# Schwarz triangle mappings and Teichmüller curves:

the Veech-Ward-Bouw-Möller curves

###### Abstract.

We study a family of Teichmüller curves constructed by Bouw and Möller, and previously by Veech and Ward in the cases . We simplify the proof that is a Teichmüller curve, avoiding the use Möller’s characterization of Teichmüller curves in terms of maximally Higgs bundles. Our key tool is a description of the period mapping of in terms of Schwarz triangle mappings.

We prove that is always generated by Hooper’s lattice surface with semiregular polygon decomposition. We compute Lyapunov exponents, and determine algebraic primitivity in all cases. We show that frequently, every point (Riemann surface) on covers some point on some distinct .

The arise as fiberwise quotients of families of abelian covers of branched over four points. These covers of can be considered as abelian parallelogram-tiled surfaces, and this viewpoint facilitates much of our study.

## 1. Introduction

A Teichmüller curve is an isometrically immersed curve in the moduli space of genus curves , with respect to the Teichmüller metric. Teichmüller curves give rise to billiards and translation surfaces with optimal dynamical properties [V], and have a rich and interesting algebro-geometric theory [M, M2]. In analogy with lattices in Lie groups they are either arithmetic (generated by square-tiled surfaces) or not, and come in groups analogous to commensurability classes. In the non-arithmetic case each such commensurability class of Teichmüller curves contains a unique extremal element which is called primitive [M2]. Also in analogy with lattices (say in ) non-arithmetic Teichmüller curves seem to be quite rare. The only currently known primitive examples in genus greater than four are the subject of this paper.

The Veech-Ward-Bouw-Möller curves. The flat pillowcase is obtained by gluing two isometric squares together, giving a flat metric on . Its symmetry group is the Klein four group .

More generally, given any four points , there is a flat metric on obtained by gluing two isometric parallelograms whose corners are the . The symmetry group is again the Klein four group, which acts on by Möbius transformations.

An abelian parallelogram-tiled surface is an abelian cover of branched over at most four points and equipped with a lift of the parallelogram-tiled metric [W1]. In this paper the parallelogram-tiled flat structure is not essential, but it is natural for the flat geometer to keep it in mind.

Given any cover of branched over we may vary to obtain a family of Riemann surfaces over the base .

For each pair with we will consider a very special family of this type,

The fibers are abelian parallelogram-tiled surfaces which are exceptionally symmetric in that they admit a very nice lift of the pillowcase symmetry group .

Informally the Veech-Ward-Bouw-Möller curve is the closure in moduli space of the image of the fiberwise quotient map

More formally, is a map from a curve to moduli space (and moreover a base change is required to define ). It is the same curve in moduli space that is considered in different language by Bouw-Möller, who show that it is a Teichmüller curve. Moreover, Bouw-Möller show that the and are the Teichmüller curves considered by Veech and Ward respectively.

We will give a simplified proof that is a Teichmüller curve. The key ideas are presented in Section 2, but we will hint briefly at the proof before proceeding to describe our new results.

Royden’s Theorem asserts that the Teichmüller metric is the same as the Kobayashi metric [Hub, IT]. The magic of the Kobayashi metric is that holomorphic maps are distance nonincreasing, so in particular if the composition of two maps is an isometry, then each of the two maps is also. So to show is an isometry, it will suffice to show that a single period coefficient on is an isometry. More precisely, will be lifted to Torelli space, and a specific entry in the period matrix will shown to be an isometry.

This program is feasible because the period mapping of may be completely described in terms of Schwarz triangle mappings, which are biholomorphisms from the upper half plane to (in our case hyperbolic) triangles [W1].

Covering relations. We have discovered that

###### Theorem 1.1.

Suppose divides , and divides . If and are even suppose also that is even. Then every point (Riemann surface) on covers a point (Riemann surface) on .

For example, every point on the Veech Teichmüller curve generated by the regular –gon covers some point on the Veech Teichmüller curve generated by the regular –gon. In other words, given any translation surface in the –orbit of the regular –gon, there is some translation surface in the –orbit of the regular –gon, so that there is a covering of Riemann surfaces .

This is surprising because there is no hint that such a result should be true from the flat geometry.

Arithmetic origins. It is also surprising that arithmetic Teichmüller curves (generated by the abelian square-tiled surfaces ) can be used to construct the non-arithmetic . A direct consequence of the construction is

###### Theorem 1.2.

A quadratic differential with simple poles may be assigned to all but finitely many points (Riemann surfaces) on , giving each of these Riemann surfaces the structure of a parallelogram-tiled surface.

is the closure a Hurwitz curve of covers of branched over four points.

Here a Hurwitz curve is the closure of a space of covers of branched over four points, where all the covers are topologically the same. That is, a Hurwitz curve results from taking a cover of branched over four points and varying the location of the branch points.

The first part of Theorem 1.2 is rather surprising: it is unexpected that so many Riemann surfaces (the points of ) could support the structure of a lattice surface in two different ways (one square-tiled and one not).

###### Proof..

The image of in moduli space is equal to closure of the image of the fiberwise quotient map , by construction. We will show in Proposition 2.8 and Theorem LABEL:T:BM that all but finitely many points on are in the image of the fiberwise quotient map and hence are of the form .

The pillowcase symmetry group preserves the parallelogram-tiling, so the quotient of the exceptionally symmetric square-tiled surface by the pillowcase group is again parallelogram-tiled. The parallelogram-tiled metric of the quotient has cone angles of .

Each point in the image of admits a cover

This map is branched over four points, and it follows easily that is (up to closure) the space of such covers. Hence is the closure of a Hurwitz curve. ∎

Real multiplication of Hecke type. Möller has shown that Techmüller curves parameterize Riemann surfaces whose Jacobians have a factor with real multiplication (an inclusion of a totally real number field into the endomorphism algebra) [M].

If is a Riemann surface, endomorphisms of can be considered as “hidden symmetries” of . Sometimes, they arise from honest symmetries, that is, they are induced by automorphisms of . Ellenberg has studied situations in which has no automorphisms, but admits real multiplication which arises from automorphisms of a finite cover of [E]. In such cases he defines the real multiplication on to be of Hecke type. See Section LABEL:S:Hecke for definitions.

###### Theorem 1.3.

The real multiplication on a factor of the Jacobians of all but finitely many points (Riemann surfaces) on guaranteed by [M] is of Hecke type. That is, the endomorphisms of the Jacobian, which together form real multiplication, come from deck transformations of the exceptionally symmetric square-tiled surfaces which cover points (Riemann surfaces) on .

Generators. In Section LABEL:S:eqns, extending work of Bouw-Möller in the case when and are relatively prime, we compute holomorphic one forms which generate each ; the formulas and corollaries are given in Section LABEL:S:grn.

Hooper has given an elementary construction of Teichmüller curves which are generated by translation surfaces with a particularly beautiful flat structure having a semiregular polygon decomposition [H]. These flat surfaces were discovered independently by Ronen Mukamel. Comparing our generators to Hooper’s, we obtain the next theorem, which again was previously known in the case where and are relatively prime.

###### Theorem 1.4.

The Teichmüller curves constructed by Hooper are the same as the Veech-Ward-Bouw-Möller curves.

Theorem 1.4 answers a question of Hooper [H, Question 19]. As a corollary of Theorem 1.4, we list some of Hooper’s results, which were obtained by Hooper using the semiregular polygon decomposition. Let be the orientation-preserving part of the group generated by reflections in the sides of a hyperbolic triangle with angles .

The uniformizing group of a Teichmüller curve is frequently called its Veech group. Definitions of primitivity appear below.

###### Corollary 1.5.

Let with , and set . . is arithmetic if and only if or is in the set

is geometrically primitive when it is not arithmetic.

The uniformizing group of is:

if and or is odd; | |
---|---|

an index two subgroup | if are both even; |

if is odd; | |

if is even. |

The curve , where the genus is

In all but the last case, is generated by an abelian differential with zeros of equal order; in the last case, there are .

Some of these results can also be obtained by other means and were obtained by [BM], but at least when and are not relatively prime the only known proof of primitivity and exact calculation of the uniformizing group are due to Hooper.

Lyapunov exponents. In flat geometry the significance of Lyapunov exponents is twofold: they describe both the dynamics of the Teichmüller geodesic flow on moduli space, and the deviation of ergodic averages for straight line flow on the translation surface [F]. For background and motivation on Lyapunov exponents in flat geometry, see for example [Fex1, EKZbig]. In parallel to the case of abelian square-tiled surfaces handled in [W1], at the end of Section 3 we determine all the Lyapunov exponents of , and clarify the relationship to the Schwarz triangles used to describe the period mappings. Previously the Lyapunov exponents were given by Bouw-Möller when and are odd and relatively prime. A corollary of our computation is

###### Corollary 1.6.

The Lyapunov spectrum of consists of nonzero multiples of . In particular, there are never any zero Lyapunov.

The Lyapunov exponents however often are not all distinct. See for example the tables in figure LABEL:F:tables.

Primitivity. A Teichmüller curve is called (geometrically) primitive if it does not arise from a translation covering construction. A Teichmüller curve in is called algebraically primitive when the trace field of the uniformizing group has degree over . This is exactly the case when there is real multiplication on the Jacobian, instead of only a factor of the Jacobian. The are always geometrically primitive but are usually not algebraically primitive.

###### Theorem 1.7.

Assuming the Teichmüller curve is not arithmetic, it is algebraically primitive if and only if one of is and the other is a prime, twice a prime, or a power of two.

See [E] and [CM] for a summary of the very few known families of curves with real multiplication, and known curves with complex multiplication. Any cone point of an algebraically primitive Teichmüller curve has complex multiplication.

Notes and references. There are only very few examples of primitive Teichmüller curves known. They are the Prym curves in genus 2, 3 and 4 [Ca, Mc, Mc4]; the Veech-Ward-Bouw-Möller curves [BM]; and two sporadic examples, one due to Vorobets in , and another due to Kenyon-Smillie in [HS, KS]. These sporadic examples correspond to billiards in the and triangles respectively, and both Teichmüller curves are algebraically primitive.

Teichmüller curves generated by abelian differentials are classified in , and there are some finiteness results in higher genus [Bam, M3], but classification even in appears difficult.

The Bouw-Möller construction is quite novel, and originally used Möller’s characterization of Teichmüller curves involving maximally Higgs bundles. Our proof that is isometrically immersed is a simplification of theirs; our contributions are to avoid Möller’s characterization, and to use Schwarz triangle mappings, which are not used in [BM] but allow for a geometric understanding. We also ground the arguments in more elementary language, and point out the connection to the geometry and combinatorics of square-tiled surfaces.

For those results that were previously only known in some cases (for example, and relatively prime), no new ideas are required to extend the results to all cases. Our contribution here is to use notation which avoids the case distinctions which pervade [BM]. This being said, often we do not use the methods of [BM] to obtain these results, preferring new approaches of a more geometric flavor.

Veech and Ward gave flat geometry proofs that is a Teichmüller curve in the case . In light of Theorem 1.4, Hooper has done the same for all and . These flat geometry proofs are more elementary than the proof we present, but the proof we present also has a number of advantages. First and foremost, our proof is closer to how Bouw and Möller discovered in the first place, and it is gratifying to understand their leap of intuition that cyclic or abelian covers of might be the building blocks for some Teichmüller curves uniformized by triangle groups. Second, it allows for the computation of Lyapunov exponents, and an understanding of the period map and monodromy. Third, it allowed us to discover many of the results in this paper.

We use a number of results from [W1], where we have developed the theory of abelian square-tiled surfaces. Most readers wishing to understand all the details of this paper will wish to consult this source.

Table of Schwarz triangles. The reader is encouraged to study figure LABEL:F:tables, where a description of the period map of many is presented. Many results in this paper are reflected in these tables.

Acknowledgements. This research was supported in part by the National Science and Engineering Research Council of Canada, and was partially conducted during the Hausdorff Institute’s trimester program “Geometry and dynamics of Teichmüller space.” The author thanks Alex Eskin, Howard Masur, Martin Möller, and Anton Zorich for their instruction and encouragement, and Matt Bainbridge, Irene Bouw, Jordan Ellenberg and Madhav Nori for helpful and interesting discussions. The author is grateful to Anton Zorich and Pat Hooper for allowing us to reproduce figures, and Jennifer Wilson for producing figures.

## 2. Key ideas for the study of

Here we give the main ingredients in the proof that is a Teichmüller curve, and hint at the proof. This section is intended as an extension of the introduction.

### 2.1. Exceptionally symmetry.

In the notation of [W1], is defined as

That is, given in the algebraic closure of such that

is defined to be the cover of with function field . The base has function field . The dependance on is suppressed in this notation. The natural flat structure on with singularities at may be lifted to , giving it the structure of a parallelogram-tiled surface.

We begin with the exceptional symmetry of , which is visible in the flat geometry. Denote by the pillowcase symmetry that sends to . Covering space theory guarantees that each involution can be lifted to an involution of . However, we will require commuting lifts (a lift of pillowcase symmetry group ), which moreover have special properties.

Let be the oriented loop about . In the standard square-tiled metric, is the core curve of the horizontal cylinder, and is the core curve of the vertical cylinder. So, “lifts” of to the square-tiled are the core curves of the horizontal cylinders on . (We will also use parallelogram-tiled , where this language does not apply. By a “lift” of we mean any unoriented simple closed curve that projects to a multiple of .)

###### Proposition 2.2.

Up to the action of the deck group, has a unique lift of the pillowcase symmetry group so that both and each have at least one fixed point, unless and are both even, in which case there are two such lifts.

In any such lift, the involution maps each lift of the unoriented curve to itself, and the involution maps each lift of to itself.

It is not at all obvious that Proposition 2.2 should be true. The proof is straightforward and unenlightening, and is deferred to Section LABEL:S:Snm.

### 2.3. Schwarz triangle mappings.

Recall that is branched over . By varying , we obtain a family over the base . The families are chosen so that the following result holds. By the row span of , we mean the abelian subgroup of generated by the rows of the matrix

whose entries are modulo . (In [W1], we saw that this row span is in bijection to a basis of the function field of over , which is why its use is pervasive). For

in the row span of , we defined

where denotes fractional part.

###### Proposition 2.4.

Consider the bundle over whose fiber over is . There exists a direct sum decomposition of this bundle

with the following properties. The summation is over in the row span of . The subbundle is nonzero if and only if .

Each nonzero is a flat rank two subbundle whose and parts each have dimension . Set and

The part of each admits a global section , and homology classes may be chosen so that the period mapping

is a Schwarz triangle mapping which maps to a hyperbolic triangle with angles at respectively.

This follows directly from [W1, Propositions LABEL:P:dim and LABEL:P:pmap]. We always use (co)homology with coefficients in (not ).

The relationship between Propositions 2.2 and 2.4 is given by the following lemma, whose proof is also deferred to Section LABEL:S:Snm following the proof of Proposition 2.2.

###### Lemma 2.5.

The Klein group action on the is given by

When for some involution , then some non-trivial involution in acts by negation on .

### 2.6. The fiberwise quotient map.

Each fiber is an and admits a lift of by Proposition 2.2. After a base change (see Section 3.1 for details), a continuous choice of is possible, and we achieve an action of on the entire family of covers of .

By Lemma 2.5, after taking fiberwise quotients, the bundle is still the sum of rank two bundles , each of whose period map is a still a Schwarz triangle mapping with the same angles, one each for each group of four distinct over which are permuted by the Klein four group. (See Lemma LABEL:L:decomp for details.)

Note that we use the notation to denote both a subbundle of the cohomology of fibers of and also . We hope this will not be too misleading, despite the fact that each for corresponds to a group of four isomorphic for which are permuted by the action. The notation is justified because as a bundle (rank two complex VHS) over , always denotes the same object.

If is the first row of ,

in Proposition 2.4 we can calculate . Lemma 2.5 gives that the four bundles are distinct. They yield a single rank 2 bundle after fiberwise quotient, whose period mapping is again described via Schwarz mapping onto a triangle with angles . The period map of will be shown to be an isometry.

### 2.7. Removable singularities.

We must of course use the orbifold structure on induced from the orbifold structure of , and assign to the unique hyperbolic metric guaranteed by uniformization. The main subtlety is that this does not correspond to the hyperbolic metric on . Indeed, the fiberwise quotient map has removable singularities; it sends some points at infinity to interior points, which turn out to be orbifold points. We remind the reader that if some of the punctures on are filled in, many possible orbifold hyperbolic metrics might result, depending on the cone angle assigned to the points which have been filled in. In particular, the hyperbolic metrics appear.

To see why the fiberwise quotient map has removable singularities, consider for a moment the flat pillowcase. The horizontal core curve may be pinched, giving a noded Riemann surface with two genus zero components. This is a degeneration (point at infinity) of . One of the involutions in the pillowcase symmetric group preserves the pinched curve and interchanges the two components, so the quotient is a smooth genus zero surface.

Proposition 2.2 allows a similar discussion for .

###### Proposition 2.8.

As , in the Deligne-Mumford compactification converges to the noded Riemann surface resulting from pinching the core curves of all horizontal cylinders on the square-tiled surface . The quotient of this noded Riemann surface by is smooth.

Similarly as , converges to the noded Riemann surface resulting from pinching the core curves of all vertical cylinders on the square-tiled surface . The quotient of this noded Riemann surface by is smooth.

To be more precise, we should say as goes to any lift in of the puncture at in , instead of saying .

###### Proof..

As , in the base converges to two ’s glued together at a node. One contains the first and fourth marked points ( and ) and the other contains the second and third. This noded Riemann surface is the result of pinching the core curve of the horizontal cylinder on the flat pillowcase.

As , the cover converges to a cover of the this noded Riemann surface. This cover is with all lifts of core curves of horizontal cylinders pinched. By Proposition 2.2, preserves each of these curves while exchanging the two sides of the curve, so in the limit fixes each node and exchanges the two sides of each node. Hence the quotient by and also all of is smooth.

The situation is similar as . ∎

The upshot is that as, for example, (or more accurately a lift of the puncture at in to ), the fiberwise quotients converge to a (smooth) Riemann surface, a point on the interior of moduli space. Hence, at lifts of to , the fiberwise quotient map has removable singularities.

## 3. is isometrically immersed

In this section, we construct the Veech-Ward-Bouw-Möller Teichmüller curves , and prove that they are isometrically immersed.

### 3.1. Base change.

Given a family over a base , a base change is the result of taking a finite cover , and pulling back to obtain a family over .

Proposition 2.2 guarantees that to each fiber of there is at least one and at most finitely many lifts of the pillowcase symmetry group so that both have fixed points.

Hence after applying a base change to the family over the base , we may obtain a family over some larger base to which a continuous assignment of such lifts of the pillowcase symmetries is possible.

As we will see, exactly what base change is required is not relevant to our arguments.

### 3.2. Construction.

We define the fiberwise quotient map by . The new base covers the old base ; both are punctured Riemann surfaces. Proposition 2.8 gives that the map has some removable singularities. We wish to “fill in” these removable singularities, but there are some technical issues because is an orbifold, and the image of the removable singularities might be orbifold points.

For this reason we pass at once to a finite cover which is a manifold. We consider the minimal cover to which there is a map which covers the map . We now consider the unique Riemann surface through which the map factors as

with generically one-to-one and without removable singularities. The procedure for producing from is quite explicit: First, pass to the space covered by , from which the induced map to is generically one-to-one. Then, fill in the removable singularities to obtain .

The Riemann surface is equipped with the hyperbolic metric given by uniformization, and it is our goal to show that is an isometry, showing that the induced generically one-to-one map is a Teichmüller curve. We refer to as the Veech-Ward-Bouw-Möller curve.

The orbifold structure on is determined by the requirement that the natural branched cover is an isometry. The situation thus far is summarized in Figure LABEL:F:smalldiagram.