DEFORMATIONS OF FUCHSIAN ADS REPRESENTATIONS ARE QUASI-FUCHSIAN

Let Gamma be a finitely generated group, and let Rep(Gamma, SO(2, n)) be the moduli space of representations of Gamma into SO(2, n) (n >= 2). An element rho : Gamma -> SO(2, n) of Rep(Gamma, SO(2, n)) is quasi-Fuchsian if it is faithful, discrete, preserves an acausal (n - 1)-sphere 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 Fuchsian representations, i.e., composition of a faithful, discrete, and cocompact representation rho(f) : Gamma -> SO(1, n) and the inclusion SO(1, n) subset of SO(2, n). In [10] we proved that quasi-Fuchsian representations are precisely representations that are Anosov as defined in [29]. 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 Rep (r, 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 AdS(n), and let rho : Gamma -> SO0(2, n) be the holonomy representation. Then, if Gamma is Gromov hyperbolic, the rho(Gamma)-invariant achronal limit set in 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 Ein(n).