  We study Patterson-Sullivan measures for a class of discrete subgroups of
higher rank semisimple Lie groups, called transverse groups, whose limit set is
well-defined and transverse in a partial flag variety. This class of groups
includes both Anosov and relatively Anosov groups, as well as all discrete
subgroups of rank one Lie groups. We prove an analogue of the
Hopf-Tsuji-Sullivan dichotomy and then use this dichotomy to prove a variant of
Burger's Manhattan curve theorem. We also use the Patterson-Sullivan measures
to obtain conditions for when a subgroup has critical exponent strictly less
than the original transverse group. These gap results are new even for Anosov
groups.
