Compact anti-de sitter 3-manifolds and folded hyperbolic structures on surfaces

Abstract

We prove that any non-Fuchsian representation ρ of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface, unless the image of ρ is virtually abelian. Using this idea, we establish that any non-Fuchsian representation ρ is strictly dominated by some Fuchsian representation j , in the sense that the hyperbolic translation lengths for j are uniformly larger than for ρ. Conversely, any Fuchsian representation j strictly dominates some non-Fuchsian representation ρ, whose Euler class can be prescribed. This has applications to the theory of compact anti-de Sitter 3-manifolds.