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

Abstract

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

4 pages