Reference : Variétés lorentziennes plates vues comme limites de variétés anti-de Sitter, d'après ...
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Physical, chemical, mathematical & earth Sciences : Mathematics
Variétés lorentziennes plates vues comme limites de variétés anti-de Sitter, d'après Danciger, Guéritaud et Kassel
[fr] Flat Lorentzian manifolds as limits of anti-de Sitter manifolds, following Danciger, Guéritaud and Kassel
Schlenker, Jean-Marc mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Société mathématique de France
[en] A survey on the recent work of Danciger, Gu\'eritaud and Kassel on Margulis space-times and complete anti-de Sitter space-times. Margulis space-times are quotients of the 3-dimensional Minkowski space by (non-abelian) free groups acting propertly discontinuously. Goldman, Labourie and Margulis have shown that they are determined by a convex co-compact hyperbolic surface S along with a first-order deformation of the metric which uniformly decreases the lengths of closed geodesics. Danciger, Gu\'eritaud and Kassel show that those space-times are principal ℝ-bundles over S with time-like geodesics as fibers, that they are homeomorphic to the interior of a handlebody, and that they admit a fundamental domain bounded by crooked planes. To obtain those results they show that those Margulis space-times are "infinitesimal" versions of 3-dimensional anti-de Sitter manifolds, and are lead to introduce a new parameterization of the space of deformations of a hyperbolic surface that increase the lengths of all closed geodesics.

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