Anosov AdS representations are quasi-Fuchsian
Summary: Let \(\Gamma\) be a cocompact lattice in \(\mathrm{SO}(1,n)\). A representation \(\rho\colon \Gamma \to \mathrm{SO}(2,n)\) is called {\textit{quasi-Fuchsian}} if it is faithful, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space. A special case are {\textit{Fuchsian representations}}, i.e., compositions of the inclusions \(\Gamma \subset \mathrm{SO}(1,n)\) and \(\mathrm{SO}(1,n) \subset \mathrm{SO}(2,n)\). We prove that quasi-Fuchsian representations are precisely those representations which are Anosov in the sense of \textit{F. Labourie} [Invent. Math. 165, No. 1, 51--114 (2006; Zbl 1103.32007)]. The study involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsian representations are holonomy representations of globally hyperbolic spacetimes diffeomorphic to \(\mathbb{R} \times \Gamma\backslash\mathbb{H}^{n}\) locally modeled on \(\mathrm{AdS}_{n+1}\).
scientific article; zbMATH DE number 6088868