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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