Nilmanifolds of dimension $\leq 8$ admitting Anosov diffeomorphisms

JORGE LAURET

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2009 vol. 361 p. 2377 - 2395

0002-9947

After more than thirty years, the only known examples of Anosov diffeomorphisms aretopological conjugated to hyperbolic automorphisms of infranilmanifolds, and eventhe existence of an Anosov automorphism is a really strong condition on aninfranilmanifold. Any Anosov automorphism determines an automorphism of the rationalLie algebra determined by the lattice, which is hyperbolic and unimodular (andconversely ...). These two conditions together are strong enough to make of suchrational nilpotent Lie algebras (called Anosov Lie algebras) very distinguishedobjects. In this paper, we classify Anosov Lie algebras of dimension less or equalthan 8.As a corollary, we obtain that if an infranilmanifold of dimension $nleq 8$ admitsan 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.