  Let $G$ be a connected semisimple real algebraic group. The class of
transverse subgroups of $G$ includes all discrete subgroups of rank one Lie
groups and any subgroups of Anosov or relative Anosov subgroups. Given a
transverse subgroup $\Gamma$, we show that the $\Gamma$-action on the Weyl
chamber flow space determined by its limit set is properly discontinuous. This
allows us to consider the quotient space and define Bowen-Margulis-Sullivan
measures. We then establish the ergodic dichotomy for the Weyl chamber flow, in
the original spirit of Hopf-Tsuji-Sullivan. We also introduce the notion of
growth indicators and discuss their properties and roles in the study of
conformal measures, extending the work of Quint. We discuss several
applications as well.
