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Isometry group of Lorentz manifolds: A coarse perspective

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Abstract

We prove a structure theorem for the isometry group \({\text {Iso}}(M,g)\) of a compact Lorentz manifold, under the assumption that a closed subgroup has exponential growth. We don’t assume anything about the identity component of \({\text {Iso}}(M,g)\), so that our results apply for discrete isometry groups. We infer a full classification of lattices that can act isometrically on compact Lorentz manifolds. Moreover, without any growth hypothesis, we prove a Tits alternative for discrete subgroups of \({\text {Iso}}(M,g)\).

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References

  • J. Amoros, M. Burger, K. Corlette, D. Kotschick and D. Toledo, Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs 44, American MathematicalSociety, Providence, RI, 1996.

  • S. Adams, G. Stuck, The isometry group of a compact Lorentz manifold. I. Invent. Math, 129(2) (1997), 239–261.

  • S. Adams and G. Stuck, The isometry group of a compact Lorentz manifold. II. Invent. Math, 129(2) (1997), 263–287.

    Article  MathSciNet  Google Scholar 

  • E. Bedford and J. Dadok. Bounded domains with prescribed group of automorphisms. Comment. Math. Helv, 62(4) (1987), 561–572.

    Article  MathSciNet  Google Scholar 

  • R. Blumenthal and J. Hebda. de Rham decomposition theorems for foliated manifolds. Ann Inst. Fourier, 33 (1983), 183–198.

    Article  MathSciNet  Google Scholar 

  • R. Blumenthal and J. Hebda. An analogue of the holonomy bundle for a foliated manifold. Tohoku Math. J., 40(2) (1988), 189–197.

    Article  MathSciNet  Google Scholar 

  • B. Bekka, P. de la Harpe and A. Valette. Kazhdan’s Property (T). Cambridge University Press, Cambridge, (2008).

  • G. Bell and A. Dranishnikov. Asymptotic dimension. Topology Appl., 155(12) (2008), 1265–1296.

    Article  MathSciNet  Google Scholar 

  • S. Buyalo and V. Schroeder. Elements of asymptotic geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2007.

  • Y. Carrière, Autour de la conjecture de L. Markus sur les variétés affines. Invent. Math., 95 (1989), 615–628.

    Article  MathSciNet  Google Scholar 

  • Y. Cornulier and P. de la Harpe. Metric geometry of locally compact groups. EMS Tracts in Mathematics, 25. European Mathematical Society (EMS), Zürich, (2016).

  • S. Cantat, T. Delzant and P. Py. Kähler groups, real hyperbolic spaces and the Cremona group. Compositio Math., 148(1), 2012.

  • J.A Carlson and D. Toledo. Harmonic mappings of Kähler manifolds to locally symmetric spaces, Inst. Hautes Études Sci. Publ. Math. (1989) 173–201.

  • G. D’Ambra. Isometry groups of Lorentz manifolds. Invent. Math, 92(3) (1988), 555–565.

    Article  MathSciNet  Google Scholar 

  • C. Drutu and M. Kapovich. Geometric Group Theory. American Mathematical Society Colloquium Publications. 63, (2018).

  • T. Delabie, J. Koivisto, F. Le Maître and R. Tessera. Quantitative measure equivalence between finitely generated groups. Preprint available from ArXiv, arXiv:2002.00719v2

  • C. Frances. Lorentz dynamics on closed 3-manifolds. Ann. H. Lebesgue, 3 (2020), 407–471.

    Article  MathSciNet  Google Scholar 

  • C. Frances. Variations on Gromov’s open-dense orbit theorem. Bulletin de la SMF, 146(4) (2018), 713–744.

  • D. Fried and W. Goldman. Three-dimensional affine crystallographic groups. Adv. Math., 47(1) (1983), 1–49.

    Article  MathSciNet  Google Scholar 

  • D. Fisher and R.J Zimmer. Geometric lattice actions, entropy and fundamental groups. Comment. Math. Helv, 77(2) (2002), 326–338.

    Article  MathSciNet  Google Scholar 

  • S. Hurtado, A. Kocksard and F. Rodriguez Hertz. The Burnside problem for \(Diff(S^2)\). To appear in Duke Mathematical Journal (2020).

  • D. Hume, J. Mackay and R. Tessera. Poincaré profiles of groups and spaces. Rev. Mat. Iberoamericana, 6, (2019).

  • D. Hume and A. Sisto, Groups with no coarse embeddings into hyperbolic groups. New York J. Math., 23 (2017), 1657–1670.

    MathSciNet  MATH  Google Scholar 

  • C. Godbillon. Feuilletages. (French) Études géométriques. Progress in Mathematics, 98. Birkhäuser Verlag, Basel, (1991).

  • W. Goldman and M. Hirsch. The radiance obstruction and parallel forms on affine manifolds. Trans. Amer. Math. Soc, 286(2) (1984), 629–649

    Article  MathSciNet  Google Scholar 

  • M. Gromov. Rigid transformation groups, Géométrie Différentielle, (D. Bernard et Choquet-Bruhat Ed.), Travaux en cours, Hermann, Paris, 33, (1988), 65–141.

  • M. Gromov. Asymptotic invariants of infinite groups, in: Geometric Group Theory, vol. 2. London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, (1993), pp. 1–295.

  • D. Hume, J. Mackay and R. Tessera. Poincaré profiles of groups and spaces. Rev. Mat. Iberoamericana, 6, 2019.

  • D.L. Johnson and L.B. Whitt. Totally geodesic foliations. J. Differential Geom, 15 (1980), 225–235.

    MathSciNet  MATH  Google Scholar 

  • B. Klingler. Complétude des variétés lorentziennes à courbure constante. Math. Ann, 306(2) (1996), 353–370

    Article  MathSciNet  Google Scholar 

  • S. Kobayashi. Transformation groups in differential geometry. Reprint of the 1972 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

  • S. Kobayashi and K. Nomizu. Foundations of differential geometry I (New York : Interscience Publishers, 1963).

    MATH  Google Scholar 

  • N.Kowalsky. Noncompact simple automorphism groups of Lorentz manifolds and other geometric manifolds. Ann. of Math. (2) 144 (1996) 3, 611–640

    Article  MathSciNet  Google Scholar 

  • R. Kulkarni and F. Raymond. \(3\)-dimensional Lorentz space-forms and Seifert fiber spaces. J. Differential Geometry, 21(2) 1985, 231–268.

    Article  MathSciNet  Google Scholar 

  • C. Le Coz and A. Gournay. Separation profiles, isoperimetry, growth and compression. Preprint available from ArXiv, arXiv:1910.11733, 2019.

  • F. Ledrappier. Quelques propriétés des exposants caractéristiques. Lecture Notes in Math., 1097, Springer, Berlin, 1984.

  • A. Lubotzky, S. Mozes and M.S. Raghunathan. The word and Riemannian metrics on lattices of semisimple groups. Publications Mathématiques de l’Institut des Hautes Scientifiques, 91 2000, 5–53

    Article  MathSciNet  Google Scholar 

  • K. Matsuzaki and M. Taniguchi. Hyperbolic Manifolds and Kleinian groups, Oxford Science Publications, Clarendon Press, 1998.

    MATH  Google Scholar 

  • K. Melnick. Isometric actions of Heisenberg groups on compact Lorentz manifolds. Geom. Dedicata, 126 (2007), 131–154.

    Article  MathSciNet  Google Scholar 

  • K. Melnick. A Frobenius theorem for Cartan geometries, with applications. L’Enseignement Mathématique (Série II), 57 (2011), no. 1-2, 57–89

  • S.B. Myers and N.E. Steenrod. The group of isometries of a Riemannian manifold. Ann. of Math. (2) 40 (1939), no. 2, 400–416.

    Article  MathSciNet  Google Scholar 

  • I. Mundet i Riera. Isometry groups of closed Lorentz 4-manifolds are Jordan. Geom. Dedicata, 207 (2020), 201–207.

  • K. Nomizu. On local and global existence of Killing fields. Ann. of Math, 72 (1960), no. 2, 105–112.

    Article  MathSciNet  Google Scholar 

  • V. Pécastaing. On two theorems about local automorphisms of geometric structures. Ann. Inst. Fourier, 66 (2016), no. 1, 175–208.

    Article  MathSciNet  Google Scholar 

  • F. Rampazzo. Frobenius-type theorems for Lipschitz distributions, J.Differential Equations, 243 (2007), no. 2, 270–300.

    Article  MathSciNet  Google Scholar 

  • M. Rosenlicht. A remark on quotient spaces. An. Acad. Brasil. Cienc, 35(4) (1963), 487–489.

    MathSciNet  MATH  Google Scholar 

  • M. Rosenlicht. Quotient varieties and the affine embedding of certain homogeneous spaces. Trans. of the AMS, 101 (1961), 211–223.

    Article  MathSciNet  Google Scholar 

  • A. Sard. Hausdorff Measure of Critical Images on Banach Manifolds. Amer. Journ. of Math, 87(1), 158–174.

  • R. Saerens and W.R. Zame. The isometry groups of manifolds and the automorphism groups of domains. Trans. Amer. Math. Soc, 301 (1987), no. 1, 413–429.

    Article  MathSciNet  Google Scholar 

  • R.W. Sharpe. Differential Geometry: Cartan’s generalization of Klein’s Erlangen Program. New York, Springer, 1997.

  • S. Simić. Lipschitz distributions and Anosov flows. Proc. Amer. Math. Soc, 124(6) (1996), 1869–1877.

    Article  MathSciNet  Google Scholar 

  • S. Sternberg. Lectures on Differential Geometry. Prentice-Hall, 1964.

  • R.A. Struble. Metrics in locally compact groups. Compositio Math, 28 (1974), 217–222.

    MathSciNet  MATH  Google Scholar 

  • R. Tessera. On coarse embeddings of amenable groups into hyperbolic graphs. Preprint available from ArXiv, arXiv:2010.07205.

  • W. Thurston. Three dimensional geometry and topology. Vol 1. Princeton University Press, Edited by Silvio Levy, 1997.

  • A. Zeghib. Isometry groups and geodesic foliations of Lorentz manifolds. I. Foundations of Lorentz dynamics. Geom. Funct. Anal, 9 (1999), no. 4, 775–822.

    Article  MathSciNet  Google Scholar 

  • A. Zeghib. Isometry groups and geodesic foliations of Lorentz manifolds. II. Geometry of analytic Lorentz manifolds with large isometry group. Geom. Funct. Anal, 9 (1999), no. 4, 823–854.

    Article  MathSciNet  Google Scholar 

  • A. Zeghib. Sur les espaces-temps homogènes. (French. English summary) [On homogeneous space-times] The Epstein birthday schrift, 551–576, Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry, 1998.

  • A. Zeghib. The identity component of the isometry group of a compact Lorentz manifold. Duke Math. J, 92 (1998), no. 2, 321–333.

    Article  MathSciNet  Google Scholar 

  • C. Boubel and A. Zeghib. Isometric actions of Lie subgroups of the Moebius group. Nonlinearity, 17 (2004), no. 5, 1677–1688.

    Article  MathSciNet  Google Scholar 

  • A. Zeghib and P. Piccione. Actions of discrete groups on stationary Lorentz manifolds. Ergodic Theory and Dynamical Systems, 34 (2014), no. 5, 1640–1673.

    Article  MathSciNet  Google Scholar 

  • R.J. Zimmer. Kazhdan groups acting on compact manifolds. Invent. Math, 75 (1984), 425–436.

    Article  MathSciNet  Google Scholar 

  • R.J. Zimmer. On the automorphism group of a compact Lorentz manifold and other geometric manifolds. Invent. Math, (3)83 (1986), 411–424

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I warmly thank Pierre Py, Romain Tessera and Abdelghani Zeghib for enlightning conversations, and the referee for very valuable remarks.

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  1. IRMA, Université de Strasbourg, 7 rue René Descartes, 67000, Strasbourg, France

    Charles Frances

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Frances, C. Isometry group of Lorentz manifolds: A coarse perspective. Geom. Funct. Anal. 31, 1095–1159 (2021). https://doi.org/10.1007/s00039-021-00585-1

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