  Tessellations of the hyperbolic spaces by regular polygons are becoming
popular because they support discrete quantum and classical models displaying
unique spectral and topological characteristics. Resolving the true bulk
spectra and the thermodynamic response functions of these models requires
converging periodic boundary conditions and our work delivers a practical
solution for this open problem on generic {p,q}-tessellations. This enables us
to identify the true spectral gaps of bulk Hamiltonians and, as an application,
we construct all but one topological models that deliver the topological gaps
predicted by the K-theory of the lattices. We demonstrate the emergence of the
expected topological spectral flows whenever two such bulk models are deformed
into each other and, additionally, we prove the emergence of topological
channels whenever a soft physical interface is created between different
topological classes of Hamiltonians.
