A Short Introduction to Translation Surfaces, Veech Surfaces, and Teichmüller Dynamics

We review the different notions of translation surfaces that are necessary to understand McMullen’s classification of GL 2 + ( ℝ ) $$\mathrm {GL}_2^+({\mathbb R})$$ -orbit closures in genus two. We start by recalling the different definitions of a translation surface, in increasing order of abstraction, starting with cutting and pasting plane polygons, ending with Abelian differentials. We then define the moduli space of translation surfaces and explain its stratification by the type of zeroes of the Abelian differential, the local coordinates given by the relative periods, its relationship with the moduli space of complex structures and the Teichmüller geodesic flow. We introduce the GL 2 + ( ℝ ) $$\mathrm {GL}_2^+({\mathbb R})$$ -action, and define the related notions of Veech group, Teichmüller disk, and Veech surface. We explain how McMullen classifies GL 2 + ( ℝ ) $$\mathrm {GL}_2^+({\mathbb R})$$ -orbit closures in genus 2: we have orbit closures of dimension 1 (Veech surfaces, of which a complete list is given), 2 (Hilbert modular surfaces, of which again a complete list is given), and 3 (the whole moduli space of complex structures). In the last section, we review some recent progress in higher genus.