Frieze patterns and Farey complexes

Frieze patterns have attracted significant attention recently, motivated by their relationship with cluster algebras. A longstanding open problem has been to provide a combinatorial model for frieze patterns over the ring of integers modulo $n$ akin to Conway and Coxeter's celebrated model for positive integer frieze patterns. Here we solve this problem using the Farey complex of the ring of integers modulo $n$; in fact, using more general Farey complexes we provide combinatorial models for frieze patterns over any rings whatsoever. Our strategy generalises that of the first author and of Morier-Genoud et al. for integers and that of Felikson et al. for Eisenstein integers. We also generalise results of Singerman and Strudwick on diameters of Farey graphs, we recover a theorem of Morier-Genoud on enumerating friezes over finite fields, and we classify those frieze patterns modulo $n$ that lift to frieze patterns over the integers in terms of the topology of the corresponding Farey complexes.