Crooked surfaces and anti-de Sitter geometry
Crooked planes were defined to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They where extended to closed polyhedral surfaces, called crooked surfaces, in the conformal compactification of Minkowski space (Einstein space). The conformal model of anti-de Sitter space is the interior of the quotient of Einstein space by an involution fixing an Einstein plane. Crooked surfaces have been defined in the 3-dim anti-de Sitter space \(AdS^3\), and have been used to construct fundamental domains for discrete groups of isometries of \(AdS^3\). (Note that not any complete anti-de Sitter 3-manifold admits a fundamental domain bounded by crooked planes, in contrast with the case of Margulis spacetimes.) The purpose of this work is to interpret \(AdS\)-crooked planes in terms of crooked surfaces in Einstein space. Last but not least, the paper contains a very nice, detailed and useful description of the anti-de Sitter geometry.
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