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# Mathematical Research Letters

## Volume 23 (2016)

### Number 3

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

Pages: 735 – 770

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n3.a10

#### Authors

#### 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 $\mathrm{AdS}$ crooked planes, in the $3$-dimensional anti-de Sitter space $\mathrm{AdS}^3$; we showed that many properly discontinuous actions of free groups on $\mathrm{AdS}^3$ admit fundamental domains bounded by $\mathrm{AdS}$ crooked planes. Here we study further the question of which proper actions on $\mathrm{AdS}^3$ admit crooked fundamental domains, and show that some do not, in contrast to the Minkowski setting.