Deformations of Fuchsian AdS representations are quasi-Fuchsian

Abstract

Let $\Gamma$ be a finitely generated group, and let $\mathrm{Rep}(\Gamma, \mathrm{SO}(2, n))$ be the moduli space of representations of $\Gamma$ into $\mathrm{SO}(2, n) (n \geq 2)$. An element $\rho : \Gamma \to \mathrm{SO}(2, n)$ of $\mathrm{Rep}(\Gamma , \mathrm{SO}(2, n))$ is quasi-Fuchsian if it is faithful, discrete, preserves an acausal $(n-1)$-sphere in the conformal boundary $\mathrm{Ein}_n$ 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 Fuchsian representations, i.e., composition of a faithful, discrete, and cocompact representation $\rho_f : \Gamma \to \mathrm{SO}(1, n)$ and the inclusion $\mathrm{SO}(1, n) \subset \mathrm{SO}(2, n)$.

In Anosov AdS representations are quasi-Fuchsian we proved that quasi-Fuchsian representations are precisely representations that are Anosov as defined in Anosov flows, surface groups and curves in projective space. In the present paper, we prove that the space of quasi-Fuchsian representations is open and closed, i.e., that it is a union of connected components of $\mathrm{Rep}(\Gamma, \mathrm{SO}(2, n))$.

The proof involves the following fundamental result: Let $\Gamma$ be the fundamental group of a globally hyperbolic spatially compact spacetime locally modeled on $\mathrm{AdS}_n$, 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 $\mathrm{Ein}_n$ is acausal.

Finally, we also provide the following characterization of representations with zero-bounded Euler class: they are precisely the representations preserving a closed achronal subset of $\mathrm{Ein}_n$.