Geometry and topology of complete Lorentz spacetimes of constant curvature
Abstract
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.
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