Surface group representations with maximal Toledo invariant

We develop the theory of maximal representations of the fundamental group 1 . / of a compact connected oriented surface (possibly with boundary) into Lie groups G of Hermitian type. For any homomorphism W 1 . / ! G, we define the Toledo invariant T. ; /, a numerical invariant which has both topological and analytical interpretations. We establish important properties of T. ; /, among which continuity, uniform boundedness on the representation variety, additivity under connected sum of surfaces and congruence relations mod . We thus obtain information about the representation variety as well as striking geometric properties of maximal representations, that is representations whose Toledo invariant achieves the maximum value.