Nilmanifolds of dimension $\leq 8$ admitting Anosov diffeomorphisms

After more than thirty years, the only known examples of Anosov diffeomorphisms are topologically conjugated to hyperbolic automorphisms of infranilmanifolds, and even the existence of an Anosov automorphism is a really strong condition on an infranilmanifold. Any Anosov automorphism determines an automorphism of the rational Lie algebra determined by the lattice, which is hyperbolic and unimodular (and conversely ...). These two conditions together are strong enough to make of such rational nilpotent Lie algebras (called Anosov Lie algebras) very distinguished objects. In this paper, we classify Anosov Lie algebras of dimension less than or equal to 8.

As a corollary, we obtain that if an infranilmanifold of dimension $n\leq 8$ admits an Anosov diffeomorphism $f$ and it is not a torus or a compact flat manifold (i.e. covered by a torus), then $n=6$ or 8 and the signature of $f$ necessarily equals $\{ 3,3\}$ or $\{ 4,4\}$, respectively.