signature=36abb3540298ef9a06f37555ca88ab5e,Deformations of Fuchsian AdS representations are Quasi-Fu...

摘要：

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 $ho: \\Gamma o \\SO(2,n)$ of $\\op{Rep}(\\Gamma,\\SO(2,n))$ is extit{quasi-Fuchsian} if it is faithful, discrete, preserves anacausal subset in the conformal boundary $\\Ein_n$ of the anti-de Sitter space;and if the associated globally hyperbolic anti-de Sitter space is spatiallycompact - a particular case is the case of extit{Fuchsian representations},i.e. composition of a faithfull, discrete and cocompact representation $ho_f:\\Gamma o \\SO(1,n)$ and the inclusion $\\SO(1,n) \\subset \\SO(2,n)$. In\\cite{merigot} we proved that quasi-Fuchsian representations are preciselyrepresentations which are Anosov as defined in \\cite{labourie}. In the presentpaper, we prove that quasi-Fuchsian representations form a connected componentof $\\op{Rep}(\\Gamma, \\SO(2,n))$. This is an almost direct corollary of thefollowing result: let $\\Gamma$ be the fundamental group of a globallyhyperbolic spacetime locally modeled on $\\AdS_n$, and let $ho: \\Gamma o\\SO_0(2,n)$ be the holonomy representation. Then, if $\\Gamma$ is Gromovhyperbolic, the $ho(\\Gamma)$-invariant achronal limit set in $\\Ein_n$ isacausal.

展开