Global Rigidity of Codimension One Actions

TL;DR

This paper classifies certain smooth, codimension-one actions of higher-rank simple Lie groups on manifolds, showing they are either suspensions of actions on parabolic subgroups or covers of actions on product spaces, extending previous results.

Contribution

It provides a smooth classification of codimension-one, higher-rank Lie group actions under mixing conditions, generalizing Nevo and Zimmer's results to a smooth setting.

Findings

Actions are either suspensions of parabolic subgroup actions or covers of actions on G/Γ×S^1.

The classification relies on integration of Pesin manifolds.

Extends rigidity results to smooth actions, not just measurable ones.

Abstract

Consider a smooth, locally free, codimension-one action of a higher-rank, simple, split Lie group $G$ on a closed manifold $M$. Let $P$ be a minimal parabolic subgroup of $G$. If the action admits a $P$-invariant probability measure that is mixing, then the action is either equivariantly diffeomorphic to the suspension of a codimension one, locally free action on a closed manifold of a parabolic subgroup of $G$; or, it is finitely and equivariantly covered by the action of $G$ on $G/\Gamma\times S^1$, where the action on $G/\Gamma$ is the coset action, and $G$ acts trivially on $S^1$. We prove this by doing a jointly integration argument of stable and center unstable Pesin manifolds. This is a smooth version of results by Nevo and Zimmer.