  Periodic lattices in hyperbolic space are characterized by symmetries beyond
Euclidean crystallographic groups, offering a new platform for classical and
quantum waves, demonstrating great potentials for a new class of topological
metamaterials. One important feature of hyperbolic lattices is that their
translation group is nonabelian, permitting high-dimensional irreducible
representations (irreps), in contrast to abelian translation groups in
Euclidean lattices. Here we introduce a general framework to construct wave
eigenstates of high-dimensional irreps of infinite hyperbolic lattices, thereby
generalizing Bloch's theorem, and discuss its implications on unusual
mode-counting and degeneracy, as well as bulk-edge correspondence in hyperbolic
lattices. We apply this method to a mechanical hyperbolic lattice, and
characterize its band structure and zero modes of high-dimensional irreps.
