Geometry and topology of complete Lorentz spacetimes of constant curvature

Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 1-56.

Nous étudions les actions propres, par isométries, de groupes discrets non virtuellement résolubles $\Gamma$ sur l'espace de Minkowski ${ℝ}^{2,1}$, en les voyant comme limites d'actions sur l'espace anti-de Sitter ${\mathrm{AdS}}^3$. À une telle action sur ${ℝ}^{2,1}$ est associée une déformation infinitésimale, dans $\mathrm{SO}(2,1)$, du groupe fondamental d'une surface hyperbolique $S$. Lorsque $S$ est convexe cocompacte, nous montrons que $\Gamma$ agit proprement sur ${ℝ}^{2,1}$ si et seulement si cette déformation au niveau du groupe est réalisée par une déformation de $S$ qui contracte uniformément ou dilate uniformément toutes les distances. Nous donnons deux applications dans ce cas. (1) Sagesse topologique : un espace-temps plat complet est homéomorphe à l'intérieur d'une variété compacte à bord. (2) Transition géométrique : un espace-temps plat complet est la limite renormalisée d'espaces-temps $\mathrm{AdS}$ qui dégénèrent.

We study proper, isometric actions of non virtually solvable discrete groups $\Gamma$ on the 3-dimensional Minkowski space ${ℝ}^{2,1}$, viewing them as limits of actions on the 3-dimensional anti-de Sitter space ${\mathrm{AdS}}^3$. To each such action on ${ℝ}^{2,1}$ is associated an infinitesimal deformation, inside $\mathrm{SO}(2,1)$, of the fundamental group of a hyperbolic surface $S$. When $S$ is convex cocompact, we prove that $\Gamma$ acts properly on ${ℝ}^{2,1}$ if and only if this group-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 $\mathrm{AdS}$ spacetimes.

Classification : 20H10, 53C50, 57M50, 57M60
Keywords: Lorentzian geometry, anti-de Sitter manifolds, Margulis spacetimes, affine geometry, topological tameness, geometric transition.
Mot clés : Géométrie lorentzienne, variétés anti-de Sitter, espaces-temps de Margulis, géométrie affine, sagesse topologique, transition géométrique

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