Infinite friezes and triangulations of annuli

TL;DR

This paper explores the relationship between infinite friezes and triangulations of annuli, establishing a near-unique correspondence and analyzing the algebraic structures and growth properties involved.

Contribution

It demonstrates that each periodic infinite frieze uniquely determines a triangulation of an annulus and studies the mutual determination of frieze pairs and their algebraic properties.

Findings

Each periodic infinite frieze corresponds to a unique annulus triangulation.

Pairs of friezes determine each other and relate to module categories.

The growth coefficient of frieze pairs is expressed via modules and quiddity sequences.

Abstract

It is known that any infinite frieze comes from a triangulation of an annulus by Baur, Parsons and Tschabold. In this paper we show that each periodic infinite frieze determines a triangulation of an annulus in essentially a unique way. Since each triangulation of an annulus determines a pair of friezes, we study such pairs and show how they determine each other. We study associated module categories and determine the growth coefficient of the pair of friezes in terms of modules as well as their quiddity sequences.