Let be a cocompact lattice in . A representation 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 and . 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 locally modeled on .
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Quentin Mérigot, Thierry Barbot, Anosov AdS representations are quasi-Fuchsian. Groups Geom. Dyn. 6 (2012), no. 3, pp. 441–483DOI 10.4171/GGD/163