SL_2-Tilings Do Not Exist in Higher Dimensions (mostly)

We define a family of generalizations of ${\mathrm{SL}}_2$-tilings to higher dimensions called $\mathit{ϵ}$-${\mathrm{SL}}_2$-tilings. We show that, in each dimension 3 or greater, $\mathit{ϵ}$-${\mathrm{SL}}_2$-tilings exist only for certain choices of $\mathit{ϵ}$. In the case that they exist, we show that they are essentially unique and have a concrete description in terms of odd Fibonacci numbers.