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

Let [math] be a closed oriented surface of negative Euler characteristic and [math] a complete contractible Riemannian manifold. A Fuchsian representation [math] strictly dominates a representation [math] if there exists a [math] –equivariant map from [math] to [math] that is [math] –Lipschitz for some [math] . In a previous paper by Deroin and Tholozan, the authors construct a map [math] from the Teichmüller space [math] of the surface [math] to itself and prove that, when [math] has sectional curvature at most [math] , the image of [math] lies (almost always) in the domain [math] of Fuchsian representations strictly dominating [math] . Here we prove that [math] is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations [math] from [math] to [math] with [math] Fuchsian strictly dominating [math] . In particular, we obtain that its connected components are classified by the Euler class of [math] . 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 [math] –manifolds.