Anosov AdS representations are quasi-Fuchsian - Mathematics > Differential Geometry

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Abstract: Let Gamma be a cocompact lattice in SO1,n. A representation rho: Gamma \toSO2,n is quasi-Fuchsian if it is faithfull, discrete, and preserves anacausal subset in the boundary of anti-de Sitter space - a particular case isthe case of Fuchsian representations, ie. composition of the inclusions ofGamma in SO1,n and of SO1,n in SO2,n. We prove that if a representationis Anosov in the sense of Labourie then it is also quasi-Fuchsian. We also showthat Fuchsian representations are Anosov : the fact that all quasi-Fuchsianrepresentations are Anosov will be proved in a second part by T. Barbot. Thestudy involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsianrepresentations are holonomy representations of globally hyperbolic spacetimesdiffeomorphic to the product R \times Gamma\H^n and locally modeled on theanti-de Sitter space.

Autor: Quentin Merigot

Fuente: https://arxiv.org/