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 3 ; we showed that many properly discontinuous actions of free groups on AdS 3 admit fundamental domains bounded by AdS crooked planes. Here we study further the

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... estion of which proper actions on AdS 3 admit crooked fundamental domains, and show that some do not, in contrast to the Minkowski setting.