Lorentzian Kleinian Groups

Abstract

Classical Kleinian groups are discrete subgroups of isometries of H n. The well-known theory of Kleinian groups starts with the definition of their associated limit set in the boundary of H n , and includes the geometric properties of the quotient hyperbolic space. This approach, naively applied, fails in the Lorentzian analogue anti-de Sitter space: discrete subgroups do not act properly discontinuously, and in many cases the set of accumulation points of orbits at the conformal boundary at infinity depends on the orbit. In this survey, we point out a way to extend this classical theory by introducing causality notions: the theory of limit sets and regularity domains extend naturally to achronal subgroups. This is closely related to the notions of globally hyper-bolic spacetimes, and we present what is known about the classification of globally hyperbolic spacetimes of constant curvature. We also review the close connection revealed by G. Mess ([79]) between globally hyperbolic spacetimes of dimension 2 + 1 and Teichm{\"u}ller space. This link can be understood via the space of time-like geodesics of anti-de Sitter space, and this space has also an interesting role, presented here, in the recent works about proper group actions on spacetimes of constant curvature.