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We define a family of generalizations of $\operatorname{SL}_2$-tilings tohigher dimensions called $\boldsymbol{\epsilon}$-$\operatorname{SL}_2$-tilings.We show that, in each dimension 3 or greater,$\boldsymbol{\epsilon}$-$\operatorname{SL}_2$-tilings exist only for certainchoices of $\boldsymbol{\epsilon}$. In the case that they exist, we show thatthey are essentially unique and have a concrete description in terms of oddFibonacci numbers.

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