We generalize the hyperbolic fracton model from the $\{5,4\}$ tessellation to generic {$p,q$} tessellations, and investigate its core properties—ground-state degeneracy, fracton mobility, and holographic correspondence—bringing to light a richer and more intricate structure than the original case. The ground-state degeneracy and subsystem symmetries are computed exactly through the inflation rule, but do not admit a simple, symmetric pattern. For small connectivity $q=3$, we find that the degeneracy is finite, either unique or fourfold. For the {4,4} square lattice as the flat limit of $q=4$, the degeneracy is well known to be subextensive. But for all other tessellations of $q>3$, the degeneracy becomes extensive, including for the {3,6} lattice on flat space, which maps onto the honeycomb color code. The fracton excitation number follows exponential-in-distance and algebraic-in-lattice-size growing patterns when moving outward and depends sensitively on the tessellation geometry, differing qualitatively from both type-I and type-II fracton models on flat lattices. Despite this increased complexity, the hallmark holographic features—subregion duality via Rindler reconstruction, the Ryu-Takayanagi formula for mutual information, and effective black hole entropy scaling with horizon area—remain valid. These results demonstrate that the holographic correspondence in fracton models persists in generic tessellations and provides a natural platform to explore more intricate subsystem symmetries and fracton physics.

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• Fractons
• Spin liquid
• Exact solutions for many-body systems
• String theory techniques in condensed matter