Wildest ${\mathrm{S}\mathrm{L}}_2$-tilings

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 $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ with wild density 1.