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

Abstract

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.