Classifying SL₂-tilings

Short, Ian (2023). Classifying SL₂-tilings. Transactions of the American Mathematical Society, 376 pp. 1–38.

DOI: https://doi.org/10.1090/tran/8296

Abstract

Recently there has been significant progress in classifying integer friezes and SL₂-tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame SL₂-tilings, positive integer SL₂-tilings, and tame integer friezes - both finite and infinite. In so doing, we obtain geometric analogues of certain known combinatorial models for tilings involving triangulations, and we prove several new results of a similar type too. For instance, we determine those bi-infinite sequences of positive integers that are the quiddity sequence of some positive infinite frieze, and we give a simple combinatorial model for classifying tame integer friezes, which generalises the classical construction of Conway and Coxeter for positive integer friezes.

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