Surface Group Representations with Maximal Toledo Invariant

We develop the theory of maximal representations of the fundamental group π1(Σ) of a compact connected oriented surface Σ with boundary ∂Σ, into the isometry group of a Hermitian symmetric space X or, more generally, a group of Hermitian type G. For any homomorphism ρ : π1(Σ) → G, we define the Toledo invariant T(Σ, ρ), a numerical invariant which is in general not a characteristic number, but which has both topological and analytical interpretations: we establish important properties, among which uniform boundedness on the representation variety Hom ( π1(Σ), G ) , additivity under connected sum of surfaces and congruence relations mod Z. We thus obtain information about the representation variety as well as striking geometric properties of the maximal representations, that is representations whose Toledo invariant achieves the maximum value: we show that maximal representations have discrete image, are faithful and completely reducible and they always preserve a maximal tube type subdomain of X . This extends to the case of a general Hermitian group some of the properties of the representations in Teichmüller space, as well as results due to Goldman [32, 34], Toledo [61], Hernández [41], Bradlow, Garćıa-Prada and Gothen [6, 5]. The congruence relations for T(Σ, ρ) involve a rotation number function which we define and study for any Hermitian group G; this is related to a continuous homogeneous quasimorphism on G̃, which in turn gives an explicit way to compute the Toledo invariant. This rotation number generalizes constructions due to Ghys [31], Barge and Ghys [1], and Clerc and Koufany [20]. We establish moreover properties of boundary maps associated to maximal representations which generalize naturally, for the causal structure of the Shilov boundary, monotonicity properties of quasiconjugations of the circle. This, together with the congruence relations leads to the result that the subset of maximal representations is always real semialgebraic. The theory developed here relies on bounded cohomology techniques developed in [55], [17], [11], and uses results from [13], and [14]. An announcement of some of these results in the case of surfaces without boundary can be found in [15] – where the role of tube type domains had already been emphasized – and a survey of them, as well as related properties of maximal representations, can be found in [8]. Date: 27th February 2009. A.I. and A.W. were partially supported by FNS grant PP002-102765. A.W. was partially supported by the National Science Foundation under agreement No. DMS-0111298 and No. DMS-0604665. 1 2 M. BURGER, A. IOZZI, AND A. WIENHARD