Quasi-Fuchsian AdS representations are Anosov

Thierry Barbot (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

In a recent paper, Q. Mérigot proved that representations in SO(2,n) of uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in the boundary of anti-de Sitter space. In the present paper, we prove the reverse implication. It also includes: -- A construction of Dirichlet domains in the context of anti-de Sitter geometry, -- A proof that spatially compact globally hyperbolic anti-de Sitter spacetimes with acausal limit set admit locally CAT(-1) Cauchy hypersurfaces.

Keywords: Anti-de Sitter space; quasi-Fuchsian representation; globally hyperbolic spacetime; Anosov representation

Domains: Mathematics [math]; Differential Geometry [math.DG]

Thierry Barbot. Quasi-Fuchsian AdS representations are Anosov. 2007. ⟨ensl-00176526⟩