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Translation Surfaces

By Jayadev S. Athreya and Howard Masur. GSM/242, 2024, 179 pp.,

A translation surface is a two-dimensional manifold obtained from a collection of polygons in the Euclidean plane, where the sides of the polygons are grouped into pairs that are parallel and of the same length. The sides in each pair are oriented so that the Euclidean translation taking one to the other preserves this orientation, and the polygon of one of the oriented sides lies to the left of the oriented side and the polygon corresponding to the other side to the right. The archetypical example of a translation surface is a “flat torus” which is obtained from a parallelogram by identifying the opposite sides by translations, which inherits the flat structure from the plane. A translation surface has its origins in the study of billiards, and has connections to algebraic geometry, number theory, low-dimensional topology, and dynamics. The study of translation surfaces and their moduli spaces has seen enormous growth in recent decades, and results in related areas have been recognized with prestigious awards such as Fields Medals, Breakthrough Prizes, and Clay Research Awards. There are several excellent surveys on this topic, but this is the first book that gives a comprehensive introduction to the subject.

In Chapter 1, the authors analyze the geometry of the flat torus and associated dynamical and counting problems. The results about flat tori and their moduli spaces described in Chapter 1 motivate the discussion in higher-genus settings. Chapter 2 introduces three different definitions of translation surfaces as polygons in the Euclidean plane, holomorphic differentials on Riemann surfaces, and (singular) geometric structures on two-dimensional manifolds. The equivalence of these three perspectives is at the heart of the study of the geometry and dynamics of translation surfaces and their moduli spaces but is often opaque in the literature. Athreya and Masur do a wonderful job explaining this equivalence in a clear way.

Chapter 3 starts with a brief discussion of Teichmüller and moduli spaces of Riemann surfaces and delves into the moduli space of translation surfaces where the equivalence of two translation surfaces is carefully defined in the previous section. This moduli space of translation surfaces admits a natural “stratification” by topological data (the genus, the number of cone points, and the excess angle at each cone point). Fixing this data, one obtains “strata of translation surfaces.” The components of strata admit natural coordinates, and a natural Lebesgue measure for which the total volume of the space of unit-area surfaces is finite.

The linear action of the group of real matrices with positive determinant on the Euclidean plane, induces an action of on the space of translation surfaces which preserves strata, and where the subgroup preserves the area 1 locus. The dynamics and ergodic theory of subgroups of acting on strata, and their role as renormalization dynamics for flows on translation surfaces, takes up most of Chapter 4 and Chapter 5.

The remaining chapters give a broad overview of the field since the 1990s while concealing technical details in well-chosen “Black Boxes.” Instead, precise references and exercises are given to lead the reader to the correct places. The book ends with a discussion of “Lattice surfaces” which are translation surfaces with “maximal symmetry” and “optimal dynamics,” originally studied by Veech. Although they form a measure zero subset in any stratum, they are nevertheless dense in each stratum and many natural examples studied in the literature have this property.

Translation Surfaces by Athreya and Masur is a wonderful addition to the literature with lots of beautiful pictures and clear exposition. I expect it to become the standard reference book for researchers and graduate students working in the area.