Anosov representations are discrete, faithful (or finite-kernel) representations of word hyperbolic groups into semisimple Lie groups, with strong dynamical properties. They were introduced by Labourie in 2006 for fundamental groups of closed negatively-curved manifolds, and generalized by Guichard and Wienhard in 2012. They have been much studied in the past few years, and play an important role in higher Teichmüller-Thurston theory and in recent developments in the theory of discrete subgroups of Lie groups. We will introduce these representations, give examples, and discuss some characterizations.

Anosov representations of word hyperbolic groups into reductive Lie groups provide a generalization of convex cocompact representations to higher real rank. I will explain how these representations can be used to construct properly discontinuous actions on homogeneous spaces. For a rank-one simple group G, this construction covers all proper actions on G, by left and right multiplication, of quasi-isometrically embedded discrete subgroups of G×G; in particular, such actions remain proper after small deformations, and we can describe them explicitly. This is joint work with F. Guéritaud, O. Guichard, and A. Wienhard.