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2023•Discrete and Continuous Dynamical SystemsOpen access

Square-tiled tori

Angel Pardo, Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile

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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, {\bf{Z}})} $-action on square-tiled tori and we classify $ {\mathrm{SL}(2, {\bf{Z}})} $-orbits using two numerical invariants that can be easily computed. We deduce the exact size of every $ {\mathrm{SL}(2, {\bf{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, {\bf{Z}})} $-orbits of square-tiled tori. We also give the asymptotic behavior of the number of cyclic square-tiled tori.

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What this paper is about

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, {\bf{Z}})} $-action on square-tiled tori and we classify $ {\mathrm{SL}(2, {\bf{Z}})} $-orbits using two numerical invariants that can be easily computed. We deduce the exact size of every $ {\mathrm{SL}(2, {\bf{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, {\bf{Z}})} $-orbits of square-tiled tori. We also give the asymptotic behavior of the number of cyclic square-tiled tori.

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Available 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, {\bf{Z}})} $-action on square-tiled tori and we classify $ {\mathrm{SL}(2, {\bf{Z}})} $-orbits using two numerical invariants that can be easily computed. We deduce the exact size of every $ {\mathrm{SL}(2, {\bf{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, {\bf{Z}})} $-orbits of square-tiled tori. We also give the asymptotic behavior of the number of cyclic square-tiled tori.

Key concepts: Torus, Square (algebra), Action (physics), Combinatorics, Unit square, Natural number, Mathematics, Physics

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