This survey is based on a series of lectures that we gave at MSRI in Spring
2015 and on a series of papers, mostly written jointly with Joan Porti. Our
goal here is to:
1. Describe a class of discrete subgroups Γ<G of higher rank
semisimple Lie groups, which exhibit some "rank 1 behavior".
2. Give different characterizations of the subclass of Anosov subgroups,
which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of
various equivalent dynamical and geometric properties (such as asymptotically
embedded, RCA, Morse, URU).
3. Discuss the topological dynamics of discrete subgroups Γ on flag
manifolds associated to G and Finsler compactifications of associated
symmetric spaces X=G/K. Find domains of proper discontinuity and use them to
construct natural bordifications and compactifications of the locally symmetric
spaces X/Γ.Comment: 77 page