Transverse groups and relatively Anosov groups

Transverse subgroups of semi-simple Lie groups include all discrete subgroups of rank one Lie groups and Anosov and relatively Anosov subgroups of higher rank Lie groups. Every transverse group comes equipped with a  family of “geodesic flows’’ associated to linear functionals on the Cartan subspace. The assumption of transversality allows for the construction of well-behaved Patterson-Sullivan measures on the limit set and associated Bowen-Margulis-Sullivan measures on the flows.  We discuss a Hopf-Tsuji-Sullivan dichotomy in this context. Finally, we show that relatively Anosov groups live on the divergent side of this dichotomy (Joint work with Tengren Zhang and Andy Zimmer)