Supra-maximal representations from fundamental groups of punctured spheres to PSL(2,R)

We study a particular class of representations from the fundamental groups of punctured spheres Σ0,n to the group PSL (2,R), which we call supra-maximal. Though most of them are Zariski dense, we show that supra-maximal representations are totally non hyperbolic, in the sense that every simple closed curve is mapped to an elliptic or parabolic element. They are also shown to be geometrizable (apart from the reducible ones) in the following very strong sense: for any element of the Teichmüller space T 0,n, there is a unique holomorphic equivariant map with values in the lower half-plane H^-. In the relative character varieties, the components of supra-maximal representations are shown to be compact and symplectomorphic (with respect to the Atiyah-Bott-Goldman symplectic structure) to the complex projective space of dimension n - 3 equipped with a certain multiple of the Fubini-Study form that we compute explicitly. This generalizes a result of Benedetto-Goldman [3] for the sphere minus four points.