Anosov diffeomorphisms on nilmanifolds up to dimension 8

TL;DR

This paper classifies all Anosov Lie algebras of dimension up to 8, providing a complete understanding of nilmanifolds with Anosov diffeomorphisms in these dimensions and identifying specific signatures for such manifolds.

Contribution

It offers the first classification of low-dimensional Anosov Lie algebras and nilmanifolds, expanding the understanding of Anosov diffeomorphisms beyond known examples.

Findings

Classified all Anosov Lie algebras up to dimension 8.

Identified that non-torus infranilmanifolds with Anosov diffeomorphisms in dimensions less than 9 are limited to specific signatures.

Established that such manifolds occur only in dimensions 6 and 8 with signatures {3,3} and {4,4} respectively.

Abstract

After more than thirty years, the only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. It is also important to note that 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 of the Mal'cev completion of the corresponding lattice which is hyperbolic and unimodular. 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 or equal than 8, which also classify nilmanifolds admitting an Anosov diffeomorphism in those dimensions. As a corollary we obtain that if an infranilmanifold of dimension n<9 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. We had to study the set of all rational forms up to isomorphism for many real Lie algebras, which is a subject on its own and it is treated in a section completely independent of the rest of the paper.