On Anosov automorphisms of nilmanifolds

Abstract

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that $\mathfrak{n}\oplus \cdots \oplus \mathfrak{n}$ ($s$ times, $s\ge 2$) has an Anosov rational form for any graded real nilpotent Lie algebra $\mathfrak{n}$ having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types $(5,3)$ and $(3,3,2)$ are not possible for Anosov Lie algebras.