Rotational component spaces for infinite-type translation surfaces

TL;DR

This paper explores the complex structure of infinite translation surfaces by analyzing rotational components and linear approaches, revealing their high variability and limitations in fully characterizing such surfaces.

Contribution

It introduces new insights into the flexibility of rotational components and linear approaches in infinite translation surfaces, showing their limitations in classifying these surfaces.

Findings

Every finite topological space can be realized as a space of rotational components.

Spaces of linear approaches can differ even when rotational components are the same.

The space of rotational components alone does not fully determine an infinite translation surface.

Abstract

Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduced by Bowman and Valdez: linear approaches and rotational components. We show that there is a large flexibility in the spaces of rotational components and even more in the spaces of linear approaches. In particular, we prove that every finite topological space arises as space of rotational components. However, this space will still not contain enough information to describe an infinite translation surface. We showcase this through an uncountable family with the same space of rotational components but different spaces of linear approaches. Additionally, we study several known and new examples to illustrate the concept of linear approaches and rotational components.