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Fundamental domains in the Einstein universe. (English) Zbl 1297.53051

Summary: We discuss fundamental domains for actions of discrete groups on the 3-dimensional Einstein Universe. They are bounded by crooked surfaces, which are conformal compactifications of surfaces that arise in the construction of Margulis space-times. We show that there exist pairwise disjoint crooked surfaces in the 3-dimensional Einstein Universe. As an application, we can construct explicit examples of groups acting properly on an open subset of that space.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
57S30 Discontinuous groups of transformations
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[1] Barbot, Thierry; Charette, Virginie; Drumm, Todd A.; Goldman, William M.; Melnick, Karin, A primer on the \((2 + 1)\) Einstein Universe, (Alekseevsky, Dmitri V.; Baum, Helga, Recent Developments Pseudo-Riemannian Geometry (2008), European Mathematical Society), 179-225 · Zbl 1154.53047
[2] Burelle, Jean-Philippe; Charette, Virginie; Drumm, Todd A.; Goldman, William M., Crooked halfspaces, in press · Zbl 1308.53040
[4] Drumm, Todd A., Fundamental polyhedra for Margulis space-times, Topology, 31, 4, 677-683 (1992) · Zbl 0773.57008
[5] Drumm, Todd A.; Goldman, William M., The geometry of crooked planes, Topology, 38, 2, 323-351 (1999) · Zbl 0941.51029
[6] Frances, Charles, Géometrie et dynamique lorentziennes conformes (2002), E.N.S.: E.N.S. Lyon, Ph.D. thesis
[7] Frances, Charles, The conformal boundary of Margulis space-times, C. R. Acad. Sci. Paris, Sér. I, 332, 751-756 (2003) · Zbl 1040.53078
[8] Frances, Charles, Lorentzian Kleinian groups, Comment. Math. Helv., 80, 4, 883-910 (2005) · Zbl 1083.22007
[9] Goldman, William M., Crooked surfaces and anti-de Sitter geometry · Zbl 1350.57001
[10] Kulkarni, Ravi S., Groups with domains of discontinuity, Math. Ann., 237, 3, 253-272 (1978) · Zbl 0369.20028
[11] Kulkarni, Ravi S., Proper actions and pseudo-Riemannian space forms, Adv. Math., 40, 1, 10-51 (1981) · Zbl 0462.53041
[12] Margulis, Gregory, Free properly discontinuous groups of affine transformations, Dokl. Akad. Nauk SSSR, 272, 937-940 (1983)
[13] Margulis, Gregory, Complete affine locally flat manifolds with a free fundamental group, J. Sov. Math., 134, 129-134 (1987) · Zbl 0611.57023
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