The Zimmer Program for partially hyperbolic actions

TL;DR

This paper advances the Zimmer program by classifying certain volume-preserving actions of semisimple Lie groups as smoothly conjugate to algebraic models under specific hyperbolicity and accessibility conditions.

Contribution

It proves smooth conjugacy to bi-homogeneous models for a class of partially hyperbolic actions, extending the classification within the Zimmer program.

Findings

Smooth conjugacy to algebraic models established

Classification results for higher-rank abelian group actions

Progress in understanding actions with partial hyperbolicity and accessibility

Abstract

Zimmer's superrigidity theorems on higher rank Lie groups and their lattices launched a program of study aiming to classify actions of semisimple Lie groups and their lattices, known as the {\it Zimmer program}. When the group is too large relative to the dimension of the phase space, the Zimmer conjecture predicts that the actions are all virtually trivial. At the other extreme, when the actions exhibit enough regular behavior, the actions should all be of algebraic origin. We make progress in the program by showing smooth conjugacy to a bi-homogeneous model (up to a finite cover) for volume-preserving actions of semisimple Lie groups without compact or rank one factors, which have two key assumptions: partial hyperbolicity for a large class of elements ({\it totally partial hyperbolicity}) and accessibility, a condition on the webs generated by dynamically-defined foliations. We also obtain classification for actions of higher-rank abelian groups satisfying stronger assumptions.