Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices

TL;DR

This paper classifies smooth volume-preserving Cartan actions of higher rank semisimple Lie groups and their lattices, extending to certain Anosov actions when the real rank is at least 3.

Contribution

It provides a new smooth classification for Cartan actions of higher rank semisimple Lie groups and their lattices, including Anosov actions in higher rank cases.

Findings

C^ classification of Cartan actions

Classification of Anosov actions for rank  or higher

Extension to volume-preserving actions

Abstract

Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let \Gamma \subset G be an irreducible lattice. We provide a C^\infty classification for volume-preserving Cartan actions of \Gamma and G. Also, if G has real rank at least 3, we provide a C^\infty classification for volume-preserving, multiplicity free, trellised, Anosov actions on compact manifolds.