The Hopf-Tsuji-Sullivan dichotomy in higher rank and applications to Anosov subgroups

Abstract

We establish an extension of the Hopf-Tsuji-Sullivan dichotomy to any Zariski dense discrete subgroup of a semisimple real algebraic group $G$. We then apply this dichotomy to Anosov subgroups of $G$, which surprisingly presents a different phenomenon depending on the rank of the ambient group $G$.