Department of Mathematics

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.