On 3-manifolds that support partially hyperbolic diffeomorphisms

TL;DR

This paper classifies 3-manifolds supporting partially hyperbolic diffeomorphisms based on their fundamental group properties and explores conditions for dynamical coherence, including new results on hyperbolic manifolds and higher dimensions.

Contribution

It provides a classification of 3-manifolds supporting partially hyperbolic diffeomorphisms and establishes new conditions for dynamical coherence, including for hyperbolic manifolds and higher dimensions.

Findings

Manifolds with nilpotent fundamental group have partially hyperbolic actions on homology.

Certain fundamental group types imply the manifold is finitely covered by specific fiber bundles.

Existence of hyperbolic 3-manifolds without dynamically coherent partially hyperbolic diffeomorphisms.

Abstract

Let M be a closed 3-manifold that supports a partially hyperbolic diffeomorphism f. If $\pi_1(M)$ is nilpotent, the induced action of f on $H_1(M, R)$ is partially hyperbolic. If $\pi_1(M)$ is almost nilpotent or if $\pi_1(M)$ has subexponential growth, M is finitely covered by a circle bundle over the torus. If $\pi_1(M)$ is almost solvable, M is finitely covered by a torus bundle over the circle. Furthermore, there exist infinitely many hyperbolic 3-manifolds that do not support dynamically coherent partially hyperbolic diffeomorphisms; this list includes the Weeks manifold. If f is a strong partially hyperbolic diffeomorphism on a closed 3-manifold M and if $\pi_1(M)$ is nilpotent, then the lifts of the stable and unstable foliations are quasi-isometric in the universal of M. It then follows that f is dynamically coherent. We also provide a sufficient condition for dynamical coherence in any dimension. If f is center bunched and if the center-stable and center-unstable distributions are Lipschitz, then the partially hyperbolic diffeomorphism f must be dynamically coherent.