Handbook of c Higher Education Press Group Actions (Vol. IV) and International Press ALM 41, Ch. 5, pp. 191–290 Beijing-Boston Discrete Isometry Groups of Symmetric Spaces Michael Kapovich∗ and Bernhard Leeb† Abstract We survey our recent work, mostly with Joan Porti, on discrete subgroups of higher rank semisimple Lie groups exhibiting rank one behavior, with the main focus on Anosov subgroups. 2000 Mathematics Subject Classification: 22E40, 20F65, 53C35. Keywords and Phrases: symmetric spaces, buildings, Finsler geometry, discrete groups, Anosov subgroups. 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 [KLP2a, KLP2b, KLP3, KLP4, KLP1, KL1]. A shorter survey of our work appeared in [KLP5]. Our goal here is to: 1. Describe a class of discrete subgroups Γ <Gof higher rank semisimple Lie groups, to be called RA (regular antipodal) subgroups, which exhibit some “rank 1behavior”. 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 = ∗Department of Mathematics, University of California, Davis, CA 95616, USA and Korea Institute for Advanced Study, 207-43 Cheongnyangri-dong, Dongdaemun-gu, Seoul, South Korea. E-mail:
[email protected] †Mathematisches Institut, Universit¨at M¨unchen, Theresienstr. 39, D-80333 M¨unchen, Ger- many, E-mail:
[email protected] 192 Michael Kapovich and Bernhard Leeb G/K.