Codes for Square-Tiled Surfaces

TL;DR

This paper introduces a permutation-based method to analyze square-tiled surfaces, enabling enumeration of their $SL_2(Z)$ orbits and determination of affine diffeomorphisms.

Contribution

It develops a novel permutation encoding approach for cylinder decompositions, facilitating orbit enumeration and affine diffeomorphism detection for square-tiled surfaces.

Findings

Permutation elements effectively record cylinder decompositions.

Method allows enumeration of $SL_2(Z)$ orbits.

Provides a criterion to identify affine diffeomorphisms.

Abstract

In this paper, we use permutation elements to record cylinder decompositions of a square-tiled surface $X$. Collecting all such possible permutation elements that record cylinder decompositions, we can enumerate the $SL_2(\mathbb{Z})$ orbit of a given surface $X$ and give a method to determine whether or not a matrix $A\in SL_2(\mathbb{Z})$ is the differential of an affine diffeomorphism of $X$.