Anosov diffeomorphisms on Thurston geometric 4-manifolds

TL;DR

This paper proves that most closed 4-manifolds with Thurston geometries do not support transitive Anosov diffeomorphisms, confirming a conjecture in a specific geometric setting.

Contribution

It establishes the non-existence of transitive Anosov diffeomorphisms on certain 4-manifolds with Thurston geometries, advancing understanding of dynamical systems on geometric manifolds.

Findings

Most Thurston 4-manifolds do not support transitive Anosov diffeomorphisms.

Manifolds covered by a product of two aspherical surfaces are exceptions.

Supports the conjecture relating Anosov diffeomorphisms to nilpotent automorphisms.

Abstract

A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov diffeomorphisms.