Introduction to origamis in Teichmüller space

Abstract

Origamis are translation surfaces that arise from certain coverings of elliptic curves. We give four different characterizations of them, in combinatorial as well as in algebro-geometric terms. By affine variation of the translation structure, an origami defines an embedding of the upper half plane into Teichmüller space; its image in moduli space is an algebraic curve. As a Riemann surface, this algebraic curve is uniformized by the Veech group of the origami, a group closely related to the affine homeomorphisms of the translation surface. We explain these concepts and their interrelations, and illustrate them by examples. In particular we discuss the very helpful example of the quaternion origami or “Eierlegende Wollmilchsau”.