Vari\'et\'es lorentziennes plates vues comme limites de vari\'et\'es anti-de Sitter, d'apr\`es Danciger, Gu\'eritaud et Kassel

TL;DR

This survey discusses recent advances in understanding Margulis space-times as limits of anti-de Sitter manifolds, highlighting new geometric structures, parameterizations, and fundamental domain constructions.

Contribution

It introduces a new perspective by viewing Margulis space-times as infinitesimal limits of anti-de Sitter spaces and describes their geometric and topological properties.

Findings

Margulis space-times are principal a0-bundles over hyperbolic surfaces.

They are homeomorphic to the interior of a handlebody.

Existence of fundamental domains bounded by crooked planes.

Abstract

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 $\mathbb R$-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.