Square-tiled tori

TL;DR

This paper classifies square-tiled tori using invariants, counts them, and analyzes the orbits under SL(2,Z), providing exact sizes and asymptotic behavior, and answering a question about cyclic covers of the torus.

Contribution

It introduces a classification of SL(2,Z)-orbits of square-tiled tori using computable invariants and determines their sizes and asymptotic counts.

Findings

Classified SL(2,Z)-orbits of square-tiled tori.

Derived exact sizes of each orbit.

Established asymptotic behavior of cyclic square-tiled tori.

Abstract

We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})$-action on square-tiled tori and we classify $\mathrm{SL}(2,\mathbf{Z})$-orbits using two numerical invariants that can be easily computed. We deduce the exact size of every $\mathrm{SL}(2,\mathbf{Z})$-orbit. In particular, this answers a question by M. Bolognesi on the number of cyclic covers of the torus, which corresponds to particular $\mathrm{SL}(2,\mathbf{Z})$-orbits of square-tiled tori. We also give the asymptotic behavior of the number of cyclic square-tiled tori.