Wildest ${\mathrm{SL}}_2$-tilings

Abstract

Tame SL${}_2$-tilings are related to Farey graph and friezes; much less is known about wild (not tame) SL${}_2$-tilings. In this note, we demonstrate SL${}_2$-tilings that are maximally wild: we prove that the maximum wild density of an integer SL${}_2$-tiling is $\frac{2}{5}$ and present SL${}_2$-tilings over $\mathbb{Z}/N\mathbb{Z}$ with wild density 1.

7 pages