Quasi-Fuchsian AdS representations are Anosov

TL;DR

This paper establishes a bi-conditional relationship between quasi-Fuchsian AdS representations and Anosov representations, and introduces new geometric constructions and properties of anti-de Sitter spacetimes.

Contribution

It proves that Anosov representations in SO(2,n) are quasi-Fuchsian, complementing previous results, and develops Dirichlet domain constructions and geometric properties of AdS spacetimes.

Findings

Proved the reverse implication of Merigot's result.

Constructed Dirichlet domains in anti-de Sitter geometry.

Showed that certain AdS spacetimes admit CAT(-1) Cauchy hypersurfaces.

Abstract

In a recent paper, Q. M\'erigot 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.