Abstract. The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n, R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmüller spaces, and for lattices in SO(1, n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation Γ → G we explicitly construct open subsets of compact G-spaces, on which Γ acts properly discontinuously and with compact quotient.As a consequence we show that higher Teichmüller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford-Klein forms and compactifications of locally symmetric spaces of infinite volume.

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We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.Comment: 73 pages, 4 figures; to appear in Geometry & Topolog

Abstract. Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Γ g be the fundamental group of a compact orientable surface of genus g ≥ 2.We survey the study of maximal representations of Γ g into G, that is the subset of Hom(Γ g , G) characterized by the maximality of the Toledo invariant ([17] and [15]). Then we concentrate on the particular case G = Sp(2n, R), and we show that if ρ is any maximal representation then the image ρ(Γ g ) is a discrete, faithful realizations of Γ g as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.

We develop the theory of maximal representations of the fundamental group