Geometry and Topology Seminar
The Toledo number is a numerical invariant associated to representations of fundamental groups of surfaces into Lie groups of Hermitian type. Maximal representations are those representations for which the Toledo number is maximal. They form connected components of the representation variety. In the case when the Lie group is SL(2,R)= Sp(2,R) they correspond precisely to holonomy representations of hyperbolic structures. Maximal representations into the symplectic group Sp(2n,R) generalize this situation with a lot of new features appearing. I will describe some of these new features and explain how maximal representations arise as homonym representations of projective structures on iterated sphere bundles over surfaces.