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.
Anosov diffeomorphisms are an important class of dynamical systems with many peculiar properties. Ever since they were introduced in the sixties, it has been an open question which manifolds can admit them, with for example a positive answer for tori of dimension greater than or equal to two. A natural generalization for tori is to consider nilmanifolds of nilpotency class two, for which the existence of Anosov diffeomorphisms is completely determined by the associated rational 2-step nilpotent Lie algebra. From a given simple undirected graph, one can construct a complex 2-step nilpotent Lie algebra, which in general contains different non-isomorphic rational forms. In this paper we use the theory of Galois descent to give a description of all these rational forms in Lie algebras associated to graphs and determine precisely which ones correspond to a nilmanifold admitting an Anosov diffeomorphism. This is the first class of complex nilpotent Lie algebras having several non-isomorphic rational forms and for which all the ones that are Anosov are described.
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