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Periodic lattices in hyperbolic space are characterized by symmetries beyondEuclidean crystallographic groups, offering a new platform for classical andquantum waves, demonstrating great potentials for a new class of topologicalmetamaterials. One important feature of hyperbolic lattices is that theirtranslation group is nonabelian, permitting high-dimensional irreduciblerepresentations (irreps), in contrast to abelian translation groups inEuclidean lattices. Here we introduce a general framework to construct waveeigenstates of high-dimensional irreps of infinite hyperbolic lattices, therebygeneralizing Bloch's theorem, and discuss its implications on unusualmode-counting and degeneracy, as well as bulk-edge correspondence in hyperboliclattices. We apply this method to a mechanical hyperbolic lattice, andcharacterize its band structure and zero modes of high-dimensional irreps.

Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices