Infinite Translation Surfaces in the Wild

TL;DR

This book provides an introductory yet comprehensive exploration of infinite-type translation surfaces, covering definitions, topological classification, symmetries, and complex dynamics, serving as a key resource for researchers and students.

Contribution

It offers a detailed, systematic study of infinite translation surfaces, including new classifications, symmetry analyses, and dynamical properties, filling gaps in existing literature.

Findings

Equivalence of three definitions of translation surfaces

Classification of infinite-type surfaces and their coverings

Complex dynamics of translation flows in infinite-type surfaces

Abstract

This book explores infinite-type translation surfaces and is intended as an introductory text for graduate and PhD students, as well as a reference for more advanced researchers. Chapter 1 introduces the three definitions of translation surfaces and meticulously proves their equivalence. It is enriched with numerous examples that are revisited throughout the book. Chapter 2 provides a detailed examination of the topological classification of infinite-type surfaces, the construction of infinite coverings of finite-type translation surfaces, and the structure of points within the metric completion. Chapter 3 investigates the affine symmetries of infinite-type translation surfaces, with special emphasis on infinite coverings of finite-type surfaces, the Hooper-Thurston-Veech construction, and affine homeomorphisms of finite-area infinite-type translation surfaces. Chapter 4 introduces infinite interval exchange transformations and employs them to demonstrate that the dynamics of translation flows are significantly more complex in the infinite-type context. The two appendices address hyperbolic geometry and the spectra of infinite graphs, respectively.