# Classifying SL$_2$-tilings

## TL;DR

This paper introduces a unified geometric approach using hyperbolic plane tessellations to classify various types of SL2-tilings and integer friezes, extending classical combinatorial models with new results.

## Contribution

It develops a geometric framework based on Farey graph tessellations for classifying tame and positive SL2-tilings and integer friezes, providing new insights and models.

## Key findings

- Classifies bi-infinite sequences as quiddity sequences of positive infinite friezes.
- Provides a geometric analogue for classical combinatorial models of friezes.
- Determines conditions for sequences to correspond to positive infinite friezes.

## Abstract

Recently there has been significant progress in classifying integer friezes and $\text{SL}_2$-tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame $\text{SL}_2$-tilings, positive integer $\text{SL}_2$-tilings, and tame integer friezes -- both finite and infinite. In so doing, we obtain geometric analogues of certain known combinatorial models for tilings involving triangulations, and we prove several new results of a similar type too. For instance, we determine those bi-infinite sequences of positive integers that are the quiddity sequence of some positive infinite frieze, and we give a simple combinatorial model for classifying tame integer friezes, which generalises the classical construction of Conway and Coxeter for positive integer friezes.