Infinite Translation Surfaces in the Wild

Vincent Delecroix; Pascal Hubert; Ferrán Valdez

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arXiv · 2403.05424

Infinite Translation Surfaces in the Wild

Vincent Delecroix·Pascal Hubert·Ferrán Valdez

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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.

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BibTeX RIS

Vincent Delecroix, Pascal Hubert, Ferrán Valdez. 2024-03-08. Infinite Translation Surfaces in the Wild. https://arxiv.org/abs/2403.05424

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