# Deformations of Fuchsian AdS representations are Quasi-Fuchsian

Abstract : Let $\Gamma$ be a finitely generated group, and let $\op{Rep}(\Gamma, \SO(2,n))$ be the moduli space of representations of $\Gamma$ into $\SO(2,n)$ ($n \geq 2$). An element $\rho: \Gamma \to \SO(2,n)$ of $\op{Rep}(\Gamma, \SO(2,n))$ is \textit{quasi-Fuchsian} if it is faithful, discrete, preserves an acausal subset in the conformal boundary $\Ein_n$ of the anti-de Sitter space; and if the associated globally hyperbolic anti-de Sitter space is spatially compact - a particular case is the case of \textit{Fuchsian representations}, \ie composition of a faithfull, discrete and cocompact representation $\rho_f: \Gamma \to \SO(1,n)$ and the inclusion $\SO(1,n) \subset \SO(2,n)$. In \cite{merigot} we proved that quasi-Fuchsian representations are precisely representations which are Anosov as defined in \cite{labourie}. In the present paper, we prove that quasi-Fuchsian representations form a connected component of $\op{Rep}(\Gamma, \SO(2,n))$. This is an almost direct corollary of the following result: let $\Gamma$ be the fundamental group of a globally hyperbolic spacetime locally modeled on $\AdS_n$, and let $\rho: \Gamma \to \SO_0(2,n)$ be the holonomy representation. Then, if $\Gamma$ is Gromov hyperbolic, the $\rho(\Gamma)$-invariant achronal limit set in $\Ein_n$ is acausal.
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https://hal-univ-avignon.archives-ouvertes.fr/hal-00777721
Contributor : Thierry Barbot <>
Submitted on : Wednesday, May 29, 2013 - 3:30:57 PM
Last modification on : Tuesday, January 14, 2020 - 10:38:02 AM
Long-term archiving on: : Tuesday, April 4, 2017 - 12:52:41 PM

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• HAL Id : hal-00777721, version 2
• ARXIV : 1301.4309

### Citation

Thierry Barbot. Deformations of Fuchsian AdS representations are Quasi-Fuchsian. 2013. ⟨hal-00777721v2⟩

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