Square-tiled tori

Angel Pardo

doi:10.3934/dcds.2022185

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arXiv · 1506.02826

Square-tiled tori

Angel Pardo

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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 $\mathrm{SL}(2,\mathbf{Z})$-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.

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Angel Pardo. 2015-06-09. Square-tiled tori. https://doi.org/10.3934/dcds.2022185

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