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The pressure metric on the Margulis multiverse

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Abstract

This paper defines the pressure metric on the Moduli space of Margulis spacetimes without cusps and shows that it is positive definite on the constant entropy sections. It also demonstrates an identity regarding the variation of the cross-ratios.

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References

  1. Abels, H.: Properly discontinuous groups of affine transformations, A survey. Geom. Dedic. 87, 309–333 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abels, H., Margulis, G.A., Soifer, G.: Properly discontinuous groups of affine transformations with orthogonal linear part. C. R. Acad. Sci. Paris Sr. I Math. 324(3), 253–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abels, H., Margulis, G.A., Soifer, G.: On the Zariski closure of the linear part of a properly discontinuous group of affine transformations. J. Differ. Geom. 60(2), 315–344 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonahon, F.: The geometry of Teichmüller space via geodesic currents. Invent. Math. 92, 139–162 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bowen, R.: Periodic orbits of hyperbolic flows. Am. J. Math. 94, 1–30 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bowen, R.: Symbolic dynamics for hyperbolic flows. Am. J. Math. 95, 429–460 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bowen, R., Ruelle, D.: The ergodic theory of axiom A flows. Invent. Math. 29, 181–202 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bridgeman, M.: Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space. Geom. Top. 14, 799–831 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bridgeman, M., Canary, R., Labourie, F., Sambarino, A.: The pressure metric for Anosov representations. arXiv:1301.7459v4 (2013)

  10. Bridgeman, M., Taylor, E.: An extension of the Weil–Petersson metric to quasi-Fuchsian space. Math. Ann. 341, 927–943 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Champetier, C.: Petite simplification dans les groupes hyperboliques. Ann. Fac. Sci. Toulouse Math. 3, 161221 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Charette, V., Drumm, T.: Strong marked isospectrality of affine Lorentzian groups. J. Differ. Geom. 66(3), 437–452 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Charette, V., Goldman, W.M., Jones,C.A.: Recurrent geodesics in flat Lorentz 3-manifolds. Canad. Math. Bull. 47, 332–342 (2004)

  14. Choi, S., Goldman, W.M.: Topological tameness of Margulis spacetimes. arxiv:1204.5308

  15. Danciger, J., Guéritaud, F., Kassel, F.: Geometry and topology of complete Lorentz spacetimes of constant curvature. Annales scientifiques de l’ENS 49(1), 1–56 (2016)

  16. Danciger, J., Gueritaud, F., Kassel, F.: Margulis spacetimes via the arc complex. Invent. Math. 204, 133–193 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Drumm, T.: Fundamental polyhedra for Margulis space-times. Doctoral Dissertation, University of Maryland (1990)

  18. Drumm, T.: Linear holonomy of Margulis space-times. J. Differ. Geom. 38(3), 679–690 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Drumm, T.A., Goldman, W.M.: Crooked planes. Electron. Res. Announc. Am. Math. Soc. 1(1), 10–17 (1995)

  20. Drumm, T., Goldman, W.M.: Isospectrality of flat Lorentz 3-manifolds. J. Differ. Geom. 58, 457–465 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Ghosh, S.: Anosov structure on Margulis space time. arXiv:1412.8211v2 (2014)

  23. Goldman, W.M.: The Margulis invariant of isometric actions on Minkowski (2+1)-space. In: Rigidity in Dynamics and Geometry. Contributions from the Programme Ergodic Theory, Geometric Rigidity and Number Theory, Isaac Newton Institute for the Mathematical Sciences Cambridge, United Kingdom, 5 January – 7 July 2000, pp.187–201 (2002). doi:10.1007/978-3-662-04743-9_9

  24. Goldman, W.M., Labourie, F.: Geodesics in Margulis spacetimes. Ergod. Theory Dyn. Syst. 32, 643–651 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Goldman, W.M., Labourie, F., Margulis, G.: Proper affine actions and geodesic flows of hyperbolic surfaces. Ann. Math. 170(3), 1051–1083 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Goldman, W.M., Margulis, G.: Flat Lorentz 3-manifolds and cocompact Fuchsian groups. Contemp. Math. 262, 135–146 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory, vol. 8, pp. 75–263. Springer (1987)

  28. Guichard, O., Wienhard, A.: Anosov representations: domains of discontinuity and applications. Invent. Math. 190(2), 357–438 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583 (1977). doi:10.1007/BFb0092042

  30. Kim, I.: Affine action and Margulis invariant. Funct. Anal. 219, 205–225 (2005). MR 2005k: 22017 Zbl 1085.57024

  31. Labourie, F.: Fuchsian affine actions of surface groups. J. Differ. Geom. 59(1), 15–31 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Labourie, F.: Anosov flows, surface groups and curves in projective space. Invent. Math. 165(1), 51–114 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Livsic, A.N.: Cohomology of dynamical systems. Math. USSR Izvestija 6, 1278–1301 (1972)

  34. Margulis, G.A.: Free completely discontinuous groups of affine transformations. Dokl. Akad. Nauk SSSR 272, 785–788 (1983)

    MathSciNet  Google Scholar 

  35. Margulis, G.A.: Complete affine locally flat manifolds with a free fundamental group. J. Sov. Math. 134, 129–134 (1987)

    Article  MATH  Google Scholar 

  36. McMullen, C.: Thermodynamics, dimension and the Weil–Petersson metric. Invent. Math. 173, 365–425 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mess, G.: Lorentz spacetimes of constant curvature. Geom. Dedicata 126, 3–45 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Milnor, J.: On fundamental groups of complete affinely flat manifolds. Adv. Math. 25, 178–187 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mineyev, I.: Flows and joins of metric spaces. Geom. Top. 9, 403482 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque, vol. 187–188. http://homepages.warwick.ac.uk/~masdbl/PP.pdf (1990)

  41. Pollicott, M.: Symbolic dynamics for smale flows. Am. J. Math. 109, 183–200 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ruelle, D.: Thermodynamic Formalism. Addison-Wesley, London (1989)

    MATH  Google Scholar 

  43. Sambarino, A.: Quantitative properties of convex representations. Commun. Math. Helv. 89, 443–488 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Shub, M.: Global Stability of Dynamical Systems. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  45. Smilga, I.: Fundamental domains for properly discontinuous affine groups. Geom. Dedicata 171, 203–229 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Smilga, I.: Proper affine actions on semisimple Lie algebras. arXiv:1406.5906

  47. Smilga, I.: Proper affine actions in non-swinging representations. arXiv:1605.03833

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Acknowledgements

I would like to express my gratitude towards my advisor Prof. Francois Labourie for his guidance. I would like to thank Dr. Andres Sambarino for the many helpful discussions that we had. I would also like to thank the organisers of the Aarhus conference on pressure metric and DMS program at MSRI for giving me the opportunity to discuss with Prof. Martin Bridgemann, Prof. Richard Canary, Prof. Olivier Guichard, Prof. Mark Pollicott and Dr. Maria Beatrice Pozzetti. Finally, I would like to thank the referee for helping me improve the exposition of this article.

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Authors and Affiliations

  1. Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, 69120, Heidelberg, Germany

    Sourav Ghosh

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  1. Sourav Ghosh
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Correspondence to Sourav Ghosh.

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The research leading to these results has received funding from the European Research Council under the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC grant agreement.

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Ghosh, S. The pressure metric on the Margulis multiverse. Geom Dedicata 193, 1–30 (2018). https://doi.org/10.1007/s10711-017-0260-y

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