Existence of Anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups

TL;DR

This paper investigates the existence of Anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups, providing a complete classification for this specific class.

Contribution

It completely characterizes which infra-nilmanifolds modeled on free nilpotent Lie groups admit Anosov diffeomorphisms, resolving a key open question in the field.

Findings

Classified infra-nilmanifolds based on free nilpotent Lie groups that admit Anosov diffeomorphisms

Provided necessary and sufficient conditions for the existence of Anosov diffeomorphisms in this setting

Extended understanding of the structure of infra-nilmanifolds supporting hyperbolic dynamics

Abstract

An infra-nilmanifold is a manifold which is constructed as a quotient space $\Gamma\backslash G$ of a simply connected nilpotent Lie group $G$, where $\Gamma$ is a discrete group acting properly discontinuously and cocompactly on $G$ via so called affine maps. The manifold $\Gamma\backslash G$ is said to be modeled on the Lie group $G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov diffeomorphism. However, it is far from obvious which of these infra--nilmanifolds actually do admit an Anosov diffeomorphism. In this paper we completely solve this question for infra-nilmanifolds modeled on a free $c$--step nilpotent Lie group.