Geometric Invariants and Asymptotics of Translation Surfaces

Abstract:

By gluing together polygons along parallel edges in a well-defined matter, we obtain translation surfaces, which are two-dimensional manifolds with rich geometric  structures.

This thesis examines the geometry of translation surfaces with a focus on  understanding the properties of random translation surfaces of high genus in order to gain a broader understanding of the moduli space of translation surfaces.

One of the primary challenges in this field is bridging the gap between finite and infinite translation surfaces corresponding to the compact and non-compact cases. We address this challenge by examining the convergence of sequences of finite translation surfaces from different strata and approximating infinite translation surfaces with them.

We use different methods to achieve this approximation. On the one hand, we use some sense of convergence in the underlying Veech group to approach infinite translation surfaces; on the other hand, we want to understand geometric invariants for translation surfaces of large genus. In particular, we explore the behavior of the Cheeger constant, a measure of the inverse of bottleneckedness.

By advancing existing constructions, introducing new perspectives, and analyzing key geometric invariants, this thesis enhances our understanding of translation surfaces for future research.