  Thurston conjectured that a closed triangulated 3-manifold in which every
edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell,
has word-hyperbolic fundamental group. We establish Thurston's conjecture by
proving that such a manifold admits a piecewise Euclidean metric of
non-positive curvature and the universal cover contains no isometrically
embedded flat planes. The proof involves a mixture of computer computation and
techniques from small cancellation theory.
