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