Cutting sequences on square-tiled surfaces

TL;DR

This paper characterizes the cutting sequences of infinite geodesics on square-tiled surfaces using interval exchanges, extending understanding of these sequences to a dense subset of translation surfaces.

Contribution

It introduces a new symbolic coding method for cutting sequences on square-tiled surfaces via interval exchanges, expanding the class of surfaces with understood cutting sequences.

Findings

Cutting sequences correspond to symbolic trajectories of interval exchanges.

Special lifts of Sturmian sequences fully describe all cutting sequences.

Results apply to a dense subset of the moduli space of translation surfaces.

Abstract

We characterize cutting sequences of infinite geodesics on square-tiled surfaces by considering interval exchanges on specially chosen intervals on the surface. These interval exchanges can be thought of as skew products over a rotation, and we convert cutting sequences to symbolic trajectories of these interval exchanges to show that special types of combinatorial lifts of Sturmian sequences completely describe all cutting sequences on a square-tiled surface. Our results extend the list of families of surfaces where cutting sequences are understood to a dense subset of the moduli space of all translation surfaces.