  A long-standing conjecture asserts that any Anosov diffeomorphism of a closed
manifold is finitely covered by a diffeomorphism which is topologically
conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper,
we show that any closed 4-manifold that carries a Thurston geometry and is not
finitely covered by a product of two aspherical surfaces does not support
(transitive) Anosov diffeomorphisms.
