  Wave functions on periodic lattices are commonly described by Bloch band
theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic
lattices support non-Abelian Bloch states that have so far eluded analytical
treatments. By adapting the solid-state-physics notions of supercells and zone
folding, we devise a method for the systematic construction of non-Abelian
Bloch states. The method applies Abelian band theory to sequences of
supercells, recursively built as symmetric aggregates of smaller cells, and
enables a rapidly convergent computation of bulk spectra and eigenstates for
both gapless and gapped tight-binding models. Our supercell method provides an
efficient means of approximating the thermodynamic limit and marks a pivotal
step towards a complete band-theoretic characterization of hyperbolic lattices.
