SL$_2 (\mathbb Z)$-tilings of the torus, Coxeter–Conway friezes and Farey triangulations

Sophie Morier-Genoud; Valentin Ovsienko; Serge Tabachnikov

doi:10.4171/lem/61-1/2-4

JournalslemVol. 61, No. 1/2pp. 71–92

SL $_{2} (Z)$ -tilings of the torus, Coxeter–Conway friezes and Farey triangulations

Sophie Morier-Genoud
Université Pierre et Marie Curie, Paris, France
Valentin Ovsienko
U.F.R. Sciences Exactes et Naturelles, Reims, France
Serge Tabachnikov
Penn State University, University Park, United States

$SL$_2 (\mathbb Z)$-tilings of the torus, Coxeter–Conway friezes and Farey triangulations cover$

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Abstract

The notion of SL $_{2}$ -tiling is a generalization of that of classical Coxeter–Conway frieze pattern. We classify doubly antiperiodic SL $_{2}$ -tilings that contain a rectangular domain of positive integers. Every such SL $_{2}$ -tiling corresponds to a pair of frieze patterns and a unimodular $2 \times 2$ -matrix with positive integer coefficients. We relate this notion to triangulated $n$ -gons in the Farey graph.

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Sophie Morier-Genoud, Valentin Ovsienko, Serge Tabachnikov, SL $_{2} (Z)$ -tilings of the torus, Coxeter–Conway friezes and Farey triangulations. Enseign. Math. 61 (2015), no. 1/2, pp. 71–92

DOI 10.4171/LEM/61-1/2-4