Fundamental domains for free groups acting on anti-de Sitter $3$-space

Abstract

Crooked planes are piecewise linear surfaces that were introduced by Drumm in the early 1990s to construct fundamental domains for properly discontinuous actions of free groups on Minkowski 3-space. In a previous paper, we introduced analogues of these surfaces, called AdS crooked planes, in the 3-dimensional anti-de Sitter space AdS; we showed that many properly discontinuous actions of free groups on AdS admit fundamental domains bounded by AdS crooked planes. Here we study further the question of which proper actions on AdS admit crooked fundamental domains, and show that some do not, in contrast to the Minkowski setting.