$SL_2(\mathbb{Z})$-tilings of the torus, Coxeter-Conway friezes and   Farey triangulations
  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 $2\times2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph. 
Sophie Morier-Genoud, Valentin Ovsienko, Serge Tabachnikov
math.CO math.NT