Dominating surface group representations by Fuchsian ones

We prove that a representation from the fundamental group of a closed surface of negative Euler characteristic with values in the isometry group of a Riemannian manifold of sectional curvature bounded by -1 can be dominated by a Fuchsian representation. Moreover, we prove that the domination can be made strict, unless the representation is discrete and faithful in restriction to an invariant totally geodesic 2-plane of curvature -1. When applied to representations into PSL(2,R) of non-extremal Euler class, our result is a step forward in understanding the space of closed anti-de Sitter 3-manifolds.