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}, i.e. 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.