Examples of Anosov diffeomorphisms

We prove that if is any graded rational Lie algebra, then the simply connected nilpotent Lie group N×N with Lie algebra has a lattice Γ such that the corresponding nilmanifold (N×N)/Γ admits an Anosov diffeomorphism. This gives a procedure to construct easily several explicit examples of nilmanifolds admitting an Anosov diffeomorphism, and shows that a reasonable classification up to homeomorphism (or even up to commensurability) of such nilmanifolds would not be possible.