On the geometry and arithmetic of infinite translation surfaces

TL;DR

This paper explores the invariants of infinite translation surfaces, especially focusing on the associated number fields, and demonstrates that classical results for precompact surfaces often do not extend to the general case, with a special focus on square-tiled surfaces.

Contribution

It constructs explicit examples showing the failure of classical invariants for general translation surfaces and characterizes infinite origamis.

Findings

Classical invariants like number fields often fail for general translation surfaces.

Explicit examples demonstrate the divergence from classical results.

Characterization of infinite origamis provided.

Abstract

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.