Global hyperbolicity is the strongest causality condition one may place on a Lorentzian spacetime M. It ensures that M does not admit any causal loop on one hand and that it is not possible to go to the boundary of the spacetime "in finite time" through a causal path on the other hand. One nice property of globally hyperbolic spacetimes is the existence of a Cauchy surface, a positive hypersurface through which every inextendible causal curve must pass exactly once; this is, in a sense, the "space part" of M. The set of negative vector lines in R^p,q+1 is the pseudo-hyperbolic space H^p,q; it is a homogeneous space for the action of SO(p,q+1). A faithful discrete representation rho from a finitely generated torsion-free group to SO(p,q+1) is said to be spacelike cocompact if it acts cocompactly on a p-dimensional positive submanifold of H^p,q. It was proven by Beyrer-Kassel that the set of spacelike cocompact representations is a union of so-called "higher Teichmüller spaces", meaning connected components of the character variety made of faithful and discrete representations, making them particularly interesting. For q=1, H^p,q is a Lorentzian space and it was proven by Mess and Barbot that the spacelike cocompact representations in SO(p,2) were exactly the holonomies of maximal globally hyperbolic spacetimes of curvature -1 admitting a compact Cauchy surface. This talk will be dedicated to proving a generalization of this theorem in the case where q is greater than 1.

Café culturel assuré à 13:05 par Daniel Monclair.