Tractor calculus, BGG complexes, and the cohomology of cocompact Kleinian groups

For a compact, oriented, hyperbolic n-manifold (M,g), realised as M=Γ\Hn where Γ 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 Γ. In particular for F an irreducible SO(n,1)-module, we show that the group cohomology with coefficients H
- (Γ,F) arises from the cohomology of an appropriate projective BGG complex on M. This yields the geometric interpretation that H
- (Γ,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 H1(Γ,S0kRn+1)≠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.