Unitary representations of the four-generator relation group approximate complex curves

Let $\Phi_1,\Phi_2,\Phi_3,\Phi_4\in U(N)$ satisfy the three relations defining the group in the source, and for an even permutation $(i j k l)$ impose the equivalent commutator relation

$$\Phi_i\Phi_j\Phi_i^{-1}\Phi_j^{-1}=\Phi_k\Phi_l\Phi_k^{-1}\Phi_l^{-1}.$$

Unitary representation conjecture. Representations of this group approximate, as $N\to+\infty$, complex curves in the abelian variety $(\mathbb{C}/(\mathbb{Z}+i\mathbb{Z}))^2$. This is presented as the first-order, calibrated counterpart of the second-order approximation of minimal surfaces in the flat torus $(S^1)^4$. The source provides no resolution or precise meaning of approximation.