J. A. Arthreya, H. Masur: “Translation Surfaces”

The book under review is an introductory textbook to the world of translation surfaces. The topic emerged roughly two decades ago from the special, seemingly simple (and fascinating) dynamical system of a single ball in a polygonal billiard table – the reader could think of a triangle. Nowadays translation surfaces can be considered as a field of its own, with strong ties to dynamics on homogeneous spaces but also with bridges to algebraic geometry and number theory.

One core topic of the book are counting problems. Are there periodic trajectories on every polygonal billiard table whose angles are rational multiples of \(\pi \)? If so, how many of them are there up to a given length, how does this number grow asymptotically? What happens if one imposes additionally constraints on the directions of the trajectories?

All these questions become tractable thanks to the hypothesis of rational angles: Unfolding the billiard table is the key: Take the billard table, reflect along its sides, and keep on reflecting. If we kept doing this forever, certain tables, e.g. for an isosceles triangle or a rectangle this will tesselate the plane, while for others e.g. an \(L\)-shaped table, the reflection copies will overlap. Tesselating is not the goal, if we have billiard trajectories in mind. The unfolding process rather identifies the reflection tiles whenever they differ by translation. The result of the unfolding procress is a compact Riemann surface, a torus both for the isosceles triangle and for the rectangle while the \(L\)-shaped table will unfold to genus two. A right-angled triangle with an angle \(\pi /n\) will unfold for even \(n\) to a regular \(n\)-gon with opposite sides identified, and for odd \(n\) it will unfold to two regular \(n\)-gons attached to each other, again with the remaining sides identified by parallel translation to form a compact Riemann surface.

Unfolding turns the above questions on billiard trajectories into questions about just straight lines on a Riemann surface equipped with a flat metric with cone angle singularities. Alternatively, we may obtain such a surface as glued from polygons by identifying the sides of the polygons pairwise by translations – a translation surface. Most of the translation surfaces however do not arise from unfolding constructions, since unfolded billiards always have a non-trivial group of symmetries.

Translation surfaces come with a moduli space, just as their older siblings – plain Riemann surfaces – come with a moduli space. No highbrow algebraic geometry is needed to construct this moduli space and the book under review consequently takes the existence of this moduli space for granted. Instead, the moduli space of translation surfaces admits as bonus a nice coordinate system: in fact a structure of a non-compact linear manifold, a manifold that admits charts whose transition functions are linear maps. This linear structure, given by the periods of a basis of homology, measured in the translation structure, is key for many applications. Moreover, the moduli space of translation surfaces admits an action by the group \(\mathrm{SL}_{2}(\mathbb{R})\) that is easily visualized by letting this group act on the individual polygons that formed the translation surfaces. This is well-defined independently of the planar embedding of the polygons since sides that were identified by parallel translation are still parallel after applying any linear transformation.

The book contains some prime examples for the slogan that in order to understand the dynamics of the individual object, the billiard table or the translation surface, one should understand the dynamics of renormalization on moduli space. For example, the question whether the straight line flow in a given direction on a translation surface is ergodic can be read off from the orbit of the translation surface under the diagonal matrices in \(\mathrm{SL}_{2}(\mathbb{R})\), the Teichmüller geodesic flow. These questions about (unique) ergodicity and weak mixing of the straight line flow form the second core topic of the book. The current knowledge about them is fairly complete and covered in full detail in the book.

Examples of translation surfaces without directions where the flow fails to be uniquely ergodic are rare. They are called lattice surfaces (or Veech surfaces), since they also admit a characterization by having a large group of automorphisms, namely a Fuchsian group which is a lattice in \(\mathrm{SL}_{2}(\mathbb{R})\). The construction of such lattice surfaces is a topic of ongoing research and this is one of the reasons while the last chapter of the book is significantly more sketchy and the reader referred to research paper for a lot more details than in the rest of the book.

The book starts out with a gentle introduction by solving with bare hands all the problems sketched above in the basic case, the square billiard table or its unfolding, a square torus.

The prerequisites for the book are merely some basic knowledge on manifolds and Riemann surfaces, a tiny bit of topology and some standard material from measure theory. The book is ideally suited for a topics course for Master students, possible also for advanced Bachelor students. It should then be complemented by a textbook on Riemann Surfaces and one on Ergodic Theory.

Currently this is the only textbook available on this topic aimed at master or graduate students. The topic can also be presented from its purely combinatorial side at a level suitable for motivated high school students as the book by D. Davis [1] illustrates. The book of V. Delecroix, P. Hubert and F. Valdez [2] focuses on a subtopic, the special case of infinite translation surfaces. These arise for example by unfolding a triangle whose angles are irrational multiples of \(\pi \). As remarked in the introduction, some aspects of dynamical systems on translation surfaces, notably the concept of Lyapunov exponents and as a tool the Kontsevich-Zorich cocycle, are not included. The survey by G. Forni and C. Matheus [3] may serve as useful complement in this direction. A textbook on translation surfaces aimed at graduate level and taking the viewpoint of complex algebraic geometry, comparable to the two volumes of “Geometry of algebraic curves” [4], is currently not available.