Dominating surface group representations and deforming closed AdS 3-manifolds

Abstract

In a previous paper by Deroin-Tholozan, the authors construct a map $\mathbfΨ_ρ$ from the Teichmüller space of $S$ to itself and prove that, when $M$ has sectional curvature $\leq -1$, the image of $\mathbfΨ_ρ$ lies (almost always) in the domain $\mathrm{Dom}(ρ)$ of Fuchsian representations stricly dominating $ρ$. Here we prove that $\mathbfΨ_ρ: \mathrm{Teich}(S) \to \mathrm{Dom}(ρ)$ is a homeomorphism. As a consequence, we are able to describe the topology of the deformation space of anti-de Sitter structures on closed 3-manifolds.