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

TL;DR

This paper introduces translation surfaces, their moduli spaces, and the significance of their orbit closures under the GL(2,R) action, aiming to make the subject accessible to mathematicians from diverse backgrounds.

Contribution

It provides an accessible overview of the recent advances in the study of orbit closures of translation surfaces, highlighting new algebraic and geometric insights.

Findings

Orbit closures are algebraic varieties with special properties.

Recent years have seen rapid progress due to new tools and interdisciplinary approaches.

The survey aims to connect the subject with broader mathematical areas.

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 Abelian 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. 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.