  Hyperbolic lattices are a revolutionary platform for tabletop simulations of
holography and quantum physics in curved space and facilitate efficient quantum
error correcting codes. Their underlying geometry is non-Euclidean, and the
absence of Bloch's theorem precludes the straightforward application of the
often indispensable energy band theory to study model Hamiltonians on
hyperbolic lattices. Motivated by recent insights into hyperbolic band theory,
we initiate a crystallography of hyperbolic lattices. We show that many
hyperbolic lattices feature a hidden crystal structure characterized by unit
cells, hyperbolic Bravais lattices, and associated symmetry groups. Using the
mathematical framework of higher-genus Riemann surfaces and Fuchsian groups, we
derive, for the first time, a list of example hyperbolic $\{p,q\}$ lattices and
their hyperbolic Bravais lattices, including five infinite families and several
graphs relevant for experiments in circuit quantum electrodynamics and
topolectrical circuits. This dramatically simplifies the computation of energy
spectra of tight-binding Hamiltonians on hyperbolic lattices, from exact
diagonalization on the graph to solving a finite set of equations in terms of
irreducible representations. The significance of this achievement needs to be
compared to the all-important role played by conventional Euclidean
crystallography in the study of solids. We exemplify the high potential of this
approach by constructing and diagonalizing finite-dimensional Bloch wave
Hamiltonians. Our work lays the foundation for generalizing some of the most
powerful concepts of solid state physics, crystal momentum and Brillouin zone,
to the emerging field of hyperbolic lattices and tabletop simulations of
gravitational theories, and reveals the connections to concepts from topology
and algebraic geometry.
