On the geometry and arithmetic of infinite translation surfaces

There are only a few invariants one classically associates with precompact translation surfaces, among them certain numberfields, i.e. fields which are finite extensions of the field Q of rational numbers. These fields are closely related to each other; they are often even equal. We prove by constructing explicit examples that most of the classical results for number fields associated to precompact translation surfaces fail in the realm of general translation surfaces and investigate the relations among these fields. A very special class of translation surfaces are so called square-tiled surfaces or origamis. We give a characterization for infinite origamis.