# Anosov AdS representations are quasi-Fuchsian

### Quentin Mérigot

Université Joseph Fourier, Grenoble, France### Thierry Barbot

Université d'Avignon, France

## Abstract

Let $\Gamma$ be a cocompact lattice in $\mathrm{SO}(1,n)$. A representation $\rho\colon \Gamma \to \mathrm{SO}(2,n)$ 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 $\mathrm{AdS}_{n+1}$.

## Cite this article

Quentin Mérigot, Thierry Barbot, Anosov AdS representations are quasi-Fuchsian. Groups Geom. Dyn. 6 (2012), no. 3 pp. 441–483

DOI 10.4171/GGD/163