  Motivated by recent theoretical and experimental developments in the physics
of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian
groups that we call the hyperbolic Bloch transform. First, we prove that the
hyperbolic Bloch transform is injective and "asymptotically unitary" already in
the simplest case, that is when the Hilbert space is the regular representation
of the Fuchsian group, $\Gamma$. Second, when $\Gamma \subset \mathrm{PSU} (1,
1)$ acts isometrically on the hyperbolic plane, $\mathbb{H}$, and the Hilbert
space is $L^2 \left( \mathbb{H} \right)$, then we define a modified, geometric
Bloch transform, that sends wave functions to sections of stable, flat bundles
over $\Sigma = \mathbb{H} / \Gamma$ and transforms the hyperbolic Laplacian
into the covariant Laplacian.
