  For a compact, oriented, hyperbolic $n$-manifold $(M,g)$, realised as $M=
\Gamma \backslash \mathbb{H}^{n}$ where $\Gamma$ is a torsion-free cocompact
subgroup of $SO(n,1)$, we establish and study a relationship between
differential geometric cohomology on $M$ and algebraic invariants of the group
$\Gamma$. In particular for $\mathbb{F}$ an irreducible $SO(n,1)$-module, we
show that the group cohomology with coefficients $H^{\bullet}(\Gamma,
\mathbb{F})$ is isomorphic to the cohomology of an appropriate projective BGG
complex on $M$. This yields the geometric interpretation that
$H^{\bullet}(\Gamma, \mathbb{F})$ parameterises solutions to certain
distinguished natural PDEs of Riemannian geometry, modulo the range of suitable
differential coboundary operators. Viewed in another direction, the
construction shows one way that non-trivial cohomology can arise in a BGG
complex, and sheds considerable light on its geometric meaning. We also use the
tools developed to give a new proof that $H^{1} (\Gamma, S_{0}^{k}
\mathbb{R}^{n+1}) \neq 0$ whenever $M$ contains a compact, orientable, totally
geodesic hypersurface. All constructions use another result that we establish,
namely that the canonical flat connection on a hyperbolic manifold coincides
with the tractor connection of projective differential geometry.
