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

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 coeffi cients. We relate this notion to triangulated $n$-gons in the Farey graph.