  It is conjectured that every manifold admitting an Anosov diffeomorphism is,
up to homeomorphism, finitely covered by a nilmanifold. Motivated by this
conjecture, an important problem is to determine which nilmanifolds admit an
Anosov diffeomorphism. The main theorem of this article gives a general method
for constructing Anosov diffeomorphisms on nilmanifolds. As a consequence, we
give counterexamples to a corollary of the classification of low-dimensional
nilmanifolds with Anosov diffeomorphisms and a correction to this statement is
proven. This method also answers some open questions about the existence of
Anosov diffeomorphisms which are minimal in some sense.
