Dominating surface group representations and deforming closed anti-de Sitter 3-manifolds

Let S be a closed oriented surface of negative Euler characteristic and M a complete contractible Riemannian manifold. A Fuchsian representation j : pi(1) (S) -> Isom(+) (H-2) strictly dominates a representation rho: pi(1) (S) ->! Isom (M) if there exists a (j, rho)-equivariant map from H-2 to M that is lambda-Lipschitz for some lambda < 1. In a previous paper by Deroin and Tholozan, the authors construct a map Psi(rho) from the Teichmuller space T(S) of the surface S to itself and prove that, when M has sectional curvature at most-1, the image of psi(rho) lies (almost always) in the domain Dom(rho) of Fuchsian representations strictly dominating rho. Here we prove that psi(rho): T(S) -> Dom(rho) is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations (j, rho) from pi(1) (S) to Isom(+) (H-2) with j Fuchsian strictly dominating rho. In particular, we obtain that its connected components are classified by the Euler class of rho. The link with anti-de Sitter geometry comes from a theorem of Kassel, stating that those pairs parametrize deformation spaces of anti-de Sitter structures on closed 3-manifolds.