Anosov AdS representations are quasi-Fuchsian

  • Quentin Mérigot

    Université Joseph Fourier, Grenoble, France
  • Thierry Barbot

    Université d'Avignon, France


Let Γ\Gamma be a cocompact lattice in SO(1,n)\mathrm{SO}(1,n). A representation ρ ⁣:ΓSO(2,n)\rho\colon \Gamma \to \mathrm{SO}(2,n) is called quasi-Fuchsian if it is faithful, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space. A special case are Fuchsian representations, i.e., compositions of the inclusions ΓSO(1,n)\Gamma \subset \mathrm{SO}(1,n) and SO(1,n)SO(2,n)\mathrm{SO}(1,n) \subset \mathrm{SO}(2,n). We prove that quasi-Fuchsian representations are precisely those representations which are Anosov in the sense of Labourie (cf. (Lab06]). The study involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsian representations are holonomy representations of globally hyperbolic spacetimes diffeomorphic to R×Γ\Hn\mathbb{R} \times \Gamma\backslash\mathbb{H}^{n} locally modeled on AdSn+1\mathrm{AdS}_{n+1}.

Cite this article

Quentin Mérigot, Thierry Barbot, Anosov AdS representations are quasi-Fuchsian. Groups Geom. Dyn. 6 (2012), no. 3, pp. 441–483

DOI 10.4171/GGD/163