All $SL_2$-tilings come from infinite triangulations

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.