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

TL;DR

This paper introduces higher-dimensional generalizations of SL_2-tilings, demonstrating their existence only under specific conditions in dimensions three or higher, and providing a concrete description when they do exist.

Contribution

It defines epsilon-SL_2-tilings in higher dimensions and proves their existence criteria, uniqueness, and explicit description using Fibonacci numbers.

Findings

Higher-dimensional epsilon-SL_2-tilings exist only for certain epsilon choices.

Such tilings are essentially unique when they exist.

They can be explicitly described using odd Fibonacci numbers.

Abstract

We define a family of generalizations of $\operatorname{SL}_2$-tilings to higher 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 certain choices of $\boldsymbol{\epsilon}$. In the case that they exist, we show that they are essentially unique and have a concrete description in terms of odd Fibonacci numbers.