Statistics of Square-Tiled Surfaces: Symmetry and Short Loops

Abstract

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this article, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between these properties and as well as the frequencies of these properties in each stratum of translation surfaces.

KEYWORDS:

• Square-tiled surfaces
• translation surfaces
• Veech group
• saddle connection
• holonomy vectors

Declaration of interest

No potential conflict of interest was reported by the author(s).

Acknowledgments

We are extremely grateful to Moon Duchin for organizing the Polygonal Billiards Research Cluster where this project originated, for suggesting this project, and for her constant support and valuable feedback. We are also grateful for the feedback and conversations with Samuel Lelièvre. We are grateful for Madeleine Elyze and Luis Kumanduri for helping during the initial explorations. We thank Vincent Delecroix and Justin Lanier for the prompt and helpful feedback on our first version. We also express our many thanks to the participants and visitors of the Billiards research cluster for numerous helpful discussions. Finally, we would like to thank the referee for many helpful comments and suggestions.

Additional information

Funding

This work began in the Polygonal Billiards Research Cluster held at Tufts University in Summer 2017 and was supported by the National Science Foundation under grant [DMSCAREER-1255442]. The second author was also supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374.