The relative cohomology of abelian covers of the flat pillowcase

We calculate the action of the group of affine diffeomorphisms on the relative cohomology of square-tiled surfaces that are normal abelian covers of the flat pillowcase, and as an application, answer a question raised by Smillie and Weiss.


Introduction
In a paper in preparation by Smillie and Weiss on horocycle orbit closures they ask for the existence of a square-tiled surface M with cone point sets Σ constructed as a normal abelian branched cover of the flat pillowcase, such that there is a direct sum decomposition H 1 (M, Σ; C) = N ⊕ H preserved by the action of the group of orientation preserving affine diffeomorphisms, there is a positive or negative definite hermitian norm on N invariant under the affine diffeomorphism group action, and the affine diffeomorphism group action on N does not factor through a discrete group. We will construct such examples in Section 6.
In answering this question, we give a comprehensive treatment on relative cohomology of branched abelian covers of the flat pillowcase, the affine diffeomorphism group action on it, as well as the invariant subspaces and invariant Hermitian form under this action. This problem is related to the monodromy of the hypergeometric functions which dates back to Euler and is outlined in [DM86]. Wright [Wri12] described the Hodge form and invariant direct sum decomposition on the absolute cohomology of such surfaces under the action of a subgroup of the Veech group, and calculated the Lyapunov exponents of this action by showing that the projectivization of the group action factors through a triangle group. Forni-Matheus-Zorich [FMZ11], Bouw-Möller [BM10], Deligne-Mostow [DM86], McMullen [McM13] and Eskin-Kontsevich-Zorich [EKZ10] have also done similar computations in different contexts. Matheus and Yoccoz [MY09] calculated the action of the full affine group on relative cohomology for two specific abelian branched covers. By modifying some of the ideas in their articles, as well as some arguments similar to [DM86] and [Thu98], we are able to describe the action of the affine group on relative cohomology and establish the existence of abelian covers as required by Smillie and Weiss's paper. Hubert-Schmithüsen [HSb] also gave a proof of the non-discreteness in some cases through Lyapunov exponents and Galois conjugate. The author thanks his thesis advisor John Smillie for suggesting the problem and many helpful conversations.
The existence of examples answering the question of Smillie and Weiss follows from a decomposition of cohomology into invariant components, which is done in Theorem 3.1, a signature calculation of the Hodge form on each component, and a discreteness criteria. Here we give an alternative, self-contained treatment. We will give a description of the affine diffeomorphism group of these surfaces in section 2. In section 3, we describe the action of affine diffeomorphism group on relative cohomology and show the existence of direct sum decomposition. In section 4, we calculated the signature of the Hodge Hermitian form. The reader is warned that our definition of Hodge Hermitian form is different from other definitions in the literature. In secretion we described a useful subgroup of affine diffeomorphism group to work with. In section 6 construct examples that answer the question of Smillie and Weiss. If we only need to construct certain examples, it can also be done with discreteness criteria and signature calculation in [DM86].
We will now set up some notation to describe normal branched covers of the pillowcase. Let P be the unit flat pillowcase with four cone points z 1 , z 2 , z 3 and z 4 of cone angle π as follows: Let G be a finite group and g = (g 1 , . . . , g 4 ) ∈ G 4 a 4-tuple of elements in G such that g 1 g 2 g 3 g 4 = 1. Let M = M (G, g), be the connected normal branched cover of P branching at z 1 , . . . z 4 , with deck transformation group G acting on the left. The loop l j around z j in counter-clockwise direction on P based in B 1 lifts to a path from the preimage of B 1 in the g-th sheet of the cover to the preimage of B 1 in the gg j -th sheet. In other words, g gives a group homomorphism from π 1 (P − {z 1 , z 2 , z 3 , z 4 }) = l 1 , l 2 , l 3 , l 4 |l 1 l 2 l 3 l 4 = 1 to G. Here the homomorphism defined by g sends l j to g j ∈ G. The connectedness of M is equivalent to the condition that {g 1 , . . . , g 4 } generate G. Let Σ denote the set of preimages of all points z j , j = 1, . . . , 4. The surface M has a half translation structure induced by the half translation structure on P . Let Aff (M, Σ) denote the group of orientation preserving affine diffeomorphisms from M to itself that sends Σ to Σ. When the orders of g j are all even, all the holonomies are translations and M is a translation surface. When the order of g j is 2, the corresponding vertex has cone angle 2π. When none of the orders of g j is 2, Σ consists of actural cone points of M , in which case Aff is the affine diffeomorphism group.
The decomposition of P into two squares in figure 1 induces a cell decomposition on M (G, g), which can be described as |G|-copies of pair of squares labeled by elements in G as B 1 g , B 2 g , that are glued together by identifying edges e j g and e j ′ g ′ when j = j ′ and g = g ′ , so that the directions indicated by the arrows match: For example, in our notation the Wollmilchsau [For06] [HSa] is M (Z/4, (1, 1, 1, 1)), can be presented as the union of the following squares with indicated glueings : As another example, let G = Z/3 and g = (0, 1, 1, 1). In this case M = M (G, g) is a half translation surface and the gluing is as follows: Now we describe the action of the deck group G on M (G, g). An element h ∈ G sends B k g to B k hg and e k g , to e k hg . The deck group action induces a right G-action on H 1 (M, Σ; C) that makes it a right G-module.

Affine diffeomorphisms
From now on we assume that G is abelian, though many of our arguments work for any finite group. At the end of this section we will point out the modification required in the non-Abelian case.
We calculate Aff = Aff (M (G, g)) in a way inspired by the coset graph description used in [Sch04]. One distinction is that we consider the whole affine diffeomorphism group while [Sch04] considers only the Veech group. Fixing G, let V be the set of all 4-tuples h = (h 1 , h 2 , h 3 , h 4 ) such that {h 1 , h 2 , h 3 , h 4 } generates G and h 1 h 2 h 3 h 4 = 1, each is associated with a square-tiled surface M (G, h) which is equipped with a cell decomposition labeled as in figure 2. By construction, an element F in Aff induces an automorphism of the deck group G by g → F gF −1 , i.e. there is a group homomorphism Aff → Aut(G). We denote the kernel of this homomorphism as Γ. Because Aut(G) is a finite group, Γ is a subgroup of Aff with finite index.
We will show that all orientation preserving affine diffeomorphisms between various M (G, h) that preserves Σ are compositions of a finite affine diffeomorphisms, which we call basic affine diffeomorphisms, which we will describe below. In our discussion we will be dealing with both translation surfaces and half translation surface surfaces. It will be convenient to view the derivative of an affine diffeomorphism as an element of P GL(2, R) = GL(2, R)/{±I}. We will call an affine translation diffeomorphism a half translation equivalence when its derivative is 1 in P GL(2, G). Now we define four of the five classes of the basic affine diffeomorphisms: Its derivative is 1.
. Its derivative is 1. We claim that any half translation equivalence from M (G, h) to M (G, h ′ ) can be written as composition of basic affine diffeomorphisms t 2 , r, f and m. Because by our assumption they preserve Σ, they can be seen as a permutation of unit squares that tiled M and M ′ . More precisely, any half translation equivalence is completely determined by the following data: i) the induced automorphism ψ of deck group, ii) a number j = 1 or 2, i.e. whether or not we interchange B 1 and B 2 , an element g ∈ G, such that F 0 (B 1 e ) = B j g , and whether Here e is the unit in G. We can now use m to deal with ψ, then use r and f to send B 1 e to B j g , and if needed precompose with t 2 .
For general orientation preserving affine diffeomorphism F , DF will be in P SL(2, Z). We add another class of basic affine diffeomorphisms: 1 to e 3 1 and has derivative 1 −1 0 1 .
Because the derivative of s and t generate P SL(2, Z), by successively composing with s and t we can reduce to the case when derivative is identity. Hence any affine diffeomorphisms between M (G, h), h ∈ V that sends Σ to Σ, or more specifically, any element in Aff , is a composition of the five classes of maps described above. Because m commutes with other 4 classes of diffeomorphisms, i.e.
any F ∈ Aff can be written as F = F 1 m ψ where F 1 is a composition of t, s, r and f , while ψ is the automorphism of deck group induced by F . Hence, elements in Γ can be written as successive compositions of t, r, f and s.
As in [Sch04], consider the directed graph D with vertex set V , each element h) ∈ V corresponding to a surface M (G, h), the edges in the graph corresponding to basic affine diffeomorphisms. Paths starting starting and ending at M (G, g) correspond to elements in Aff . Now the fact that any affine diffeomorphism is a successive composition of t, s, r, f , and m means that the map from the set of such paths to Aff is surjective. Similarly, let D 0 be graph D with those edges corresponding to m removed, then the set of paths startng and ending at g in D 0 maps surjectively to Γ.
Consider the example G = Z/6, g = (1, 1, 1, 3). This example is the Ornithorynque [FM08]. In the following figure we give the connected component of D that contains g, with loops corresponding to deck transformation (i.e. all the r arrows) omitted: (1, 1, 1, 3) (1, 1, 3, 1) (1, 3, 1, 1) (3, 1, 1, 1) This method of calculating Aff does not use the fact that G is abelian in any important way. When G is non-abelian, we can define Γ in a slightly different way. In general, we let Γ be the elements in Aff (M ) that induce an inner automorphism on G, then elements in Γ are compositions of t, s, r, f as well as m ψ where ψ is an element of an inner automorphism of G.

Invariant decomposition of relative cohomology
We start by giving a direct sum decomposition of H 1 (M (G, g), Σ), and calculate the dimension of the summands as well as the action of the affine diffeomorphisms on them. A compatible splitting for the absolute cohomology H 1 (M ) was described in [Wri12]. In the case of absolute cohomology the summands can have dimensions 0, 1 or 2. H 1 (ρ) can also be described as cohomology with twisted coefficients as in [DM86], [Thu98].
Proof. Consider the relative cellular cochain complex We can identify C 1 (M, Σ; C) with (C[G]) 4 as a right-G module by writing m ∈ C 1 (M, Σ; C) as ( identify C 2 (M, Σ; C) with (C[G]) 2 as a right-G module by writing n ∈ C 2 (M, Σ; C) as then the coboundary map from C 1 to C 2 is (1) d 1 (a, b, c, d) = (a + b + c + d, a + g 2 b + g 2 g 3 c + g 2 g 3 g 4 d) Hence: (2) H 1 (M, Σ; C) = {(a, b, c, d) ∈ (C[G]) 4 : a + b + c + d = a + g 2 b + g 2 g 3 c + g 2 g 3 g 4 d = 0} Because C[G] is semisimple [Ser77], it splits into simple algebras C[G] = ρ∈∆ D ρ , where D ρ is the simple subalgebras of C[G] corresponding to irreducible representation ρ. The splitting of the algebra gives a splitting of the claim complex 0 → C 1 → C 2 , hence a splitting of the cohomology: where M is the right ideal generated by {g 2 − 1, g 2 g 3 − 1, g 2 g 3 g 4 − 1}. This is because if g is a product of elements in {g 2 , g 2 g 3 , g 2 g 3 g 4 } then 1 − g ∈ M . Also, because M is connected, {g 2 , g 2 g 3 , g 2 g 3 g 4 } generates G, hence M is generated by all elements of the form 1 − g for any g ∈ G, therefore C[G] = C ⊕ M , where C is the trivial sub-algebra generated by g∈G g [Ser77]. Because C[G] is semisimple, splits, hence we have In the previous section we describe elements in Aff as compositions of elementary affine diffeomorphisms t h , s h , r g,h , f h and m ψ , and elements in Γ as compositions of elementary affine diffeomorphisms t h , s h , r g,h and f h . We will show the invariance of H 1 (ρ) under Γ by explicitly describing the action of elementary affine diffeomorphisms. The induced map of t h , s h , r g,h , f h from H 1 (M (G, h), Σ; C) to some H 1 (M (G, h), Σ; C) are as follows: which, according to equation (3), would send H 1 (ρ) to H 1 (ψ −1 ρ). In other words, elements in Aff permute H 1 (ρ).
Remark 1. In certain situations Γ = Aff . This happens when the g j are all of different order, or when G is Z/n, n ≥ 4 and g = (1, 1, 1, n − 3). In these cases H 1 (ρ) are all invariant under Aff . Our argument here is similar to, but not completely the same as those used in [MY09].

The signature of the Hodge form
Now we define and calculate the signature of an invariant Hermitian form on H 1 (ρ) as in [Thu98] and [DM86].
The Hodge form A G , or area form as in [Thu98], on H 1 (M, Σ; C) is defined as 1 2i of the cup product with coefficient pairing

In other words,
where (·, ·) G is the positive definite Hermitian norm on C[G] defined as Alternatively, if elements in H 1 (M, Σ; C) are represented by closed differential forms, A G can be written as By definition A G is invariant under the Γ-action. Furthermore, from (9) and the fact that different D ρ are orthogonal under (·, ·) G , we know that H 1 (ρ) for different representation ρ are orthogonal to each other under A G . When ρ is the trivial representation, A G = 0 on H 1 (ρ). Now we assume ρ to be a non-trivial representation. Because we will deal with Hermitian forms that may be degenerate, we denote the signature of a Hermitian form as (n 0 , n + , n − ), where n 0 , n + , n − are the number of 0, positive and negative eigenvalues respectively.
We will prove the following theorem: . The number n 0 is also the number of indices j such that ρ(g j ) = 1.

A subgroup of Γ
In this section we introduce a subgroup Γ 1 of Γ of finite index, which is easier to work with than Γ. In section 6, we will give a criteria for non-discreteness of the action of Γ by analyzing the action of this subgroup of finite index.

The spherical case and polyhedral groups
The Hodge norm A G on H 1 (ρ) induces a metric, hence a geometric structure on the projectivization P(H 1 (ρ)) = CP 1 invariant under the Γ-action. When A G is positive definite or negative definite, it induces a spherical structure on CP 1 . When A G has signature (1, 0, 1), it induces a Euclidean structure on CP 1 − [0 : 1]. When A G has signature (1, 1, 0), it induces a Euclidean structure on CP 1 − [1 : 0]. Finally, when the signature of A G is (0, 1, 1), it induces a hyperbolic structure on a disc D in PH 1 (ρ), which consists of the image {α ∈ H 1 (ρ) : A(α, α) > 0}. In this section we will describe the spherical case, and in the next section we will describe the remaining cases.
When A G is positive definite or negative definite, the generators of Γ 1 , γ 1 and γ 2 , act as finite order rotations with different fixed points, and their orders are the orders of ρ(g 1 g 2 ) and ρ(g 2 g 3 ) in C * respectively, hence by the ADE classification [Dic59] of finite subgroups of SO(3) we know that if both the orders of ρ(g 1 g 2 ) and ρ(g 2 g 3 ) are greater than 5 the action of Γ on H 1 (ρ) can not factor through a discrete group.
Examples of M and ρ that satisfies the conditions in Section 1 can be built from any 4-tuple of positive rational numbers not on the above list that sum up to 1 or 3. For example, (1/8, 1/8, 1/8, 5/8) is not on the list, so let G = Z/8, g 1 = g 2 = g 3 = 1, g 4 = 5, ρ(g 1 ) = ρ(g 2 ) = ρ(g 3 ) = e πi/4 , ρ(g 4 ) = e 5πi/4 satisfies the conditions in Section 1. This is an abelian cover of flat pillowcase that satisfy the conditions in section 1 with the smallest number of squares.
When two of the four ρ(g j ) are equal to 1, then the Γ 1 action on H 1 (ρ) factors through a finite abelian group.