Finite presentation of the subgroup G ⊂ SL2(C) generated by X=βα, Y=α^{-1}γ, Z=β^{-1}γ^{-1}

Prove that the subgroup G of SL2(C) generated by X = βα, Y = α^{-1}γ, and Z = β^{-1}γ^{-1} admits the following finite presentation (equivalently stated in two forms): either G = ⟨X, Y, Z | XXX = yXyX = yzYX = YzzX = yZxYzX = −I⟩, or equivalently G = ⟨X, Y, Z | ZxxYXXX = yZyZYYzYX = ZZxZYzXzX = I, XXX = yZxYzX = yXyX = −I⟩, where I is the 2×2 identity matrix and −I is the diagonal matrix with −1 on the diagonal.

Background

After introducing X = βα, Y = α^{-1}γ, and Z = β^{-1}γ^{-1}, the authors paper the subgroup G they generate inside SL2(C), motivated by connections to tiling relations and double-dimer monodromies. Computational searches enumerate short identity words and reductions derived from tile boundaries, leading to a small candidate set of fundamental relations.

These reductions suggest that all necessary relations can be captured by a finite set corresponding to halves of bone and snake relations (and certain closed-loop relations such as the barbell and 2×2×2 region). Based on this evidence, the authors conjecture the explicit finite presentation above, given in two equivalent forms.