Frieze patterns and Farey complexes

TL;DR

The paper develops Farey complexes $F_{R,U}$ as a unifying combinatorial framework to model tame $SL_2(R)$-tilings and friezes over arbitrary rings. By exploiting group actions, path itineraries, and covering theory, it establishes bijections between $SL_2(R)$-orbits of path data and both tilings and friezes, enabling systematic construction, classification, and lifting results. It provides explicit diameter and surface-geometry results for Farey complexes, enumerates friezes over finite fields, and gives precise lifting criteria from modular to integer settings. These results extend Conway-Coxeter style models to a broad algebraic context, linking combinatorics, hyperbolic geometry, and number theory, with practical implications for lifting and counting across rings. The work thus offers a cohesive, ring-agnostic framework for understanding friezes and $SL_2$-tilings with broad applications to algebraic and geometric structures.

Abstract

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.

Figures (11)

• Figure 1.1: A frieze (left) and a diamond of four entries (right)
• Figure 1.2: A tame semiregular frieze over $\mathbb{Q}$
• Figure 1.3: Tame $\text{SL}_2$-tilings over $\mathbb{Z}$ (left) and $\mathbb{Z}/5\mathbb{Z}$ (right)
• Figure 1.4: Part of the Farey complex $\mathscr{F}_{\mathbb{Z}}$
• Figure 1.5: Farey complexes $\mathscr{F}_2$, $\mathscr{F}_3$, $\mathscr{F}_4$, $\mathscr{F}_5$, and $\mathscr{F}_6$

Theorems & Definitions (102)

• Definition
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• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6