Anosov Lie Algebras and Algebraic Units in Number Fields

The nilmanifolds admitting Anosov automorphisms correspond to Anosov Lie algebras, which in turn determine very special algebraic units in number fields. By studying properties of these algebraic units using basic field theory, we can rule out many possibilities for a Lie algebra to be Anosov. This technique is useful for the classification of Anosov Lie algebras and hence the classification of nilmanifolds admitting Anosov automorphisms. We use our methods to study 13-dimensional Anosov Lie algebras. We also note that our method gives a simpler proof of the facts that there does not exist a 7-dimensional non-toral nilmanifold admitting an Anosov automorphism, and the types (5, 3) and (3, 3, 2) are not possible types for Anosov Lie algebras with no abelian factor.