  We study the geometry and dynamics of discrete infinite covolume subgroups of
higher rank semisimple Lie groups. We introduce and prove the equivalence of
several conditions, capturing "rank one behavior'' of discrete subgroups of
higher rank Lie groups. They are direct generalizations of rank one equivalents
to convex cocompactness. We also prove that our notions are equivalent to the
notion of Anosov subgroup, for which we provide a closely related, but
simplified and more accessible reformulation, avoiding the geodesic flow of the
group. We show moreover that the Anosov condition can be relaxed further by
requiring only non-uniform unbounded expansion along the (quasi)geodesics in
the group.
  A substantial part of the paper is devoted to the coarse geometry of these
discrete subgroups. A key concept which emerges from our analysis is that of
Morse quasigeodesics in higher rank symmetric spaces, generalizing the Morse
property for quasigeodesics in Gromov hyperbolic spaces. It leads to the notion
of Morse actions of word hyperbolic groups on symmetric spaces,i.e. actions for
which the orbit maps are Morse quasiisometric embeddings, and thus provides a
coarse geometric characterization for the class of subgroups considered in this
paper. A basic result is a local-to-global principle for Morse quasigeodesics
and actions. As an application of our techniques we show algorithmic
recognizability of Morse actions and construct Morse "Schottky subgroups'' of
higher rank semisimple Lie groups via arguments not based on Tits' ping-pong.
Our argument is purely geometric and proceeds by constructing equivariant Morse
quasiisometric embeddings of trees into higher rank symmetric spaces.
