Infinite Translation Surfaces in the Wild

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.

Figures (99)

• Figure 1: A genus 2 translation surface and a neighbourhood of its conical singularity.
• Figure 2: The infinite staircase. Opposite sides are identified. There are four infinite degree vertices in the surface.
• Figure 5: A genus 1 half-translation surface and a neighbourhood of a conical singularity of angle $\pi$ (corresponding to a simple pole of the quadratic differential).
• Figure 6: The orientation double covering of the half-translation surface of Figure \ref{['fig:HalfTransSurf']}.
• Figure 7: Two infinite strips in the infinite staircase from Figure \ref{['fig:StaircaseFirst']}.

Theorems & Definitions (59)

• proof
• proof : of Corollary \ref{['cor:MalagaMapRandomRecurrent']}
• proof : Sketch of proof
• proof
• proof
• proof
• proof
• proof : Proof of Lemma \ref{['lem:triangulation']}
• proof : Proof of Lemma \ref{['lem:FiniteAreaAndSaddleConnections']}
• proof