Nil-Anosov actions

TL;DR

This paper introduces Nil-Anosov actions of Lie groups on manifolds, proves a spectral decomposition for nilpotent groups, and characterizes actions with codimension-one unstable bundles as Nil-extensions over simpler Anosov actions.

Contribution

It defines Nil-Anosov actions, proves their stable/unstable foliation invariance, establishes a spectral decomposition theorem, and classifies certain actions as Nil-extensions over abelian or flow actions.

Findings

Spectral decomposition for Nil-Anosov actions of nilpotent groups.

Actions with codimension-one unstable bundle are Nil-extensions over abelian or flow actions.

Topological transitivity for actions with dimension n ≥ 3.

Abstract

We consider Anosov actions of a Lie group $G$ of dimension $k$ on a closed manifold of dimension $k+n.$We introduce the notion of Nil-Anosov action of $G$ (which includes the case where $G$ is nilpotent) and establishes the invariance by the entire group $G$of the associated stable and unstable foliations. We then prove a spectral decomposition Theoremfor such an action when the group $G$ is nilpotent. Finally, we focus on the case where $G$ is nilpotent andthe unstable bundle has codimension one. We prove that in this case the action is a Nil-extensionover an Anosov action of an abelian Lie group. In particular:i) if $n \geq 3,$ then the action is topologically transitive,ii) if $n=2,$ then the action is a Nil-extension over an Anosov flow.