On closed anti de Sitter spacetimes

Abstract

This article deals with the structure of the fundamental group of compact anti de Sitter spacetimes, i.e. Lorentz manifolds with constant negative curvature. Algebraically such a manifold is the quotient of the universal cover of the homogeneous space $SO(2, n) / SO(1, n)$ by a discrete group $\Gamma$ acting properly and co-compactly on it. This exists if and only if $n$ is even. Indeed, as this was observed by Kulkarni, $U(1, d)$ is contained in $SO(2, 2d)$ , and acts properly transitively on $SO(2, 2d)/SO(1, 2d)$ . It then suffices to take $\Gamma$ as a co-compact lattice in $U(1, d)$ . The results of the present article give evidence to the question: in dimension $>3$ , are all compact anti de Sitter spacetimes constructed in this way?