Translation Surface

In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form.

These surfaces arise in dynamical systems where they can be used to model billiards, and in Teichmüller theory. A particularly interesting subclass is that of Veech surfaces (named after William A. Veech) which are the most symmetric ones.

Definitions

Geometric definition

A translation surface is the space obtained by identifying pairwise by translations the sides of a collection of plane polygons.

Here is a more formal definition. Let $P_1,\ldots,P_m$ be a collection of (not necessarily convex) polygons in the Euclidean plane and suppose that for every side $s_i$ of any $P_k$ there is a side $s_j$ of some $P_l$ with $j\not=i$ and $s_j = s_i + \vec v_i$ for some nonzero vector $\vec v_i$ (and so that $\vec v_j = -\vec v_i$. Consider the space obtained by identifying all $s_i$ with their corresponding $s_j$ through the map $x \mapsto x + \vec v_i$.

The canonical way to construct such a surface is as follows: start with vectors $\vec w_1, \ldots, \vec w_n$ and a permutation $\sigma$ on $\$, and form the broken lines $L = x, x+\vec w_1, \ldots, x+\vec w_1+\cdots+\vec w_n$ and $L' = x, x+\vec w_, \ldots, x+\vec w_+\cdots+\vec w_$ starting at an arbitrarily chosen point. In the case where these two lines form a polygon (i.e. they do not intersect outside of their endpoints) there is a natural side-pairing.

The quotient space is a closed surface. It has a flat metric outside the set $\Sigma$ images of the vertices. At a point in $\Sigma$ the sum of the angles of the polygons around the vertices which map to it is a positive multiple of $2\pi$, and the metric is singular unless the angle is exactly $2\pi$.

Analytic definition

Let $S$ be a translation surface as defined above and $\Sigma$ the set of singular points. Identifying the Euclidean plane with the complex plane one gets coordinates charts on $S \setminus \Sigma$ with values in $\mathbb C$. Moreover, the changes of charts are holomorphic maps, more precisely maps of the form $z \mapsto z + w$ for some $w \in \mathbb C$. This gives $S \setminus \Sigma$ the structure of a Riemann surface, which extends to the entire surface $S$ by Riemann's theorem on removable singularities. In addition, the differential $dz$ where $z : U \to \mathbb C$ is any chart defined above, does not depend on the chart. Thus these differentials defined on chart domains glue together to give a well-defined holomorphic 1-form $\omega$ on $S$. The vertices of the polygon where the cone angles are not equal to $2\pi$ are zeroes of $\omega$ (a cone angle of $2k\pi$ corresponds to a zero of order $(k-1)$).

In the other direction, given a pair $(X,\omega)$ where $X$ is a compact Riemann surface and $\omega$ a holomorphic 1-form one can construct a polygon by using the complex numbers $\int_\omega$ where $\gamma_j$ are disjoint paths between the zeroes of $\omega$ which form an integral basis for the relative cohomology.

Examples

The simplest example of a translation surface is obtained by gluing the opposite sides of a parallelogram. It is a flat torus with no singularities.

If $P$ is a regular $4g$-gon then the translation surface obtained by gluing opposite sides is of genus $g$ with a single singular point, with angle $(2g-1)2\pi$.

If $P$ is obtained by putting side to side a collection of copies of the unit square then any translation surface obtained from $P$ is called a ''square-tiled surface''. The map from the surface to the flat torus obtained by identifying all squares is a branched covering with branch points the singularities (the cone angle at a singularity is proportional to the degree of branching).

Riemann–Roch and Gauss–Bonnet

Suppose that the surface $X$ is a closed Riemann surface of genus $g$ and that $\omega$ is a nonzero holomorphic 1-form on $X$, with zeroes of order $d_1, \ldots, d_m$. Then the Riemann–Roch theorem implies that

:$\sum_^m d_j = 2g - 2.$

If the translation surface $(X,\omega)$ is represented by a polygon $P$ then triangulating it and summing angles over all vertices allows to recover the formula above (using the relation between cone angles and order of zeroes), in the same manner as in the proof of the Gauss–Bonnet formula for hyperbolic surfaces or the proof of Euler's formula from Girard's theorem.

Translation surfaces as foliated surfaces

If $(X,\omega)$ is a translation surface there is a natural measured foliation on $X$. If it is obtained from a polygon it is just the image of vertical lines, and the measure of an arc is just the euclidean length of the horizontal segment homotopic to the arc. The foliation is also obtained by the level lines of the imaginary part of a (local) primitive for $\omega$ and the measure is obtained by integrating the real part.

Moduli spaces

Strata

Let $\mathcal H$ be the set of translation surfaces of genus $g$ (where two such $(X,\omega), (X',\omega')$ are considered the same if there exists a holomorphic diffeomorphism $\phi:X \to X'$ such that $\phi^*\omega' = \omega$). Let $\mathcal M_g$ be the moduli space of Riemann surfaces of genus $g$; there is a natural map $\mathcal H \to \mathcal M_g$ mapping a translation surface to the underlying Riemann surface. This turns $\mathcal H$ into a locally trivial fiber bundle over the moduli space.

To a compact translation surface $(X, \omega)$ there is associated the data $(k_1,\ldots,k_m)$ where $k_1\le k_2\le\cdots$ are the orders of the zeroes of $\omega$. If $\alpha = (k_1,\ldots,k_m)$ is any partition of $2g - 2$ then the stratum $\mathcal H(\alpha)$ is the subset of $\mathcal H$ of translation surfaces which have a holomorphic form whose zeroes match the partition.

The stratum $\mathcal H(\alpha)$ is naturally a complex orbifold of complex dimension $2g + m - 1$ (note that $\mathcal H(0)$ is the moduli space of tori, which is well-known to be an orbifold; in higher genus, the failure to be a manifold is even more dramatic). Local coordinates are given by
:$(X,\omega) \mapsto \left(\int_ \omega, \ldots, \int_ \omega\right)$
where $n=\dim(H_1(S, \)) = 2g + m - 1$ and $\gamma_1,\ldots, \gamma_k$ is as above a symplectic basis of this space.

Masur-Veech volumes

The stratum $\mathcal H(\alpha)$ admits a $^*$-action and thus a real and complex projectivization $\to _1(\alpha) \to _2(\alpha)$. The real projectivization admits a natural section $_1(\alpha) \to (\alpha)$ if we define it as the space of translation surfaces of area 1.

The existence of the above period coordinates allows to endow the stratum $\mathcal H(\alpha)$ with an integral affine structure and thus a natural volume form $\nu$. We also get a volume form $\nu_1(\alpha)$ on $_1(\alpha)$ by disintegration of $\nu$. The Masur-Veech volume $Vol(\alpha)$ is the total volume of $_1(\alpha)$ for $\nu_1(\alpha)$. This volume was proved to be finite independently by William A. Veech and Howard Masur.

In the 90's Maxim Kontsevich and Anton Zorich evaluated these volumes numerically by counting the lattice points of $(\alpha)$. They observed that $Vol(\alpha)$ should be of the form $\pi^$ times a rational number. From this observation they expected the existence of a formula expressing the volumes in terms of intersection numbers on moduli spaces of curves.

Alex Eskin

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