Dominating surface group representations and deforming closed AdS 3-manifolds

Abstract

In a previous paper by Deroin-Tholozan, the authors construct a map ${\mathbf{\Psi }}_{\rho }$ from the Teichmüller space of $S$ to itself and prove that, when $M$ has sectional curvature $\le -1$, the image of ${\mathbf{\Psi }}_{\rho }$ lies (almost always) in the domain $\mathrm{Dom}(\rho )$ of Fuchsian representations stricly dominating $\rho$. Here we prove that ${\mathbf{\Psi }}_{\rho }:\mathrm{Teich}(S)\to \mathrm{Dom}(\rho )$ 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.

Added an Appendix proving the continuity of the minimal Lipschitz constant; simplified the statements of the theorem

References in corpus (2)

• Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces
• Ideal triangulations and geometric transitions

Cited by in corpus (2)

• Domination results in $n$-Fuchsian fibers in the moduli space of Higgs bundles
• Almost strict domination and anti-de Sitter 3-manifolds