Anosov diffeomorphisms on infra-nilmanifolds associated to graphs

TL;DR

This paper characterizes which infra-nilmanifolds associated with graphs admit Anosov diffeomorphisms, extending previous work on nilmanifolds and providing a criterion based on holonomy group actions.

Contribution

It generalizes the classification of Anosov diffeomorphisms to infra-nilmanifolds associated with graphs, offering a necessary and sufficient condition based on holonomy actions.

Findings

Derived a criterion for Anosov diffeomorphisms on infra-nilmanifolds associated to graphs.

Constructed examples with cyclic holonomy groups admitting Anosov diffeomorphisms.

Extended previous results from nilmanifolds to the broader class of infra-nilmanifolds.

Abstract

Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, i.e. a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restricted class of nilmanifolds, which are quotients of 1-connected nilpotent Lie groups by uniform lattices. For example, Dani and Mainkar studied this question for the nilmanifolds associated to graphs, which form the natural generalization of nilmanifolds modeled on free nilpotent Lie groups. This paper further generalizes their work to the full class of infra-nilmanifolds associated to graphs, leading to a necessary and sufficient condition depending only on the induced action of the holonomy group on the defining graph. As an application, we construct families of infra-nilmanifolds with cyclic holonomy groups admitting an Anosov diffeomorphism, starting from faithful actions of the holonomy group on simple graphs.