2004/06/09 by Jorge Lauret, Cynthia Will, Lauret, Jorge +2
Mathematics · #Advanced Topics in Algebra #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DG #math.DS
paper · pdf · doi:10.48550/arxiv.math/0406199
Corrected version: added Remarks 2.9, 2.11, 2.15, 5.1; new version of Proposition 3.3 and some minor changes in the proofs of Sections 4 and 5 (37 pages)
openalex publication_date 2004/06/09 · arxiv created 2004/07/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.