 − along the geodesic flow on an arithmetic Teichmüller curve in a stratum Q(d 1 , . . . , d n ), of meromorphic quadratic differentials with at most simple poles, satisfy the following relations: for the Lyapunov exponents of the bundle H 1 = H 1 + and for the Lyapunov exponents of the bundle H 1 − . Here S 0 is a square-tiled surface representing the corresponding arithmetic Teichmüller curve, and g and g eff are the genus and the effective genus (17) of S 0 .

3.2.
Sum of the Lyapunov exponents for a square-tiled cyclic cover. Now everything is ready to apply the results of the previous section to square-tiled cyclic covers. a 1 , a 2 , a 3 , a 4 satisfying the relations in formula (1).

THEOREM 22. Let us consider an even integer N and a collection of odd integers
The sum of all nonnegative Lyapunov exponents of the Hodge bundle H 1 along the geodesic flow on the arithmetic Teichmüller curve of the square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) is expressed by the formula below: Proof. We apply formula (16) taking into account the following data. The singularity pattern (m 1 , . . . , m n ) of the holomorphic 1-form corresponding to the square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) is computed in Lemma 2, see formula (5). The SL(2, Z)-orbit of the square-tiled surface M N (a 1 , a 2 , a 3 , a 4 ) is described by Theorem 18 and the cylinder decomposition for each square-tiled surface in the orbit is given in Lemma 9. By plugging the above data in formula (16) we obtain formula (20). N and a 1 , . . . , a 4 satisfying the relations in formula (1). Let us suppose, in addition, that either N is odd, or N is even and at least one of a i , i = 1, 2, 3, 4, is also even. The sum of all nonnegative Lyapunov exponents of the bundle H 1 + along the geodesic flow on the arithmetic Teichmüller curve of the square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) is expressed by the following formula:

THEOREM 23. Let us consider integers
gcd 2 (N , a 1 + a 2 ) + gcd 2 (N , a 1 + a 3 ) + gcd 2 (N , a 1 + a 4 ) The sum of all nonnegative Lyapunov exponents of the bundle H 1 − along the geodesic flow on the arithmetic Teichmüller curve of the square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) is expressed by the following formula: Proof. We apply formulae (18) and (19) taking into account the following data. The singularity pattern (d 1 , . . . , d n ) of the quadratic differential corresponding to the square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) is computed in Lemma 2, see (6). The PSL(2, Z)-orbit of the square-tiled cyclic surface M N (a 1 , a 2 , a 3 , a 4 ) is described by Theorem 13 and a cylinder decomposition for each square-tiled surface in the orbit is given in Lemma 9. By plugging the above data in (18) and (19) we obtain (21) and (22) respectively.

REMARK 24.
Actually, the Hodge bundles H 1,0 and H 1 C over the Teichmüller curve of a square-tiled cyclic cover have an explicit decomposition into a direct sum of one-and two-dimensional vector subbundles, see [5]. A similar decomposition, used also in [6,28] and [18], enables, in particular, to compute explicitly all individual Lyapunov exponents for any square-tiled cyclic cover, see [12].

Degenerate Lyapunov spectrum.
In this section we list all examples of arithmetic Teichmüller curves coming from square-tiled cyclic covers with maximally degenerate Lyapunov spectra. We recall that by elementary geometric reasons in strata of Abelian differentials λ 1 is equal to one, while in strata of quadratic differentials (which are not squares) λ − 1 is equal to one, for any ergodic invariant measure. Thus, for strata of Abelian differentials we speak of "maximally degenerate spectrum" whenever λ 2 = · · · = λ g = 0, while for strata of quadratic differentials (which are not squares) we speak of "maximally degenerate spectrum" of λ − -exponents, whenever λ − 2 = · · · = λ − g eff = 0 and "maximally degenerate spectrum" of λ + -exponents, whenever λ + 1 = · · · = λ + g = 0.
For all other cyclic covers of the form M N (a 1 , a 2 , a 3 , a 4 ) with even N and odd a i , i = 1, 2, 3, 4, one has λ 2 > 0.
It remains to prove that for all other collections N , a 1 , . . . , a 4 the right-hand side of formula (20) is strictly greater than 1. Applying formula (20) we see that this statement is valid for the remaining two collections for N = 4. Now we can assume that N ≥ 6.
Since gcd(N , a i ) is a divisor of N , and 1 ≤ a i < N , we conclude that gcd(N , a i ) might be N /2, N /3 or less. Hence, we always have This implies that if we have gcd(N , a i ) = gcd(N , a j ) = N /2 for two distinct indices i = j , then a i = a j = N /2 and at least one of the summands in gcd 2 (N , a 1 + a 2 ) + gcd 2 (N , a 1 + a 3 ) + gcd 2 (N , a 1 + a 4 ) is equal to N 2 . This means that the expression on the right of (20) is strictly greater than 1.
Thus, we can assume that there is at most one a i such that gcd(N , a i ) = N /2, while for the other indices j = i we have gcd(N , a j ) ≤ N /3. Then For N ≥ 16 we have 5N /72 > 1, and hence, for N ≥ 16 the expression on the right of (20) is strictly greater than 1. It remains to consider finite number of arrangements N , a 1 , . . . , a 4 , with 6 ≤ N ≤ 14. This can be done either by a straightforward check, or by considerations similar to the ones as above.  Figure 6 presents the square-tiled cyclic cover M 4 (1,1,1,1). This surface is also a 2-fold cover over a torus branched at four points. It has plenty of unusual properties. In particular, T. Monteil noticed that for every saddle connection there is always a twin saddle connection with the same length and the same direction, and there are no simple saddle connections joining a singularity to itself. M. Möller proved that the Teichmüller curves of the square-tiled cyclic covers M 4 (1, 1, 1, 1), presented in Figure 6, and M 6 (1, 1, 1, 3), presented in Figure 3, are also Shimura curves [30].

REMARK 30.
According to M. Möller [30], the conjecture holds in all genera different from five and for some strata in genus five. For the remaining strata in genus five the statement requires some extra verification.

PROBLEM.
Are there any closed invariant suborbifolds (of any dimension) in strata of Abelian differentials with maximally degenerate Lyapunov spectra (λ 2 = · · · = λ g = 0) different from the two Teichmüller curves presented above? REMARK 31. It is proved in [11] that there are no such regular SL(2, R)-invariant suborbifolds in any stratum of Abelian differentials of genus 7 and higher and in some strata in genera 5 and 6. REMARK 32. In genus g = 2, M. Bainbridge [4] has proved that the second exponent λ 2 equals 1/2 for all ergodic SL(2, R)-invariant measures supported in the stratum H (1, 1), corresponding to two simple zeros of the holomorphic differential, and λ 2 equals 1/3 for all ergodic SL(2, R)-invariant measures supported in the stratum H (2), corresponding to a double zero. In particular, in genus 2 the Lyapunov spectrum is always nondegenerate and simple. Bainbridge's result was already known conjecturally since [22] as a consequence of a formula for the sum of exponents for an SL(2, R)-invariant submanifold of the hyperelliptic locus in any stratum. Such a formula has now been proved in [11].
Thus, any regular SL(2, R)-invariant suborbifold with maximally degenerate Lyapunov spectrum might live in genera 3 and 4 and in some strata in genera 5 and 6 only.
It is proved in [11] that there are no regular PSL(2, R)-invariant suborbifolds with maximally degenerate Lyapunov spectra on the bundle H 1 − in any stratum of holomorphic quadratic differentials of genus 7 and higher and in some strata in genera 5 and 6. Thus, if the answer to Problem 3.3.2 is affirmative, the corresponding invariant suborbifold should correspond to genera 3 or 4 or to some particular strata in genera 5 or 6.
A formula in [11], which generalizes formula (18), shows that one always has λ + 1 > 0, hence the spectrum of Lyapunov exponents of the subbundle H 1 + over an PSL(2, R)-invariant submanifold in a stratum of holomorphic quadratic differentials can never be maximally degenerate. Theorem 35 in the section below shows that the situation is different for strata of meromorphic quadratic differentials with at most simple poles.

Meromorphic quadratic differentials.
Let us consider square-tiled cyclic covers M N (a 1 , a 2 , a 3 , a 4 ) which give rise to meromorphic quadratic differentials with simple poles. This case corresponds to collections (N ; a 1 , a 2 , a 3 , a 4 ) of parameters where at least one of the a i 's is equal to N .  M N (a 1 , a 2 , a 3 , a 4 ) is maximally degenerate: all Lyapunov exponents λ + i are equal to zero. Proof. Without loss of generality we may assume that a 1 = N . Applying formula (21) we get: is maximally degenerate, that is, λ − 2 = · · · = λ − g eff = 0 and λ + 1 = 0. For all other cyclic covers of the form M N (N , a 2 , a 3 , a 4 ) one has g eff ≥ 2 and λ − 2 > 0. Proof. Without loss of generality we may assume that a 1 = N ≥ 3. Formula (22) applied to this case can be rewritten as follows: Hence, for N > 4 we get λ − 2 > 0. Applying the above formula to remaining data for N = 2, 3, 4 we complete the proof of the Theorem. Theorem 36 have the following origin. For each of the cyclic covers as above consider a canonical ramified double cover such that the induced quadratic differential is a global square of a holomorphic 1-form. The resulting square-tiled surface is isomorphic to the square-tiled cyclic cover M 6 (3, 1, 1, 1) for the double cover over M 3 (3, 1, 1, 1) and also to the square-tiled cyclic cover M 4 (1, 1, 1, 1) for the double cover over M 4 (4, 2, 1, 1  (1, 1, 1, 1) is connected. The ambient stratum H (2, 2, 2) for the square-tiled surface corresponding to the cyclic cover M 6 (1, 1, 1, 3) contains two connected components representing even and odd parity of the spinstructure of the corresponding holomorphic 1-form, see [23]. The corresponding parity of the spin-structure of the square-tiled cyclic cover M 6 (1, 1, 1, 3) was computed in [27] by combinatorial methods. We present another calculation to illustrate an alternative analytic technique, which is less known in the dynamical community.
Proof. Let us consider a holomorphic 1-form ω with zeroes of even degrees only, that is, with a pattern of zeroes of the form (2d 1 , . . . , 2d n ) and let K (ω) = 2d 1 P 1 + · · · + 2d n P n be its zero divisor. An equivalent definition of the parity of the spin structure φ(ω) associated to ω is the dimension of the space of holomorphic 1-forms with zeroes of degrees at least d 1 , . . . , d n at P 1 , . . . , P n respectively, computed modulo 2, that is, φ(ω) := dim 1 2 K (ω) + 1 (mod 2) (see [1,21,29,31] for more information on the spin-structure). Let us compute the above dimension for the holomorphic 1-form ω defining the square-tiled flat structure ω 2 = p * q 0 on M 6 (1, 1, 1, 3). We recall that M 6 (1, 1, 1, 3) is defined by the equation Let us consider the following 1-forms on M 6 (1, 1, 1, 3), We claim that all these forms are holomorphic (in the case of α(c 1 , c 2 ), it is holomorphic for all values of parameters c 1 , c 2 ∈ C). For example, let us check this for γ. When w = 0 and z = ∞, the form γ is clearly holomorphic. In a neighborhood of any of the ramification points P i , where i = 1, 2, 3, we have w 6 ∼ (z −z i ), so d z ∼ w 5 d w, and with respect to a local coordinate w in a neighborhood of P 4 , we get γ ∼ w 2 d w. This shows that γ has a double zero at each of the points P 1 , P 2 , P 3 . In a neighborhood of z 4 we have w 2 ∼ (z − z 4 ), so with respect to a local coordinate w we get d z = w d w, and hence γ ∼ w 2 w d w w 3 = d w. Hence, each of the three preimages of z 4 on our Riemann surface is a regular point of γ. Finally, choosing a local coordinate t = 1/z in a neighborhood of z = ∞ we see that (z − z i ) ∼ t −1 , d z ∼ t −2 d t , and w ∼ t −1 , so γ ∼ d t . Hence, any preimage of z = ∞ is a regular point of γ. In a similar way we can check that α(c 1 , c 2 ) and β are holomorphic (see [5,6,12] for more details). Clearly, (24) α(c 1 , c 2 ) + c 3 β + c 4 γ is identically zero if and only if c 1 = c 2 = c 3 = c 4 = 0. By formula (9), we know that the genus of M 6 (1, 1, 1, 3) is equal to 4. Hence, we have constructed a basis of the space of holomorphic 1-forms on M 6 (1, 1, 1, 3): every holomorphic 1form can be represented as a linear combination (24), and this representation is unique. Note that the forms α, β, γ are eigenforms of the deck transformation (z, w) → (z, ζw), where ζ is a sixth primitive root of unity, ζ 6 = 1. The corresponding eigenvalues are ζ, ζ 2 , ζ 3 . In particular, since ζ 3 = −1, the form γ is anti-invariant with respect to a generator of the group of deck transformations. Our calculation shows that the eigenspace corresponding to the eigenvalue −1 is onedimensional. Hence, the form γ differs from the holomorphic 1-form ω defining our square-tiled flat structure on M 6 (1, 1, 1, 3) only by a (nonzero) multiplicative constant.
Assume that c 1 , c 2 are not simultaneously equal to zero. When −c 2 /c 1 satisfies −c 2 /c 1 ∉ {z 1 , z 2 , z 3 , z 4 , ∞}, the holomorphic 1-form α(c 1 , c 2 ) has six simple zeroes: one at each of the six preimages of the root of the polynomial (c 1 z +c 2 ). When c 1 = 0, the holomorphic 1-form α(c 1 , c 2 ) also has six simple zeroes: one at each of the six preimages of z = ∞. When −c 2 /c 1 = z i , where i = 1, 2, 3, the form α(c 1 , c 2 ) has a single zero of degree 6 at P i . Finally, when −c 2 /c 1 = z 4 , the form α(c 1 , c 2 ) has three zeroes of degree two: one at each of the three preimages of z = z 4 .
The holomorphic form β has 6 simple zeroes: one at each of the six ramification points z = z i , where i = 1, 2, 3, 4.

APPENDIX B. AN EXERCISE IN ARITHMETICS
In this appendix we present an exercise mentioned in the proof of Theorem 18. We show that if a square-tiled cyclic cover M N (a 1 , a 2 , a 3 , a 4 ) has a symmetry represented by a single cycle of length 4 or 2, then the index of the Veech group is different from 1.
We start with a symmetry represented by a cycle of length 4. If the index of the Veech group is 1, then for all permutations π ∈ S 4 the flat surfaces represented by square-tiled cyclic covers M N (a π(1) , a π(2) , a π(3) , a π(4) ) are isomorphic. By Lemma 9 this implies that for any i = j and any m = l , where i , j , m, l ∈ {1, 2, 3, 4} one has (25) gcd(N , a i + a j ) = gcd(N , a m + a l ) .
Since the symmetry group is the entire S 4 , we may assume that the ramification points are numbered in such way that the cycle π acts as a 1 → a 2 → a 3 → a 4 → a 1 , that is a i = a 1 · k i −1 (mod N ), where i = 1, 2, 3, 4 .
By (1) a 1 + a 2 + a 3 + a 4 is divisible by N . On the other hand, gcd(a 1 + a 2 + a 3 + a 4 , N ) = gcd (a 1 (1 + k)(1 + k 2 ), N = = gcd (1 + k)(1 + k 2 ), N divides the product gcd(1 + k, N ) · gcd(1 + k 2 , N ) which is equal to d 2 by (26). Hence, N is one of the integers 1, 2, 4, which contradicts the assumption that all a i , a j are pairwise distinct. Let us suppose now that M N (a 1 , . . . , a 4 ) has a symmetry represented by a cycle of length 2 and let us show that the Veech group cannot have index 1. Since the symmetry group is the entire S 4 , without loss of generality we may assume that k · a 1 (mod N ) = a 1 k · a 2 (mod N ) = a 2 k · a 3 (mod N ) = a 4 k · a 4 (mod N ) = a 3 Let := N / gcd(k −1, N ). Since (k −1)· a 1 (mod N ) = 0 and a 1 ≤ N , 1 < k < N , we have > 1, and divides a 1 . Similarly, divides a 2 . Hence divides a 1 + a 2 and thus gcd(a 1 +a 2 , N ). We have seen that, when the index of the Veech group is 1, the relations in formula (25) hold, in particular gcd(a 1 + a 2 , N ) = gcd(a 1 + a 3 , N ) .
Since is a divisor of both gcd(a 1 + a 3 , N ) and a 1 , it divides a 3 , hence it also divides a 4 = k · a 3 (mod N ). Thus, divides gcd (N , a 1 , a 2 , a 3 , a 4 ). Since > 1, this contradicts the second condition in formula (1).