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 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 be a collection of polygons in the Euclidean plane and suppose that for every side of any there is a side of some with and for some nonzero vector there is a natural side-pairing.
The quotient space is a closed surface. It has a flat metric outside the set images of the vertices. At a point in the sum of the angles of the polygons around the vertices which map to it is a positive multiple of, and the metric is singular unless the angle is exactly.

Analytic definition

Let be a translation surface as defined above and the set of singular points. Identifying the Euclidean plane with the complex plane one gets coordinates charts on with values in. Moreover, the changes of charts are holomorphic maps, more precisely maps of the form for some. This gives the structure of a Riemann surface, which extends to the entire surface by Riemann's theorem on removable singularities. In addition, the differential where 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 on. The vertices of the polygon where the cone angles are not equal to are zeroes of .
In the other direction, given a pair where is a compact Riemann surface and a holomorphic 1-form one can construct a polygon by using the complex numbers where are disjoint paths between the zeroes of 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 is a regular -gon then the translation surface obtained by gluing opposite sides is of genus with a single singular point, with angle.
If is obtained by putting side to side a collection of copies of the unit square then any translation surface obtained from 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.

Riemann–Roch and Gauss–Bonnet

Suppose that the surface is a closed Riemann surface of genus and that is a nonzero holomorphic 1-form on, with zeroes of order. Then the Riemann–Roch theorem implies that
If the translation surface is represented by a polygon then triangulating it and summing angles over all vertices allows to recover the formula above, 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 is a translation surface there is a natural measured foliation on. 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 primitive for and the measure is obtained by integrating the real part.

Moduli spaces

Strata

Let be the set of translation surfaces of genus . Let be the moduli space of Riemann surfaces of genus ; there is a natural map mapping a translation surface to the underlying Riemann surface. This turns into a locally trivial fiber bundle over the moduli space.
To a compact translation surface there is associated the data where are the orders of the zeroes of. If is any integer partition of then the stratum is the subset of of translation surfaces which have a holomorphic form whose zeroes match the partition.
The stratum is naturally a complex orbifold of complex dimension . Local coordinates are given by
where and is as above a symplectic basis of this space.

Masur-Veech volumes

The stratum admits a -action and thus a real and complex projectivization. The real projectivization admits a natural section 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 with an integral affine structure and thus a natural volume form. We also get a volume form on by disintegration of. The Masur-Veech volume is the total volume of for. 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. They observed that should be of the form 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 and Andrei Okounkov gave the first algorithm to compute these volumes. They showed that the generating series of these numbers are q-expansions of computable quasi-modular forms. Using this algorithm they could confirm the numerical observation of Kontsevich and Zorich.
More recently Chen, Möller, Sauvaget, and don Zagier showed that the volumes can be computed as intersection numbers on an algebraic compactification of. Currently the problem is still open to extend this formula to strata of half-translation surfaces.

The SL2(R)-action

If is a translation surface obtained by identifying the faces of a polygon and then the translation surface is that associated to the polygon. This defined a continuous action of on the moduli space which preserves the strata. This action descends to an action on that is ergodic with respect to.

Half-translation surfaces

Definitions

A half-translation surface is defined similarly to a translation surface but allowing the gluing maps to have a nontrivial linear part which is a half turn. Formally, a translation surface is defined geometrically by taking a collection of polygons in the Euclidean plane and identifying faces by maps of the form . Note that a face can be identified with itself. The geometric structure obtained in this way is a flat metric outside of a finite number of singular points with cone angles positive multiples of.
As in the case of translation surfaces there is an analytic interpretation: a half-translation surface can be interpreted as a pair where is a Riemann surface and a quadratic differential on. To pass from the geometric picture to the analytic picture one simply takes the quadratic differential defined locally by , and for the other direction one takes the Riemannian metric induced by, which is smooth and flat outside of the zeros of.

Relation with Teichmüller geometry

If is a Riemann surface then the vector space of quadratic differentials on is naturally identified with the tangent space to Teichmüller space at any point above. This can be proven by analytic means using the Bers embedding. Half-translation surfaces can be used to give a more geometric interpretation of this: if are two points in Teichmüller space then by Teichmüller's mapping theorem there exists two polygons whose faces can be identified by half-translations to give flat surfaces with underlying Riemann surfaces isomorphic to respectively, and an affine map of the plane sending to which has the smallest distortion among the quasiconformal mappings in its isotopy class, and which is isotopic to.
Everything is determined uniquely up to scaling if we ask that be of the form, where, for some ; we denote by the Riemann surface obtained from the polygon. Now the path in Teichmüller space joins to, and differentiating it at gives a vector in the tangent space; since was arbitrary we obtain a bijection.
In facts the paths used in this construction are Teichmüller geodesics. An interesting fact is that while the geodesic ray associated to a flat surface corresponds to a measured foliation, and thus the directions in tangent space are identified with the Thurston boundary, the Teichmüller geodesic ray associated to a flat surface does not always converge to the corresponding point on the boundary, though almost all such rays do so.

Veech surfaces

The Veech group

If is a translation surface its Veech group is the Fuchsian group which is the image in of the subgroup of transformations such that is isomorphic to. Equivalently, is the group of derivatives of affine diffeomorphisms . Veech groups have the following properties:
  • They are discrete subgroups in ;
  • They are never cocompact.
Veech groups can be either finitely generated or not.

Veech surfaces

A Veech surface is by definition a translation surface whose Veech group is a lattice in, equivalently its action on the hyperbolic plane admits a fundamental domain of finite volume. Since it is not cocompact it must then contain parabolic elements.
Examples of Veech surfaces are the square-tiled surfaces, whose Veech groups are commensurable to the modular group. The square can be replaced by any parallelogram. In fact the Veech group is arithmetic if and only if the surface is tiled by parallelograms.
There exists Veech surfaces whose Veech group is not arithmetic, for example the surface obtained from two regular pentagons glued along an edge: in this case the Veech group is a non-arithmetic Hecke triangle group. On the other hand, there are still some arithmetic constraints on the Veech group of a Veech surface: for example its trace field is a number field that is totally real.