  Motivated by the recent experimental realizations of hyperbolic lattices in
circuit quantum electrodynamics and in classical electric-circuit networks, we
study flat bands and band-touching phenomena in such lattices. We analyze
noninteracting nearest-neighbor hopping models on hyperbolic analogs of the
kagome and dice lattices with heptagonal and octagonal symmetry. We show that
two characteristic features of the energy spectrum of those models, namely the
fraction of states in the flat band as well as the number of touching points
between the flat band and the dispersive bands, can both be captured exactly by
a combination of real-space topology arguments and a reciprocal-space
description via the formalism of hyperbolic band theory. Furthermore, using
real-space numerical diagonalization on finite lattices with periodic boundary
conditions, we obtain new insights into higher-dimensional irreducible
representations of the non-Euclidean (Fuchsian) translation group of hyperbolic
lattices. First, we find that the fraction of states in the flat band is the
same for Abelian and non-Abelian hyperbolic Bloch states. Second, we find that
only Abelian states participate in the formation of touching points between the
flat and dispersive bands.
