Geometry and topology of complete Lorentz spacetimes of constant curvature

Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny

doi:10.24033/asens.2275

Geometry and topology of complete Lorentz spacetimes of constant curvature
[Géométrie et topologie des espaces-temps lorentziens complets à courbure constante]

Danciger, Jeffrey ; Guéritaud, François ; Kassel, Fanny

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

Résumé
Abstract

Nous étudions les actions propres, par isométries, de groupes discrets non virtuellement résolubles $Γ$ sur l'espace de Minkowski $ℝ^{2, 1}$ , en les voyant comme limites d'actions sur l'espace anti-de Sitter ${AdS}^{3}$ . À une telle action sur $ℝ^{2, 1}$ est associée une déformation infinitésimale, dans $SO (2, 1)$ , du groupe fondamental d'une surface hyperbolique $S$ . Lorsque $S$ est convexe cocompacte, nous montrons que $Γ$ 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 $AdS$ qui dégénèrent.

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

Publié le : 2016-11-12

MR Zbl

DOI : 10.24033/asens.2275

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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     title = {Geometry and topology of complete {Lorentz} spacetimes of constant curvature},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1--56},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
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Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny. Geometry and topology of complete Lorentz spacetimes of constant curvature. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 1-56. doi : 10.24033/asens.2275. http://www.numdam.org/articles/10.24033/asens.2275/

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