We will show that: that sends the element in π 1 (P {z 1, z 2, z 3, z 4 }) represented by l j to g j G.

Transcription

1 1. Introdution Square-tiled translation surfaes are lattie surfaes beause they are branhed overs of the flat torus with a single branhed point. Many non-square-tiled examples of lattie surfaes arise from deforming ertain square-tiled surfaes. These inludes Veeh s regular n-gon, Ward s surfae, and, more generally, the Bouw-Möller surfaes, whih were disovered by Bouw-Möller [BM10]. They arises from abelian normal overs of the flat pillowase. Hooper [Hoo13] gave a translation-surfae-theoreti desription of these examples by introduing a lass of Thurston-Veeh diagrams alled grid graphs, and Wright [Wri13]proved the equivalene of the onstrution in [Hoo13] and [BM10]. Here we formalize the onstrution of Bouw-Möller surfaes by studying the lattie surfae that arises from deforming a ertain marked square-tiled surfae M aording to a ertain ohomology lass α H 1 (M, Σ; C). Our approah lead to a new lass of Thurston-Veeh diagram similar to Hooper s but not new lattie surfaes. We hope that it an provide some insights on what makes Bouw-Möller s onstrution works. We will show that: Theorem 1.1. If a lattie surfae arises from deforming an abelian branhed over of flat pillowase M aording to a relative ohomology lass α, its affine diffeomorphism group is ommensurable with the affine diffeomorphism group of M, and it also preserves the Thurston-Veeh struture in the two diagonal diretions, then its Thurston-Veeh diagram must be one of the three types desribed in Setion 4 and 5. As a onsequene, the surfae must be one of the followings ases: 1. a branhed over of the regular 2n-gon branhed at the one point. 2. a branhed over of a Bouw-Möller surfae. 3. a branhed over of the regular n-gon branhed at the mid-point of edges. We start by desribing the square-tiled surfaes we will deform. To do so, we use the same notation as in [Wu14], whih we now review. As shown in Figure 1, let P be the flat pillowase built from a pair of unit squares by identifying edges with the same label. The two squares are labeled as B 1 and B 2, the four one points are labeled as z 1, z 2, and, and the four edges are labeled as e 1, e 2, e 3, and e 4. Let G be a finite abelian group. Let g = (g 1, g 2, g 3, g 4 ) G 4 be a 4-tuple of elements in G suh that g 1 g 2 g 3 g 4 = 1. We denote the simple loop around z j that travels in ounterlokwise diretion on P, with base point in B 1, as l j. The surfae we will deal with, M = M(G, g), is the branhed abelian over of the pillowase P with dek group G indued by the group homomorphism π 1 (P {z 1, z 2,, }) = l 1, l 2, l 3, l 4 l 1 l 2 l 3 l 4 = 1 G that sends the element in π 1 (P {z 1, z 2,, }) represented by l j to g j G. The deomposition of P into two squares in figure 1 leads to a ell deomposition on M(G, g). This ell struture an be desribed as G opies of pairs of squares, labeled 1

2 2 e 3 e 2 z 2 B 2 e 1 e 4 z 1 e 2 B 1 e 4 e 3 Figure 1. by an element g G and an index k {1, 2} as B k g, as shown in figure 2, that are glued together by identifying edges with the same label in suh a way that the arrows are in the same diretion. e 3 gg 2 g 3 z4 e 2 gg 2 B 2 g e 4 gg 2 g 3 g 4 z 2 e 1 g z 1 e 2 g B 1 g e 4 g e 3 g Figure 2. For example, the ell struture of the surfae M(Z/4, (1, 1, 1, 1)) (the Wollmilhsau [FM08, HS]) is presented as in Figure 3. Let Σ be the preimage of {z 1, z 2,, }. The flat struture on P indues a half translation struture on M with one points in Σ. When the orders of g 1, g 2, g 3 and g 4 in G are all even, this indued half translation struture is also a translation struture, whih is the only ase we will deal with in this paper. If some of the g i have odd orders, we an replae G by a larger group G and replae M with a double over M suh that the orders of all g i in G are even. As in [Wri13] and [Wu14], the ation of G on the relative

3 3 e 3 2 e 3 3 e 3 0 e 3 1 e 2 1 B 2 0 e 4 3 e 2 2 B 2 1 e 4 0 e 2 3 B 2 2 e 4 1 e 2 0 B 2 3 e 4 2 z 2 z 1 z 2 z 1 z 2 z 1 z 2 z 1 e 2 0 B 1 0 e 4 0 e 2 1 B 1 1 e 4 1 e 2 2 B 1 2 e 4 2 e 2 3 B 1 3 e 4 3 e 3 0 e 3 1 e 3 2 e 3 3 Figure 3. ohomology of marked translation surfae (M, Σ) indues a deomposition of H 1 (M, Σ; C) into irreduible representations. Hene we have deomposition H 1 (M, Σ; C) = ρ H1 (ρ), where ρ goes through all the irreduible representations of G, and H 1 (ρ) is the sum of all the irreduible G-subspaes isomorphi to ρ. This deomposition is preserved by a finite index subgroup of the Affine diffeomorphism group Aff(M), whih we denote as Γ. Given a translation surfae (M, Σ) with a polygonal deomposition, and a 1-form α H 1 (M, Σ; C), we now define the deformation of (M, Σ) aording to α. We have: Lemma 1.2. Let (M, Σ) be a square-tiled translation surfae. If a ohomology lass α H 1 (M, Σ; C) evaluated on the four sides of any square in M are the oordinates of the four sides of a onvex quadrilateral with non-negative area, then there is a translation surfae X with ertain points identified, and a degree 1 map from M to X, suh that the pull bak of the translation struture of X is defined by α. Proof. We onstrut X and the map M X as follows: every 2-ell B in M is mapped to a onvex, possibly degenerate, quadrilateral on C with non-negative area, the oordinate of its four sides are given by α evaluated on the four sides of B. Eah of the four sides of B is sent to the respetive side of the quadrilateral linearly. The sides of those quadrilaterals are then glued to eah other by isometry to form X. Definition 1.1. We all X and the degree 1 map i : M X in the previous lemma a deformation of M aording to α. For example, if we let M = M(Z/30, (1, 11, 29, 19)), α be a 1-form orresponding to (5, 3) Bouw-Möller surfae (whih we will desribe in Setion 5), then the deformation onsists of four translation surfaes of the following shape glued together at their one points as in Figure 4. Labels a... h show the gluing of edges, and numbers show the image of 2-ells in M. Some of these images are onvex quadrilaterals, some are triangles, and some (1,5,11,15 in

4 4 e f g 13 h h d 9 b 8 g 7 f a1 2 3 b 6 e d 4 5 a Figure 4. the above figure) are ollapsed into line segments. The metri ompletion of X i(σ) may have multiple onneted omponents. Also, the deformation depends not only on α but also on the hoie of ell struture on (M, Σ). We denote the deformation of M aording to α as M X(α). Definition 1.2. Let i: (M, Σ) (X, Σ ) be a deformation of (M, Σ). We say a homeomorphism g : (M, Σ) (M, Σ) ats on X through an affine diffeomorphism f : X X, if the translation struture of X pulled bak by i g and f i determine the same ohomology lass in H 1 (M, Σ; C). Remark 1.1. In the situation we will onsider, we assume that a finite-index subgroup Γ of the affine diffeomorphism group Aut(M) ats on lattie surfae X through a finite index subgroup of Aut(X). Hene by Definition 1.2, Γ preserves N(α) = span C (α, α), and the ation is through a lattie in U(1, 1).

5 The deomposition H 1 (M, Σ) = ρ H1 (ρ) is Γ-invariant. By replaing Γ with Γ Γ, we will, from now on, always assume that Γ is a subgroup with finite index of Γ without loss of generality. As a onsequene, the projetion from H 1 (M, Σ) to eah H 1 (ρ) is Γ - equivariant. Hene we have, due to Shur s Lemma: Proposition 1.3. Under the assumption of Remark 1.1, the projetion from N(α) = span C (α, α) to H 1 (ρ) is either 0, or a Γ -isomorphism. In the latter ase Γ ats on H 1 (ρ) through a lattie in U(1, 1). From this we know: Corollary 1.1. Under the assumption of Remark 1.1, N(α) H 1 (ρ) is isomorphi to N(α) as Γ -module H 1 (ρ) In setion 2 we review the onept of Thurston-Veeh struture. In setion 3 we review the disrete Fourier transform whih is needed for the proof, and in setion 4 and 5 we prove Theorem Thurston-Veeh diagram Reall that a Thurston-Veeh struture [T + 88] on a translation surfae onsists of two tuples of positive integers {r i } and {r j }, and a pair of transverse ylinder deompositions {C i } and {C j }, with moduli {M i} and {M j } respetively, suh that M i 1 /M i2 = r i1 /r i2, M j 1 /M j 2 = r j 1 /r j 2. A Thurston-Veeh struture on a translation surfae indues two paraboli affine diffeomorphisms γ and γ given by two multitwists on {C i } and {C j } respetively. The intersetion onfiguration of the two ylinder deompositions an be represented by a Thurston-Veeh diagram, whih is a ribbon graph onstruted as follows: eah vertex represents a ylinder, an edge between two verties stands for an intersetion of two ylinders, and the yli order among all edges assoiating with eah vertex enodes the order of these intersetions on the ylinder represented by this vertex. On a square-tiled surfae M, the two diagonal diretions are periodi, and ylinder deompositions on these two diretions form a Thurston-Veeh struture. We all the ylinders from the bottom-left to top-right {C i }, and the ylinders from the bottom-right to top-left {C j }. The Thurston-Veeh diagram that arises from this Thurston-Veeh struture an be onstruted expliitly as follows: we get a square-tiled oned flat surfae by gluing the squares with the same edge pairing as in M but with all the gluing diretions flipped, whih we denote as FM. Now eah vertex in the square tiling of FM orresponds to a ylinder in M formed by triangles, as illustrated in Figure 5. Here the 4 triangles I, II, III and IV that form a ylinder in the bottom-left to topright diretion beome the 4 triangles around P after the re-gluing, just as triangles that 5

6 6 M FM IV II III F IV I P III II Q I a d d re-glue b b a flip a d d d d a a b b b b a Figure 5. form the ylinder between the two red arrows beome the 4 triangles around Q after the re-gluing. Furthermore, the intersetion of these two ylinders is represented by the edge P Q. The yli orders among the edges an be seen from blue and red arrows. Hene, the Thurston-Veeh diagram is the 1-skeleton of FM, with the yli orders in lokwise and ounterlokwise diretion alternatively by olumns as illustrated in Figure 6. In our ase when M is a over of the pillowase, FM is always orientable. Beause in the gluing proess desribed above, any two adjaent squares are always glued with orientation reversed, we an define an orientation on FM as either the orientation of all the B 1 g after the regluing, or the opposite of the orientation of all B 2 g after regluing. Also, beause M is

7 7 P Q Figure 6. a normal over, all ylinders in the same diagonal diretion have the same irumferene and width hene the same moduli, i.e. r i = r j = 1 for all i, j. Definition 2.1. We say that a 1-form α H 1 (M, Σ) preserves a ylinder deomposition {C i } on M, if α evaluated on all paths ontained in all the boundary urves of these ylinders are parallel. We say α preserves a Thurston-Veeh struture, if it preserves both ylinder deompositions, and the multitwists γ, γ, whih are indued by to the Thurston- Veeh struture, preserve span C (α, α). A ohomology lass α that satisfies the assumption of Remark 1.1 must preserve at least two paraboli elements, and, thereby, two ylinder deompositions. Here, we assume that it preserves the two ylinder in the two diagonal diretions of M. We denote the set of elements of H 1 (M, Σ; C) that preserves the two ylinder deomposition in the diagonal diretions as L(M, Σ), and the set of those elements in L that further preserve the Thurston-Veeh struture in the diagonal diretions as N (M, Σ). Furthermore, we denote by L and N the set of those elements in L and N whose value on bottom-left-to-top-right diagonals are in R, and whose value on bottom-right-to-top-left diagonals are in 1R. Any element in L or N an be made into an element in L or N after an affine transformation. The irumferene, width and moduli of C i under α is defined as the base, height and moduli of the parallelogram formed by α evaluated on the boundary irle and α evaluated on a path that rosses the ylinder one from right to left. Let V = V (FM) be the spae of real valued funtions on the verties of FM. Let Ψ : L V be the map defined as follows: given any element α L, Ψ(α) send a vertex to the width under α of the ylinder it represents. It is a bijetive map. Beause r i = r j = 1, the fat that multitwists γ and γ preserve span C (α, α) implies that any C i with non-zero width have the same moduli, and those with width 0 also have 0 irumferene. After an

8 8 affine ation we an make all the ylinders to have either idential moduli λ > 0 or zero width and zero irumferene. Hene, by [T + 88], we have: Lemma 2.1. [T + 88] Let A be the adjaeny matrix of the Thurston-Veeh diagram, then up to an affine ation, α satisfies A(Ψ(α)) = λψ(α). When α L, the assumption of Lemma 1.1 an be further rewritten as a ondition on the image of Ψ: Lemma 2.2. Assuming α L, the assumption of Lemma 1.1 is equivalent to the fat that no pair of adjaent verties of FM an be assigned values of opposite signs by Ψ(α), and there is an edge suh that neither of its end points are assigned 0. Proof. If assumption of Lemma 1.1 holds, any square Bj k will be sent to a onvex quadrilateral with non-negative area under the flat struture defined by α. The two diagonals of this quadrilateral ut it into 4 small triangles of non-negative area. If two non-degenerate ylinders in the ylinder deompositions desribed above interset, its intersetion must be a retangle formed by two of those small triangles hene must have non-negative signed area, i.e. Ψ(α) an not assign values of opposite signs for any pair of adjaent verties. At least one of these retangles must have positive area, hene there is at least one pair of adjaent verties suh that Ψ(α) is non-zero on both. On the other hand, if Ψ(α) does not assign values of opposite signs for any pair of adjaent verties, we an build a translation surfae X as follows: eah parallelogram formed by the intersetion of some C i and C j are sent to a Ψ(α)(v i)-by-ψ(α)(v j ) retangle, where v i and v j are verties in FM that represent C i and C j respetively. As illustrated in Figure 7, the sides of these retangles are glued in the same order as the gluing of the parallelograms on M, whih means that α satisfies the assumption of Lemma 1.1. Figure 7.

9 Remark 2.1. When α satisfies the assumption of Lemma 1.1, we an deompose X(α) into a union of (possibly degenerated) ylinders in two ways, and hoose the map i : M X(α) suh that these two ylinder deompositions are the images of {C i } and {C j }. Let S be the ategory of square-tiled surfaes with morphisms being ontinuos maps that send squares isometrially to squares. Our onstrution of FM, L(M), N (M) and V (FM) from M are funtorial. More preisely, Lemma 2.3. A S-morphism ξ : M M indues anonially a S-morphism Fξ : FM FM, ξ : L (M ) L (M), and (Fξ) : V (FM ) V (FM) and the following diagram ommute: L (M ) L (M) Ψ Ψ V (FM ) V (FM) Furthermore, ξ sends N to N. 9 Remark 2.2. (Fξ) an be desribed more onretely as follows: for any f V (FM ), any vertex in FM, (Fξ) f(v) = f(fξ(v)) if Fξ preserves the yli order among the edges assoiated to v, and (Fξ) f(v) = f(fξ(v)) if it reverses the yli order. Remark 2.3. F 2 is the same as the identity funtor 1 S. Hene F : S S is an isomorphism. Now we onsider the question of finding α N that satisfies the assumption of Lemma 1.1 and that a subgroup of Aff(M) of finite index ats on N(α) as a lattie. We say suh α satisfies property (L). Now we analyze all possible 1-forms with property (L). Let e be the identity element in G. ( Let γ 1 be) the element in Γ that preserve the buttonright orner of Be 1 and has derivative ; let γ be the element in Γ that preserve ( ) 1 0 the button-right orner of Be 1 and has derivative ; let γ be the element in Γ that ( ) 3 2 preserve the button-right orner of Be 1 and has derivative, and let γ be the ( ) 3 2 element in Γ that preserve the button-right orner of Be 1 and has derivative. 2 1 These elements are liftings of the Dehn twists in the horizontal, vertial and the two diagonal diretions on P. Let α N. By Definition 2.1, some power of γ 3 and γ 4 ats on X(α) as paraboli affine automorphisms, hene their ation on N(α) are paraboli. Beause a power of γ 3 ats on H 1 (ρ) as paraboli map if and only if ρ(g 1 g 3 ) = 1, by Corollary 1.1, N(α) ρ {ρ:ρ(g1 )ρ(g 3 )=1}H 1 (ρ). Hene, α is the pull bak of some α N (M(G/(g 1 g 3 ), g)) by a ord G (g 1 g 3 )-fold branhed over M M 1 = M(G/(g 1 g 3 ), g). Hene X(α) is a branhed

10 10 over of X(α ) branhing at i(σ) Σ. Beause the dek group of M 1 is G 1 = G/(g 1 g 3 ), g 1 g 3 = g 2 g 4 = e in G 1. As a onsequene, FM 1 is a ompat orientable flat surfae with no one points, hene it must be a square-tiled torus R 2 /L where L Z 2. Furthermore, from the onstrution of FM we know that eah horizontal ylinder in M orresponds to a horizontal ylinder of the same width and irumferene in FM. Hene (2ord G (g 1 g 2 ), 0) L. Similarly, (0, 2ord G (g 1 g 4 )) L. Let M 2 be a (not neessarily onneted) square tiled surfae suh that FM 2 is a 2ord G (g 1 g 2 )-by-2ord G (g 1 g 4 ) torus, then M 2 is a branhed over of M 1. By the onstrution of F, M 2 is also a branhed abelian over. Furthermore, the pull bak of any α N (M, Σ) satisfying property (L) would also satisfy property (L). Hene, we have: Proposition 2.4. Any α satisfying property (L) is related to an α 0 N (M 2 ) satisfying property (L) by finite branhed overs, where FM 2 is a 2ord G (g 1 g 2 )-by-2ord G (g 1 g 4 ) torus R 2 /2ord G (g 1 g 2 )Z 2ord G (g 1 g 4 )Z Hene, from now on, we always let FM be a 2s-by-2t retangular torus. 3. Disrete Fourier Transform To apply Corollary 1.1, we need to find a deomposition of V that is ompatible with the deomposition H 1 (M, Σ; C) = ρ H1 (ρ). We an obtain suh a deomposition by Disrete Fourier Transform. Let V (FM) = V (T s,t ) be the spae of real-valued funtions on Z 2 /(2sZ 2tZ). By disrete Fourier transform, V has the following diret-sum deomposition: V s,t = λ,µ V λ,µ s,t, where λ, µ are integers, suh that 0 λ < s/2, 0 µ < t if s is even, 0 λ < s/2, 0 µ < t, or λ = (s 1)/2 and µ < t/2 if s is odd, and V λ,µ s,t = span R (f 0,0, f 0,1, f 1,0, f 1,1, g 0,0, g 0,1, g 1,0, g 1,1 ), where: { sin(2xπλ/2s) sin(2yπµ/2t), if x + y is even f 0,0 (x, y) = 0, otherwise { sin(2xπλ/2s) os(2yπµ/2t), if x + y is even f 0,1 (x, y) = 0, otherwise { os(2xπλ/2s) sin(2yπµ/2t), if x + y is even f 1,0 (x, y) = 0, otherwise { os(2xπλ/2s) os(2yπµ/2t), if x + y is even f 1,1 (x, y) = 0, otherwise { sin(2xπλ/2s) sin(2yπµ/2t), if x + y is odd g 0,0 (x, y) = 0, otherwise

11 { sin(2xπλ/2s) os(2yπµ/2t), if x + y is odd g 0,1 (x, y) = 0, otherwise { os(2xπλ/2s) sin(2yπµ/2t), if x + y is odd g 1,0 (x, y) = 0, otherwise { os(2xπλ/2s) os(2yπµ/2t), if x + y is odd g 1,1 (x, y) = 0. otherwise 11 This deomposition is related to, but not the same as the eigenspae deomposition of disrete Laplaian. By the funtorial property, the G-ation on M indues an ation on FM whih indues ations on various V λ,µ. More preisely, the ation of g i on FM, i = 1, 2, 3, 4, are translations in the four diagonal diretions. By alulation, we know that Ψ 1 (V λ,µ ) H 1 ( ++ ) + H1 ( + ) + H1 ( + ) + H1 ( ), where: ++ (g 1) = e iπ(1+λ/s+ν/t) ++ (g 2) = e iπ(1 λ/s+ν/t) ++ (g 3) = e iπ(1 λ/s ν/t) ++ (g 4) = e iπ(1+λ/s ν/t) + (g 1) = e iπ(1 λ/s+ν/t) + (g 2) = e iπ(1+λ/s+ν/t) + (g 3) = e iπ(1+λ/s ν/t) + (g 4) = e iπ(1 λ/s ν/t) + (g 1) = e iπ(1+λ/s ν/t) + (g 2) = e iπ(1 λ/s ν/t) + (g 3) = e iπ(1 λ/s+ν/t) + (g 4) = e iπ(1+λ/s+ν/t) (g 1) = e iπ(1 λ/s ν/t) (g 3) = e iπ(1+λ/s+ν/t) (g 2) = e iπ(1+λ/s ν/t) (g 4) = e iπ(1 λ/s+ν/t) From [BM10], [Wri13] or Setion 7 of [Wu14], we know that if the ation of Γ on PH 1 (ρ) for any of the ρ ±± above is ommensurable to a hyperboli triangle group with paraboli element, then either (a) λ = 0 and ν t, or ν = 0 and λ s or (b) λ s and ν t. In ase (a) it is ommensurable to the (,, n)-triangle group, while in ase (b) it is ommensurable to a (, m, n)-triangle group. By Corollary 1.1, if α satisfies property (L) then Ψ(α) must be in one of these two types of V λ,µ. 4. The (,, n) ase In this ase, without loss of generality we an assume λ = s/n and ν = 0. Figure 8 shows the sign of the funtion f 0 = f 0,0 + g 0,0 for n = 5 on verties of FM. Points on whih f 0 = f 0,0 + g 0,0 is positive or negative form vertial stripes, whih are separated by olumns of zeros, hene Ψ 1 (f 0 ) defines a singular translation struture on M by Lemma 2.2. By omputation, this translation struture is tiled by regular 2n-gons.

12 (0, 0) Figure 8. These lattie surfaes were first disovered by Veeh. Now we have: Proposition 4.1. If α N that satisfies ondition (L) suh that Ψ(α) V s/n,0, then up to an affine ation and an automorphism of the square-tiled surfae M, Ψ(α) = f 0. Hene, X(α) is tiled by regular 2n-gons up to a SL(2, R)-ation. Proof. Firstly, beause the sum of the values of Ψ(α) is 0, by Lemma 2.2, the set of verties on whih Ψ(α) is positive and the set of verties on whih Ψ(α) is negative have to be separated by verties on whih Ψ(α) is 0. Hene, Ψ(α) must reahes 0 at a ertain vertex. After a G ation as well as relabeling if neessary, we an assume this vertex to be (0, 0) without loss of generality. Hene Ψ(α)( 1, 1)Ψ(α)(1, 1) < 0 or Ψ(α)(x, y) = 0 for all x + y even. If Ψ(α)( 1, 1)Ψ(α)(1, 1) < 0, by Lemma 2.2, Ψ(α)(0, 1) = 0, hene Ψ(α)(x, y) f 0,0 R + g 0,0 R, hene α is affine equivalent to α 0. The other ase, Ψ(α)(x, y) = 0 for all x + y even, ontradits with our assumption aording to Lemma The (, m, n) ase, Bouw-Möller surfaes Now we deal with the (, m, n) ase. Let f 1 = f 0,0 + g 0,0. Points on whih f 1 is positive or negative form retangles, whih are separated by olumns and rows of zeros, hene Ψ 1 (f 1 ) defines a singular translation struture on M by Lemma 2.2. Figure 9 shows the sign of f 1 when m = 5, n = 3. By deleting verties that are assigned 0 width in the 1-skeleton on T s,t we an get the grid graph desribed in [Hoo13], hene due to Lemma 2.1 the metri ompletion of eah omponent of (X(Ψ 1 (f 1 )) Σ ) is the (m, n) Bouw-Möller surfae. For example, if we label the squares of FM in the Figure 9 above as in Figure 10, then one omponent of X(Ψ 1 (f 1 )) beomes Figure 11, and X(Ψ 1 (f 1 )) is a union of opies of suh translation surfaes glued together at the one point.

13 (0, 0) Figure Figure 10. Also, when m = n, f 2 = f 0,0 + f 1,1 + g 0,0 + g 1,1 and f 3 = f 0,1 + f 0,1 + g 1,0 + g 1,0 also define lattie surfaes. They are affine equivalent to branhed overs of the regular n-gon branhed at the midpoints of its sides. Figure 12 shows the sign of f 2 for m = n = 5. Similar to the previous setion, we have: Proposition 5.1. If α N that satisfies ondition (L) suh that Ψ(α) V s/n,t/m, then up to affine ation and the automorphism of M, Ψ(α) is either f 1 or f 2. Proof. Due to Lemma 2.1, we an further assume that AΨ(α) = λψ(α) for some onstant λ. From this and the Peron-Frobenious theorem we know that the sign of Ψ(α) ompletely

14 Figure (0, 0) determines Ψ(α) up to saling. Figure 12. By the formula for f i,j and g i,j, i, j = 0, 1 in Setion 3 and the fat that AΨ(α) = λψ(α), Ψ(α) = C 1 sin(πx/n + A) os(πy/m) + C 2 sin(πx/n + B) sin(πy/m) for some onstants C 1, C 2, A and B, hene it restrited to any row is a funtion of the form ψ(x) = C sin(πx/n+a) for some onstants C and a, and it is a funtion of the form ψ (x) = C sin(πy/m + a ) for some onstants C and a when restrited to any olumn. Furthermore, by Lemma 2.2, the

15 set of verties on whih Ψ(α) is positive and the set of verties on whih Ψ(α) is negative have to be separated by verties on whih Ψ(α) is 0, hene Ψ(α) restrited to any row of verties must be one of the following ases: (a) 0. (b) C sin(π(x k)/n), k Z. It reahes 0 2s n times. Similarly, Ψ(α) restrited to any olumn of verties must be one of the following ases: (a) 0. (b) C sin(π(x k )/m), k Z. It reahes 0 2t m times. Beause of Lemma 2.2, at least one row or olumn will be in ase (b). Furthermore, we have: Lemma 5.2. If a row or a olumn of type (a) is next to a row or a olumn of type (b), the subgraph spanned of verties on whih Ψ(α) is non-zero is a union of grid graphs as defined in [Hoo13], hene by Lemma 2.1 it must be a over of a Bouw-Möller surfae. Proof. If a row of type (a) is next to a row of type (b), without loss of generality we let the row of type (a) be the 0-th row and the row od type (b) be the 1st. Beause Ψ(α) = C 1 sin(πx/n + A) os(πy/m) + C 2 sin(πx/n + B) sin(πy/m) and Ψ(α)(x, y) = 0, C 1 = 0. Furthermore, beause the 1st row is of type (b), nb/π Z. Hene the subgraph spanned of verties on whih Ψ(α) is non-zero is a union of grid graphs as defined in [Hoo13]. The argument for olumns is the same. Now we only need to deal with the ase when Ψ(α) restrited to any row or olumn of verties are all of type (b). Beause all olumns are of type (b) there must be 4st m zeros, and beause all rows are of type (b) there must be 4st n zeros. Hene, m = n. The only way for these zero verties to separate the other verties of T s,t into positive and negative parts is by aligning in the diagonal diretion. In other words, after saling and relabeling, Ψ(α) is f 0,0 + f 1,1 + g 0,0 + g 1,1 or f 0,1 + f 0,1 + g 1,0 + g 1,0. 15 Proof of Theorem 1.1. Theorem 1.1 is a orollary of Proposition 4.1 and Proposition 5.1. Referenes [BM10] I. I. Bouw and M. Möller. Teihmüller urves, triangle groups, and Lyapunov exponents. Ann. of Math., 172(2): , [FM08] G. Forni and C. Matheus. An example of a Teihmüller disk in genus 4 with degenerate Kontsevih- Zorih spetrum arxiv preprint arxiv: [Hoo13] W. P. Hooper. Grid graphs and lattie surfaes. International Mathematis Researh Noties, 12, [HS] F. Herrlih and G. Shmithüsen. An extraordinary origami urve. Math. Nahr., (2): [T + 88] William P Thurston et al. On the geometry and dynamis of diffeomorphisms of surfaes. Bulletin (new series) of the Amerian Mathematial Soiety, 19(2): , [Wri13] A. Wright. Shwarz triangle mappings and Teihmüller urves: the Veeh-Ward-Bouw-Möller urves. Geometri and Funtional Analysis, 23(2): , [Wu14] C. Wu. Affine diffeomorphisms on the relative ohomology of abelian overs of the flat pillowase