On Anosov automorphisms of nilmanifolds

JORGE LAURET, CYNTHIA WILL

JOURNAL OF PURE AND APPLIED ALGEBRA

ELSEVIER SCIENCE BV

Año: 2008 vol. 212 p. 1747 - 1755

0022-4049

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 nilpotentLie algebra determined by the lattice, so called an Anosov Lie algebra.  We prove that $ gooplus...oplus go$ ($s$ times, $sgeq 2$) has an Anosov rational form for any graded realnilpotent Lie algebra $ go$ 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 algebraicintegers (even units) to show that the types $(5,3)$ and $(3,3,2)$ are not possible for Anosov Lie algebras.