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

Abstract

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.