  A few years ago, the first example of a closed manifold admitting an Anosov
diffeomorphism but no expanding map was given. Unfortunately, this example is
not explicit and is high-dimensional, although its exact dimension is unknown
due to the type of construction. In this paper, we present a family of concrete
12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map,
where nilmanifolds are defined as the quotient of a 1-connected nilpotent Lie
group by a cocompact lattice. We show that this family has the smallest
possible dimension in the class of infra-nilmanifolds, which is conjectured to
be the only type of manifolds admitting Anosov diffeomorphisms up to
homeomorphism. The proof shows how to construct positive gradings from the
eigenvalues of the Anosov diffeomorphism under some additional assumptions
related to the rank, using the action of the Galois group on these algebraic
units.
