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

Jeffrey Danciger (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

François Guéritaud (Laboratoire Paul Painlevé, CNRS and Université Lille 1, Villeneuve d’Ascq, France; and Wolfgang-Pauli Institute, University of Vienna, Austria)

Fanny Kassel (Laboratoire Paul Painlevé, CNRS and Université Lille 1, Villeneuve d’Ascq, France)

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.

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Published 8 July 2016