All $SL_2$-tilings come from infinite triangulations

TL;DR

This paper proves that all $SL_2$-tilings can be constructed from triangulations of the disc with a small number of accumulation points, extending previous results and including tilings with no entries equal to 1.

Contribution

It demonstrates that every $SL_2$-tiling originates from triangulations with few accumulation points, broadening the understanding of their structure beyond previous limitations.

Findings

All $SL_2$-tilings derive from triangulations via Conway-Coxeter counting.

Existence of tilings with finitely many or no entries equal to 1.

The minimal entry in certain tilings is uniquely determined.

Abstract

An $SL_2$-tiling is a bi-infinite matrix of positive integers such that each adjacent 2 by 2 submatrix has determinant 1. Such tilings are infinite analogues of Conway-Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory. We show that, by means of so-called Conway-Coxeter counting, every $SL_2$-tiling arises from a triangulation of the disc with two, three or four accumulation points. This improves earlier results which only discovered $SL_2$-tilings with infinitely many entries equal to 1. Indeed, our methods show that there are large classes of tilings with only finitely many entries equal to 1, including a class of tilings with no 1's at all. In the latter case, we show that the minimal entry of a tiling is unique.