Geometry and topology of complete Lorentz spacetimes of constant curvature

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We study proper, isometric actions of nonsolvable discrete groups Gamma on the 3-dimensional Minkowski space R²,¹ as limits of actions on the 3-dimensional anti-de Sitter space AdS³. To each such action is associated a deformation of a hyperbolic surface group Gamma₀ inside O(2,1). When Gamma₀ is convex cocompact, we prove that Gamma acts properly on R²,¹ if and only if this group-level deformation is realized by a deformation of the quotient surface that everywhere contracts distances at a uniform rate. We give two applications in this case. (1) Tameness: A complete flat spacetime is homeomorphic to the interior of a compact manifold. (2) Geometric degeneration: A complete flat spacetime is the rescaled limit of collapsing AdS spacetimes.