Maximal Representations of Surface Groups: Symplectic Anosov Structures

Let G be a connected semisimple Lie group such that the associ- ated symmetric space X is Hermitian and let Γg be the fundamental group of a compact orientable surface of genus g ≥ 2. We survey the study of maximal representations of Γg into G, that is the subset of Hom(Γg ,G ) characterized by the maximality of the Toledo invariant ((17) and (15)). Then we concen- trate on the particular case G =S p(2n, R), and we show that if ρ is any maximal representation then the image ρ(Γg) is a discrete, faithful realiza- tions of Γg 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.