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

TL;DR

This paper links the cohomology of hyperbolic manifolds with algebraic invariants of their fundamental groups through BGG complexes, revealing geometric solutions to natural PDEs and new insights into cohomology arising in geometric contexts.

Contribution

It establishes an isomorphism between group cohomology and BGG complex cohomology on hyperbolic manifolds, providing a geometric interpretation and new proofs for non-trivial cohomology.

Findings

Group cohomology is isomorphic to BGG complex cohomology on hyperbolic manifolds.

Solutions to natural PDEs are parameterized by cohomology groups.

New proof that certain cohomology groups are non-zero when the manifold contains a totally geodesic hypersurface.

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.