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Geodesics in Margulis spacetimes

Published online by Cambridge University Press:  08 November 2011

WILLIAM M. GOLDMAN  and
FRANÇOIS LABOURIE
WILLIAM M. GOLDMAN
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (email: [email protected])
FRANÇOIS LABOURIE
Affiliation:
Laboratoire de Mathématiques, Université Paris-Sud, Orsay F-91405 Cedex; CNRS, Orsay cedex, F-91405, France (email: [email protected])
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Abstract

Let M3 be a Margulis spacetime whose associated complete hyperbolic surface Σ2 has a compact convex core. Generalizing the correspondence between closed geodesics on M3 and closed geodesics on Σ2, we establish an orbit equivalence between recurrent spacelike geodesics on M3 and recurrent geodesics on Σ2. In contrast, no timelike geodesic recurs in either forward or backward time.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 643 - 651
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Abels, H.. Properly discontinuous groups of affine transformations, a survey. Geom. Dedicata 87 (2001), 309–333.CrossRefGoogle Scholar
[2]Abels, H., Margulis, G. and Soifer, G.. Properly discontinuous groups of affine transformations with orthogonal linear part. C. R. Acad. Sci. Paris Sér. I Math. 324(3) (1997), 253–258.CrossRefGoogle Scholar
[3]Abels, H., Margulis, G. and Soifer, G.. The linear part of an affine group acting properly discontinuously and leaving a quadratic form invariant. Geom. Dedicata 153 (2011), 1–46.CrossRefGoogle Scholar
[4]Charette, V.. Proper actions of discrete groups on 2+1 spacetime. Doctoral Dissertation, University of Maryland, 2000.Google Scholar
[5]Charette, V.. Affine deformations of ultraideal triangle groups. Geom. Dedicata 97 (2003), 17–31.CrossRefGoogle Scholar
[6]Charette, V., Drumm, T. and Goldman, W.. Affine deformations of a three-holed sphere. Geom. Topol. 14(3) (2010), 1355–1382.CrossRefGoogle Scholar
[7]Charette, V. and Goldman, W.. Affine Schottky groups and crooked tilings. Crystallographic Groups and Their Generalizations (Kortrijk, 1999) (Contemporary Mathematics, 262). American Mathematical Society, Providence, RI, 2000, pp. 69–97.CrossRefGoogle Scholar
[8]Charette, V., Goldman, W. and Jones, C.. Recurrent geodesics in flat Lorentz 3-manifolds. Canad. Math. Bull. 47(3) (2004), 332–342.CrossRefGoogle Scholar
[9]Drumm, T.. Linear holonomy of Margulis space-times. J. Differential Geom. 38(3) (1993), 676–690.CrossRefGoogle Scholar
[10]Drumm, T.. Fundamental polyhedra for Margulis space-times. Doctoral Dissertation, University of Maryland, 1990; Topology 31(4) (1992), 677–683.CrossRefGoogle Scholar
[11]Drumm, T. and Goldman, W.. Complete flat Lorentz 3-manifolds with free fundamental group. Internat. J. Math. 1(2) (1990), 149–161.CrossRefGoogle Scholar
[12]Eberlein, P.. Geodesic flows on negatively curved manifolds I. Ann. of Math. (2) 95(3) (1972), 492–510.CrossRefGoogle Scholar
[13]Fried, D.. Personal communication to W. Goldman (2002).Google Scholar
[14]Fried, D. and Goldman, W.. Three-dimensional affine crystallographic groups. Adv. Math. 47 (1983), 1–49.CrossRefGoogle Scholar
[15]Goldman, W., Labourie, F. and Margulis, G.. Proper affine actions and geodesic flows of hyperbolic surfaces. Ann. of Math. (2) 170(3) (2009), 1051–1083.CrossRefGoogle Scholar
[16]Goldman, W. and Margulis, G.. Flat Lorentz 3-manifolds and cocompact Fuchsian groups. Crystallographic Groups and their Generalizations (Contemporary Mathematics, 262). American Mathematical Society, Providence, RI, 2000, pp. 135–146.CrossRefGoogle Scholar
[17]Katok, A. and Hasselblatt, B.. Introduction to The Modern Theory of Dynamical Systems. With a supplementary chapter by Katok and Leonardo Mendoza (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[18]Labourie, F.. Fuchsian affine actions of surface groups. J. Differential Geom. 59(1) (2001), 15–31.CrossRefGoogle Scholar
[19]Margulis, G.. Free properly discontinuous groups of affine transformations. Dokl. Akad. Nauk SSSR 272 (1983), 937–940.Google Scholar
[20]Margulis, G.. Complete affine locally flat manifolds with a free fundamental group. J. Soviet Math. 134 (1987), 129–134.CrossRefGoogle Scholar
[21]Mess, G.. Lorentz spacetimes of constant curvature. Geom. Dedicata 126(1) (2007), 3–45, New Techniques in Lorentz Manifolds: Proceedings of the BIRS 2004 Workshop, Eds. V. Charette and W. Goldman.CrossRefGoogle Scholar
[22]Milnor, J.. On fundamental groups of complete affinely flat manifolds. Adv. Math. 25 (1977), 178–187.CrossRefGoogle Scholar