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

Morse actions of discrete groups on symmetric space