Statistics of Square-tiled Surfaces: Symmetry and Short Loops

TL;DR

This paper investigates the properties and symmetries of square-tiled surfaces, analyzing their frequency and structure through experimental and theoretical methods to deepen understanding of their geometric and dynamical features.

Contribution

It introduces new counting methods and explores the relationships between properties of square-tiled surfaces across different strata.

Findings

Identified patterns in the distribution of properties among square-tiled surfaces.

Established links between symmetry features and frequency of properties.

Provided experimental evidence supporting theoretical conjectures.

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 paper, 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 their frequency in each stratum of translation surfaces.