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

### 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.

Edward R. Goetze

Ralf J. Spatzier