Nearest-Neighbor Tight-Binding Realization of Hyperbolic Lattices with $\mathbb{Z}_2$ Gauge Structures

A systematic framework for realizing $\mathbb{Z}_2$ gauge extensions of hyperbolic lattices within the nearest-neighbor tight-binding formalism is developed. Using the triangle group $\Delta(2,8,8)$ as an example, we classify all inequivalent projective symmetry groups by computing the second cohomology group $H^2(\Delta(2,8,8),\mathbb{Z}_2)$. Each class corresponds to a distinct flux configuration and can be constructed by tight-binding models to verify the symmetry relations of the extended group. The translation subgroups of the $\mathbb{Z}_2$ extended lattices are associated with high genus surfaces, which follows the Riemann-Hurwitz formula. By applying the Abelian hyperbolic band theory, we find the all-flat dispersions along specific directions in momentum space and van Hove singularities correlated with discrete eigenenergies. Our results establish a general route to investigate gauge-extended hyperbolic lattices and provide a foundation for further studying symmetry fractionalization and spin liquid phases in non-Euclidean geometries.