Geometry and topology of complete Lorentz spacetimes of constant curvature

2013/06/10 by Danciger, Jeffrey, Guéritaud, François, Kassel, Fanny · 2 citations
#20H10 #53B30 #53C50 #57M50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1306.2240

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 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 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.

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