Anosov AdS representations are quasi-Fuchsian

Abstract

Let $\Gamma$ be a cocompact lattice in $\mathrm{SO}(1,n)$. A representation $\rho :\Gamma \to \mathrm{SO}(2,n)$ is called quasi-Fuchsian if it is faithful, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space. A special case are Fuchsian representations, i.e., compositions of the inclusions $\Gamma \subset \mathrm{SO}(1,n)$ and $\mathrm{SO}(1,n)\subset \mathrm{SO}(2,n)$. We prove that quasi-Fuchsian representations are precisely those representations which are Anosov in the sense of Labourie (cf. (Lab06]). The study involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsian representations are holonomy representations of globally hyperbolic spacetimes diffeomorphic to $\mathbb{R}\times \Gamma \backslash {\mathbb{H}}^n$ locally modeled on ${\mathrm{AdS}}_{n+1}$.