Translation surfaces and their orbit closures: An introduction for a broad audience

Abstract

Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a GL(2, R) action on each stratum, and to solve most problems about a translation surface one must first know the closure of its orbit under this action. Furthermore, these orbit closures are of fundamental interest in their own right, and are now known to be algebraic varieties that parameterize translation surfaces with extraordinary algebro-geometric and flat properties. The study of orbit closures has greatly accelerated in recent years, with an influx of new tools and ideas coming diverse areas of mathematics.Many areas of mathematics, from algebraic geometry and number theory, to dynamics and topology, can be brought to bear on this topic, and furthermore known examples of orbit closures are interesting from all these points of view.This survey is an invitation for mathematicians from different backgrounds to become familiar with the subject. Little background knowledge, beyond the definition of a Riemann surface and its cotangent bundle, is assumed, and top priority is given to presenting a view of the subject that is at once accessible and connected to many areas of mathematics.Acknowledgements: I am especially grateful to Ronen Mukamel for helpful conversations that shaped the presentation of real multiplication in genus 2, and to Curtis McMullen for helpful comments and corrections. I am also very grateful to

Holomorphic quadratic differentials in Teichmüller theory

“…For a parallel discussion of translation surfaces corresponding to holomorphic 1forms, and its relation to the theory of billiards, see the surveys [132,133] and [135]. For more about the structure, closed trajectories, length-spectra and degeneration of such singular-flat metrics, see for example, [104], [30], [23], [35] and [97].…”

Equidistribution of saddle connections on translation surfaces

“…For more information about the theory of translation surfaces, including the connection to rational billiards, many survey articles are available, such as [Zor06] and [Wri15].…”

From rational billiards to dynamics on moduli spaces

“…measure rigidity" article by Einsiedler [Ein09], as well as the eight-page note "The mathematical work of Maryam Mirzakhani" by McMullen [McM14] and the 13-page note "The magic wand theorem of A. Eskin and M. Mirzakhani" by Zorich [Zor14,Zor]. There are a large number of surveys on translation surfaces, for example [MT02,Zor06] and the author's recent introduction aimed at a broad audience [Wri15b]. Alex Eskin has given a mini-course on his paper with Mirzakhani, and notes are available on his website [Esk].…”