Billiards and Teichmüller curves

Abstract

A Teichmüller curve $V \subset \mathcal{M}_g$ is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems.

1. Introduction

The moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g$ is both a metric space and an algebraic variety.

The metric comes from the Teichmüller distance between $X,Y \in \mathcal{M}_g$, which measures the minimal conformal distortion of a map $f : X \rightarrow Y$. This metric is given by a norm on each tangent space to $\mathcal{M}_g$, but for $g>1$ it is not Riemannian; in fact the norm balls are complicated convex sets, varying so much from point to point that $\mathcal{M}_g$ is completely inhomogeneous.

The algebraic structure on $\mathcal{M}_g$ comes from a projective embedding, which provides a multitude of algebraic curves $V \subset \mathcal{M}_g$. Each curve carries a natural hyperbolic metric, coming from its uniformization $V = \mathbb{H}/\Gamma$.

We say $V \subset \mathcal{M}_g$ is a Teichmüller curve if this inclusion is an isometry. These rare and remarkable objects lie at the nexus of algebraic geometry, number theory, complex analysis, topology and automorphic forms. We focus on primitive examples, since all others are related to these by covering constructions (§2).

Teichmüller curves are elusive, but once found, they can often be viewed explicitly from many perspectives at once. For example, any primitive Teichmüller curve $V$ determines a totally real number field $K$, with $\deg (K/\mathbb{Q}) \le g$, such that:

•

$V \cong \mathbb{H}/\Gamma$, with $\Gamma \subset \operatorname {SL}_2(K)$;

•

every Riemann surface $X \in V \subset \mathcal{M}_g$ can be assembled from triangles with vertices in $K \oplus K\tau \subset \mathbb{C}$, for some $\tau \in \mathbb{H}$; and

•

a factor of the Jacobian of $X$ admits real multiplication by $K$.

The curve $V$ is rigid, so both $V$ and its map to $\mathcal{M}_g$ are also defined over a number field. In particular cases one can obtain algebraic equations for $V$, generators for $\Gamma$, and geometric models for $X$ and for endomorphisms of its Jacobian.

Billiards.

Frequently $X \in V$ can be chosen so there is a polygon $P$ and a finite reflection group $G$, adapted to $V$, such that

$$\begin{equation*} X/G \cong P \subset \mathbb{C} . \end{equation*}$$

In this case, billiards in the polygon $P$ has optimal dynamics: every trajectory is either periodic, or uniformly distributed. Moreover, $X$ and $V$ can be reconstructed from $P$.

The regular polygons provide the first examples of both optimal billiards and Teichmüller curves (see Figure 1.1 and §3). It is also possible that $P$ is immersed, rather than embedded in $\mathbb{C}$; in this case we obtain a generalized polygon with optimal dynamics (see Figure 1.2 and §7).

The current catalog.

This paper provides a survey of the known examples of Teichmüller curves, a glimpse of their multifaceted constructions, a hint of how they were discovered, and a selection of the many open questions that remain.

The known primitive Teichmüller curves are given by 6 infinite series, and 3 sporadic examples. These are:

1.

the three Weierstrass series $W_D$, in genus $2$, $3$ and $4$ (§4, §5);

2.

the sporadic examples of type $E_6$, $E_7$ and $E_8$, in genus 3 and 4 (§6);

3.

the Bouw–Möller series $V_{p,q}$, providing finitely many examples in every $\mathcal{M}_g$ (§7); and

4.

the gothic and arabesque series, $G_D$ and $A_D$, both in genus 4 (§8, §9).

The five horizontal series $W_D$, $G_D$ and $A_D$ above each lie in a single moduli space. The index $D$ is a real quadratic discriminant, i.e. an integer $D>0$, $D=0$ or $1 \operatorname {mod} 4$, with $\sqrt {D} \not \in \mathbb{Z}$.

Completeness?

In low genus, the list of primitive Teichmüller curves is almost complete.

•

In genus $g=2$, all primitive Teichmüller curves are known: they are accounted for by the series $W_D \subset \mathcal{M}_2$ and one other curve, associated to billiards in the regular decagon (Theorem 4.5).

•

In genus $g=3$, there are only finitely many primitive Teichmüller curves not accounted for by the series $W_D \subset \mathcal{M}_3$ (Theorem 5.5).

On the other hand, the vertical Bouw–Möller series $V_{pq}$ gives the only known construction of primitive Teichmüller curves in genus $g \ge 5$. Thus a central open problem is to settle:

Question 1.1.

Are there infinitely many primitive Teichmüller curves in $\mathcal{M}_5$?

The unexpected families $G_D$ and $A_D$ in genus 4 hint that similar constructions may be hidden in higher genus.

Teichmüller surfaces.

The discovery of the gothic curves $G_D$ also revealed an almost miraculous new phenomenon: there are primitive, totally geodesic Teichmüller surfaces in $\mathcal{M}_{1,3}$, $\mathcal{M}_{1,4}$ and $\mathcal{M}_{2,1}$. This survey concludes with a description of these new surfaces from the perspective of algebraic geometry in §8, and from the perspective of quadrilaterals in §9.

Notes and references.

References and commentary are collected, section by section, in §10.

Via the action of $\operatorname {SL}_2(\mathbb{R})$ on $\Omega \mathcal{M}_g$, Teichmüller curves are connected to the larger topic of dynamics on moduli spaces, which is itself patterned on the theory of homogeneous dynamics, Lie groups, lattices and ergodic theory. For a view of the broader setting, we recommend the many excellent surveys such as D, Go, HS2, Mas3, MT, Mo3, Sch2, Vo1, Wr2, Wr3, Y, and Z.

Outline.

In §2 and §3 we set the stage with definitions and basic examples regarding moduli spaces, polygons and billiards. The known families of primitive Teichmüller curves are described in §4 through §9.

Four appendices follow. The triangle groups $\Delta (p,q,\infty ) \subset \operatorname {SL}_2(\mathbb{R})$ are reviewed in Appendix A. There are six accidental isomorphisms between members of series of Teichmüller curves listed above; these are recorded in Appendix B. Tables of invariants of Teichmüller curves appear in Appendices C and D.

Notation.

The $n$th Chebyshev polynomial will be denoted by $T_n(x) \in \mathbb{Z}[x]$; it is characterized by

$$\begin{equation} T_n(\cos \theta ) = \cos (n \theta ) . {\tag{1.1}} \end{equation}$$

We let $H^1(X)$ denote cohomology with complex coefficients. The upper half-plane in $\mathbb{H} = \{z \; : \; \operatorname {Im}(z)>0\} \subset \mathbb{C}$ is endowed with the complete hyperbolic metric

$$\begin{equation*} \rho _\mathbb{H} = \frac{|dz|}{2 \operatorname {Im} z} \end{equation*}$$

of constant curvature $-4$. The group $\operatorname {SL}_2(\mathbb{R})$ acts linearly on $\mathbb{R}^2 \cong \mathbb{C}$ and by Möbius transformations on $\mathbb{H}$.

2. Moduli spaces and Teichmüller curves

This section develops background material on Riemann surfaces, polygons, and the action of $\operatorname {SL}_2(\mathbb{R})$ on the moduli space of holomorphic 1-forms $\Omega \mathcal{M}_g$. This material will allow us to formulate the main topic we aim to address:

Problem 2.1.

Construct and classify all primitive Teichmüller curves $V \rightarrow \mathcal{M}_g$.

Moduli space.

The moduli space $\mathcal{M}_g$ parameterizes the isomorphism classes of compact Riemann surfaces $X$ of genus $g$. It is naturally a complex orbifold, and an algebraic variety, of complex dimension $3g-3$ when $g \ge 2$.

The Teichmüller metric on $\mathcal{M}_g$ is defined by a norm on each tangent space; it can be characterized as the largest metric such that every holomorphic map

$$\begin{equation} f : \mathbb{H} \rightarrow \mathcal{M}_g {\tag{2.1}} \end{equation}$$

is either a contraction or an isometry. In the isometric case, we say $f$ is a complex geodesic.

Metrically, moduli space is completely inhomogeneous: the tangent spaces at $X,Y \in \mathcal{M}_g$ are isomorphic as normed vector spaces if and only if $X=Y$. Nevertheless, there exists a unique complex geodesic through every point in every possible direction.

Polygons and Riemann surfaces.

How can one specify a Riemann surface $X \in \mathcal{M}_g$?

In the case $g=1$, $X$ is a torus, thus one can write $X = \mathbb{C}/\Lambda$ for some lattice $\Lambda \subset \mathbb{C}$. Alternatively, if we choose a parallelogram $P \subset \mathbb{C}$ that is a fundamental domain for the action of $\Lambda$, we can construct $X$ by gluing together opposite sides of $P$.

More generally, if $P \subset \mathbb{C}$ is any polygon, and the edges of $P$ are identified in pairs by translations, then the result is a compact Riemann surface $X = P/{\sim }$. And in fact:

Every compact Riemann surface of genus $g \ge 1$ can be presented as a polygon $P \subset \mathbb{C}$ with its edges glued together by translations.

Note that $X$ inherits a flat metric from $P$. At first sight this may seem paradoxical: for $g \ge 2$, $X$ admits no smooth flat metric. However the metric on $X$ has, in general, isolated singularities of negative curvature arising from the vertices of $P$.

Example in genus 2.

Consider the polygon $L(a,b)$ shown in Figure 2.1. Note that we have introduced two extra vertices, so $L(a,b)$ is combinatorially an octagon. Gluing edges by vertical and horizontal translations, we obtain a Riemann surface $X$ of genus 2. The eight vertices of $L(a,b)$ descend to a single point $p \in X$; there, the induced flat metric has a cone angle of $6\pi$.

Holomorphic 1-forms.

This flat uniformization of $X$ by a polygon $P$ can be contrasted with more traditional ways of presenting a compact Riemann surface, e.g. as an algebraic curve or as a quotient of $\mathbb{H}$ by a Fuchsian group. While a polygonal presentation of $X$ is elementary, it is not canonical; moreover, it provides $X$ with additional structure.

To explain this, recall that the space $\Omega (X)$ of holomorphic $1$-forms $\omega$ on $X$ has dimension $g$; indeed, $\dim \Omega (X)$ can be taken as the definition of the genus $g$ of $X$. In local coordinates, $\omega = \omega (z) \,dz$, where $\omega (z)$ is a holomorphic function. Provided $\omega \neq 0$, its zero set $Z(\omega ) \subset X$ consists of $2g-2$ points, counted with multiplicity.

The moduli space of all nonzero 1-forms $(X,\omega )$ of genus $g$ forms a bundle

$$\begin{equation*} \Omega \mathcal{M}_g \rightarrow \mathcal{M}_g; \end{equation*}$$

in fact, it is a holomorphic vector bundle of rank $g$ with its zero section removed.

Strata.

The locus where the zeros of $\omega$ have multiplicities $p = (p_1,\ldots ,p_n)$ forms a stratum

$$\begin{equation*} \Omega \mathcal{M}_g(p_1,\ldots ,p_n) \subset \Omega \mathcal{M}_g \end{equation*}$$

of dimension $2g+n-1$. These strata decompose $\Omega \mathcal{M}_g$ into disjoint algebraic sets, indexed by the partitions $p$ of $2g-2$. We sometimes use exponential notation for repeated blocks of a partition; e.g. the unique open stratum is denoted by $\Omega \mathcal{M}_g(1^{2g-2}) = \Omega \mathcal{M}_g(1,1,\ldots ,1)$.

From polygons to 1-forms.

Let $X = P/{\sim }$. Since the 1-form $dz$ on $\mathbb{C}$ is invariant under translation, it descends to give a 1-form $\omega \in \Omega (X)$. Here is a more precise description of the relationship between Riemann surfaces and polygons.

Theorem 2.2.

Every element of $\Omega \mathcal{M}_g$ can be presented in the form

$$\begin{equation} (X,\omega ) = (P,dz) / {\sim }{\tag{2.2}} \end{equation}$$

for a suitable polygon $P \subset \mathbb{C}$.

It is often useful, as we will see below, to allow the ‘polygon’ $P$ to be disconnected. With this proviso, the proof of the result above is fairly elementary: one can construct a geodesic triangulation of the flat surface $(X,|\omega |)$, with $Z(\omega )$ among its vertices, and then present $X$ as the quotient of a collection of Euclidean triangles.

Geometry of a 1-form.

A holomorphic 1-form provides $X$ with a singular flat metric $|\omega |$. This metric has a cone angle of $2\pi (p+1)$ at each zero of order $p$. The form $|\omega |^2$ determines a smooth measure on $X$, with total mass given by

$$\begin{equation*} \operatorname {area}(X,\omega ) = \int _X |\omega |^2 = \operatorname {area}(P) . \end{equation*}$$

Near any point $p \not \in Z(\omega )$, we can choose a local flat coordinate $z(q) = \int _p^q \omega$ on $X$ such that $\omega = dz$. The geodesics on $(X,|\omega |)$ are simply straight lines in these charts. Since these flat coordinates are well–defined up to translation, each geodesic $\gamma$ has a well–defined slope $s$. In particular, $\gamma$ cannot cross itself; all geodesics are simple. We allow a geodesic to begin or end at a point of $Z(\omega )$, but never to pass through a zero. In particular, a closed geodesic is always disjoint from $Z(\omega )$.

These features are elementary to see in a polygonal model $(X,\omega ) = (P,dz)/{\sim }$; for example, the horizontal lines in $P$ descend to a foliation $\mathcal{F}(\omega )$ of $X$ by geodesics with slope zero. Intrinsically, this foliation is defined by the closed 1-form $\beta = \operatorname {Im} \omega$.

A cylinder $C \subset X$ is the closure of a maximal open set foliated by parallel closed geodesics. Every closed geodesic $\gamma$ lies in a cylinder; in particular, $\gamma$ is never unique its homotopy class. Most elements of $\pi _1(X)$ are not represented by closed geodesics; rather, the loop of minimal length in a given homotopy class is a chain of geodesic segments of varying slopes, with endpoints in $Z(\omega )$.

Action of $\operatorname {SL}_2(\mathbb{R})$.

Remarkably, upon passage to the bundle $\Omega \mathcal{M}_g$, the highly inhomogeneous space $\mathcal{M}_g$ acquires a dynamical character: namely, it admits a natural action of $\operatorname {SL}_2(\mathbb{R})$. This action is easily described in terms of a polygonal presentation 2.2: for $A \in \operatorname {SL}_2(\mathbb{R})$, we have:

$$\begin{equation*} A \cdot (X,\omega ) = (X_A, \omega _A) = (A(P),dz)/{\sim }. \end{equation*}$$

Here $A$ acts linearly on $P \subset \mathbb{C} \cong \mathbb{R}^2$, and the (combinatorial) gluing instructions remains the same.

Alternatively, given $A = \left( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix}\right)$ one can define a harmonic form on $X$ by

$$\begin{equation*} \omega _A = \left(\begin{matrix} 1 & i \end{matrix}\right) \left(\begin{matrix} a&b\\ c&d \end{matrix}\right) \left(\begin{matrix} \operatorname {Re} \omega \\ \operatorname {Im} \omega \end{matrix}\right) , \end{equation*}$$

and then change the complex structure on $X$ so $\omega _A$ is holomorphic on $X_A$. The zeros of $\omega$ and $\omega _A$ have the same order, so:

$\operatorname {SL}_2(\mathbb{R})$ leaves each stratum $\Omega \mathcal{M}_g(p)$ invariant.

Complex geodesics.

Note that if $A \in \operatorname {SO}_2(\mathbb{R})$ is simply a rotation, then $X_A = X$ and $\omega _A = \exp (i \theta ) \omega$ for some $\theta$. Thus the projection of $A \cdot (X,\omega )$ to $\mathcal{M}_g$ depends only on the coset

$$\begin{equation} [A] \in \operatorname {SO}_2(\mathbb{R}) \backslash \operatorname {SL}_2(\mathbb{R}) \cong \mathbb{H}, {\tag{2.3}} \end{equation}$$

and the map $F(A) = (X_A,\omega _A)$ covers a unique map $f : \mathbb{H} \rightarrow \mathcal{M}_g$, making the diagram

$$\begin{equation} \vcenter{\img[][81pt][55pt][{$\xymatrix{ \SL_2(\mathbb{R}) \ar[d] \ar[r]^F & \Omega\mathcal{M}_g \ar[d]^\pi\\ \mathbb{H} \ar[r]^f & \mathcal{M}_g \\ }$}]{Images/imgdf642c7ec5ae8bbbeb4e484d23efb4dd.svg}} {\tag{2.4}} \end{equation}$$

commute.

The map $f$ is a holomorphic, isometric immersion of $\mathbb{H}$ into moduli space, which we refer to as the complex geodesic generated by $(X,\omega )$. If $X = P/{\sim }$, then the image of $f$ simply consists of all Riemann surfaces of the form $X_A = A(P)/{\sim }$, $A \in \operatorname {SL}_2(\mathbb{R})$; see Figure 2.2.

Real geodesics.

Every 1-form also generates a distinguished real Teichmüller geodesic ray $\gamma : [0,\infty ) \rightarrow \mathcal{M}_g$; parameterized by arclength, it is given by

$$\begin{equation} \gamma (t) = f(i e^{2t}) = X_t, {\tag{2.5}} \end{equation}$$

where

$$\begin{equation*} (X_t,\omega _t) = \left(\begin{matrix} e^{-t} & 0 \\ 0 & e^t \end{matrix}\right) \cdot (X,\omega ) . \end{equation*}$$

The Riemann surface $X_t$ is obtained from $(X,|\omega |)$ by shrinking its horizontal geodesics and expanding its vertical ones.

Teichmüller curves.

The stabilizer of $(X,\omega )$ is a discrete subgroup

$$\begin{equation*} \operatorname {SL}(X,\omega ) \subset \operatorname {SL}_2(\mathbb{R}) . \end{equation*}$$

It is easy to see that the complex geodesic generated by $(X,\omega )$ descends to give a map $f : V \rightarrow \mathcal{M}_g$, where

$$\begin{equation} V = \mathbb{H} / \operatorname {SL}(X,\omega ) . {\tag{2.6}} \end{equation}$$

Here the action of $\operatorname {SL}(X,\omega )$ on $\mathbb{H}$ is slightly twisted, since $\operatorname {SO}_2(\mathbb{R})$ acts on the left in equation 2.3; it is given by $A(z) = (az - b)/(-cz + d)$.

Now suppose $V$ has finite hyperbolic area; equivalently, suppose $\operatorname {SL}(X,\omega )$ is a lattice in $\operatorname {SL}_2(\mathbb{R})$. Then the image of the map

$$\begin{equation*} f : V \rightarrow \mathcal{M}_g \end{equation*}$$

is a Teichmüller curve in $\mathcal{M}_g$. That is, $V$ is the normalization of a totally geodesic algebraic curve.

We refer to $(X,\omega )$ as a generator of the Teichmüller curve $V$. The generator of $V$ is not unique—for any $\lambda \in \mathbb{C}^*$ and $A \in \operatorname {SL}_2(\mathbb{R})$, $V$ is also generated by $A \cdot (X,\lambda \omega )$.

Hidden symmetries.

The pivotal group $\operatorname {SL}(X,\omega )$—which is large in the case of a Teichmüller curve—reflects hidden symmetries of the form $(X,\omega )$ itself.

More precisely, $\operatorname {SL}(X,\omega )$ can be described as follows. Let $\operatorname {Aff}^+(X,\omega )$ denote the group of orientation–preserving homeomorphisms of $X$ that stabilize $Z(\omega )$, and have the form

$$\begin{equation*} \phi (x,y) = A \left(\begin{matrix} x\\ y \end{matrix}\right) + b \end{equation*}$$

in local flat coordinates $z = x + iy$ on the domain and range satisfying $\omega = dz$. Here $A \in \operatorname {SL}_2(\mathbb{R})$ and $b \in \mathbb{R}^2$.

We refer to $\phi$ as an affine automorphism of $(X,\omega )$, since it preserves the real–affine structure on $X$ determined by $\omega$; in particular, $\phi$ sends geodesics to geodesics. The matrix $A = D\phi$ is independent of the choice of charts, and is characterized by the property that

$$\begin{equation} \phi ^*(\omega ) = \left(\begin{matrix} 1 & i \end{matrix}\right) A \left(\begin{matrix} \operatorname {Re} \omega \\ \operatorname {Im} \omega \end{matrix}\right) . {\tag{2.7}} \end{equation}$$

In particular, $A=I$ if and only if $\phi$ belongs to $\operatorname {Aut}(X,\omega )$, the group of holomorphic automorphisms of $X$ satisfying $\phi ^*(\omega ) = \omega$.

It is then easy to see we have an exact sequence:

$$\begin{equation*} 1 \rightarrow \operatorname {Aut}(X,\omega ) \rightarrow \operatorname {Aff}^+(X,\omega ) \stackrel{D}{\rightarrow } \operatorname {SL}(X,\omega ) \rightarrow 1. \end{equation*}$$

For $g>1$, the group $\operatorname {Aut}(X,\omega )$ is finite, so the stabilizer of $(X,\omega )$ in $\operatorname {SL}_2(\mathbb{R})$ is virtually the same as its affine symmetry group.

Examples.

The square torus $E = \mathbb{C}/\mathbb{Z}[i]$ generates the simplest example of a Teichmüller curve. In this case, every orientation–preserving automorphism of $E \cong \mathbb{R}^2/\mathbb{Z}^2$ as a Lie group is also an affine automorphism of $(E,dz)$. Thus $\operatorname {SL}_2(E,dz) = \operatorname {SL}_2(\mathbb{Z})$, and the map $f : V \rightarrow \mathcal{M}_1$ is an isomorphism. This is the trivial Teichmüller curve.

An example in genus two is provided by the form $(Y,\eta ) = L(2,2)/{\sim }$ (see Figure 2.1). Here we find

$$\begin{equation} \operatorname {SL}(Y,\eta ) = \langle S,T \rangle = \left\langle \left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix}\right), \left(\begin{matrix} 1 & 2\\ 0 & 1 \end{matrix}\right) \right\rangle \subset \operatorname {SL}_2(\mathbb{Z}) . {\tag{2.8}} \end{equation}$$

In these examples $\operatorname {SL}(E,dz)$ and $\operatorname {SL}(Y,\eta )$ are both triangle groups, namely $\Delta (2,3,\infty )$ and $\Delta (2,\infty ,\infty )$. See Appendix A for more on triangle groups, which will occur frequently in the discussions to follow.

Cylinders and parabolics.

The modulus of a cylinder of height $h$ and circumference $c$ is $m=h/c$. In general, if $(X,|\omega |)$ is covered by a collection of $n$ horizontal cylinders $(C_i)$ with moduli $(m_i)$, and $m>0$ divides $m_i$ for all $i$ (meaning $m_i/m$ is an integer), we can construct an affine automorphism $\phi$ of $(X,\omega )$ with

$$\begin{equation*} D \phi = \left(\begin{matrix} 1 & 1/m \\ 0 & 1 \end{matrix}\right) . \end{equation*}$$

Namely we take $\phi |C_i$ to be a linear, right Dehn twist, iterated $m_i/m$ times. The iterate is chosen so $D\phi |C_i$ is the matrix above for all $i$. Since $\phi |C_i$ is the identity on $\partial C_i$, these twists fit together to give a map $\phi \in \operatorname {Aff}^+(X,\omega )$.

Conversely, it is not hard to show:

Proposition 2.3.

Suppose $\operatorname {SL}(X,\omega )$ contains a parabolic element $A$ fixing the line of slope $s$ through the origin. Then $(X,\omega )$ is tiled by a family of cylinders $(C_1,\ldots ,C_n)$ of slope $s$, with rational ratios of moduli.

We can now explain the appearance of the parabolic matrix $T = \left( \begin{smallmatrix} 1&2\\ 0&1 \end{smallmatrix}\right)$ in $\operatorname {SL}(Y,\eta )$. Note that $L(2,2)$ is built from three copies of the unit square. The bottom two squares define a horizontal cylinder $C_1 \subset Y$, isometric to a Euclidean cylinder of height and circumference $(h_1,c_1) = (1,2)$. Similarly the top square gives a cylinder $C_2$ with $(h_2,c_2) = (1,1)$. Thus $m = \gcd (m_1,m_2) = \gcd (1/2,1) = 1/2$, and $D \phi = \left( \begin{smallmatrix} 1&2\\ 0&1 \end{smallmatrix}\right)$.

As for the generator $S$, it is easy to see that $L(a,b)/{\sim }$ has a 4-fold rotational symmetry whenever $a=b$ (see Figure 3.4).

Cusps of $V$.

We note that the Teichmüller curve $V$ generated by a 1-form $(X,\omega )$ is properly immersed in $\mathcal{M}_g$, and hence the $\operatorname {SL}_2(\mathbb{R})$ orbit of $(X,\omega )$ in $\Omega \mathcal{M}_g$ is closed.

Proposition 2.4.

A Teichmüller curve $V$ has finite hyperbolic area, but it is never compact; it always has at least one cusp.

Idea of the proof.

Assume $g \ge 2$ and $(X,\omega ) \in \Omega \mathcal{M}_g(p)$. Construct a geodesic segment $\sigma$ on $(X,|\omega |)$ with endpoints in $Z(\omega )$. After rotating $\omega$, we can assume $\sigma$ is horizontal. Now consider the Teichmüller geodesic ray $X_t = \gamma (t)$ in $V$ generated by $(X,\omega )$, as in equation 2.5. As $t \rightarrow \infty$ the length of $\delta$ on $X_t$ tends to zero. Since the $\operatorname {SL}_2(\mathbb{R})$ orbit of $(X,\omega )$ is closed in its stratum, the endpoints of $\delta$ cannot collide, so $X_t$ tends to infinity in $\mathcal{M}_g$. Therefore $V$ is noncompact, and $X_t$ tends to a cusp of $V$.■

Combined with Proposition 2.3, we find that $(X,\omega )$ has many cylinder decompositions and a dense set of periodic directions.

Square–tiled surfaces.

Let us say $(X,\omega )$ is a square–tiled 1-form if it can be obtained by gluing together a finite number of copies of the unit square $([0,1]^2,dz)$.

Generalizing the case of $(Y,\eta ) = L(2,2)/{\sim }$, one can show that $\operatorname {SL}(X,\omega )$ has finite index in $\operatorname {SL}_2(\mathbb{Z})$ for any square–tiled 1-form. One can also check that square tiled surfaces are dense in $\mathcal{M}_g$. Consequently:

Teichmüller curves are dense in $\mathcal{M}_g$.

These Teichmüller curves, however, are simply echos in higher genus of the trivial Teichmüller curve $V \cong \mathcal{M}_1$. Every Teichmüller curve generates similar echos in higher genus, via covering constructions. For example, there is a degree 3 holomorphic map $f : Y \rightarrow E$ such that $\eta = f^*(dz)$.

Primitivity.

For this reason we will focus our attention on primitive Teichmüller curves in $\mathcal{M}_g$: those that do not arise from lower genus.

To define these, let us say $(X,\omega )$ is the pullback of $(Y,\eta )$ if there is a holomorphic map $f : X \rightarrow Y$ such that $\omega = f^*(\eta )$. A 1-form with $g(X)>1$ is primitive if it is not the pullback of a form of lower genus.

Every form in $\Omega \mathcal{M}_g$, $g>1$, is the pullback of a unique primitive 1-form $(X,\omega )$ Mo1, Thm. 2.6, Mc3, Thm. 2.1. We say a Teichmüller curve is primitive if it is generated by a primitive 1-form. In this case $\operatorname {Aut}(X,\omega )$ is trivial, and hence

$$\begin{equation*} \operatorname {Aff}^+(X,\omega ) \cong \operatorname {SL}(X,\omega ) . \end{equation*}$$

Invariants.

We conclude this section by discussing three invariants of the Teichmüller curve $f : V \rightarrow \mathcal{M}_g$ generated by a 1-form $(X,\omega )$.

1.

The lattice $\operatorname {SL}(X,\omega )$, often called the Veech group, is determined by $V$ up to conjugacy in $\operatorname {SL}_2(\mathbb{R})$. Indeed, it is simply the Fuchsian group uniformizing $V$.

2.

The trace field of $\operatorname {SL}(X,\omega )$, defined by$$\begin{equation*} K = \mathbb{Q}(\operatorname {tr}A \; : \; A \in \operatorname {SL}(X,\omega )) \subset \mathbb{R}, \end{equation*}$$

is also an invariant of $V$. It is a totally real number field, of degree at most $g$ over $\mathbb{Q}$, satisfying$$\begin{equation} K = \mathbb{Q}(\operatorname {tr}A) {\tag{2.9}} \end{equation}$$

for any hyperbolic element