Geometry and topology of complete Lorentz spacetimes of constant curvature

TL;DR

This paper investigates the geometry and topology of complete Lorentz spacetimes with constant curvature, focusing on proper actions of discrete groups on Minkowski space as limits of anti-de Sitter space actions, revealing conditions for properness and applications to spacetime properties.

Contribution

It establishes a criterion for proper group actions on Minkowski space via deformations of hyperbolic surface groups, linking geometric contractions to properness and exploring implications for spacetime structure.

Findings

Proper actions correspond to uniformly contracting deformations.

Complete flat spacetimes are homeomorphic to interior of compact manifolds.

Complete flat spacetimes can be viewed as limits of collapsing AdS spacetimes.

Abstract

We study proper, isometric actions of nonsolvable discrete groups Gamma on the 3-dimensional Minkowski space R^{2,1} as limits of actions on the 3-dimensional anti-de Sitter space AdS^3. To each such action is associated a deformation of a hyperbolic surface group Gamma_0 inside O(2,1). When Gamma_0 is convex cocompact, we prove that Gamma acts properly on R^{2,1} 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.