GEOMETRY AND TOPOLOGY OF COMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE

We study proper, isometric actions of non virtually solvable discrete groups Gamma on the 3-dimensional Minkowski space R-2,R-1, viewing them as limits of actions on the 3-dimensional anti de Sitter space AdS(3). To each such action on R-2,R-1 is associated an infinitesimal deformation, inside SO(2, 1), of the fundamental group of a hyperbolic surface S. When S is convex cocompact, we prove that Gamma acts properly on R-2,R-1 if and only if this grou -level deformation is realized by a deformation of S that uniformly contracts or uniformly expands all distances. We give two applications in this case. (1) Tameness: A complete flat spacetime is homeomorphic to the interior of a compact manifold with boundary. (2) Geometric transition: A complete flat spacetime is the rescaled limit of collapsing AdS spacetimes.