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# Groups, Geometry, and Dynamics

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**Volume 6, Issue 3, 2012, pp. 441–483**

**DOI: 10.4171/GGD/163**

Published online: 2012-08-16

Anosov AdS representations are quasi-Fuchsian

Quentin Mérigot^{[1]}and Thierry Barbot

^{[2]}(1) Université Joseph Fourier, Grenoble, France

(2) Université d'Avignon, France

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
$\mathrm{AdS}_{n+1}$.

*Keywords: *Globally hyperbolic AdS spacetimes, Anosov representations

Mérigot Quentin, Barbot Thierry: Anosov AdS representations are quasi-Fuchsian. *Groups Geom. Dyn.* 6 (2012), 441-483. doi: 10.4171/GGD/163