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Can One Hear the Shape of a Group?

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Geometry and Topology of Manifolds

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 154))

Abstract

The iso-spectrum problem for marked lengnth spectrum for Riemannian manifolds of negative curvature has a rich history. We rephrased the problems for metrics on discrete groups, discussed its connection to a conjecture by Margulis, and proved some results for “total relatively hyperbolic groups” in Koji Fujiwara, Journal of Topology and Analysis, 7(2), 345–359 (2015). This is a note from my talk on that paper and mainly discuss the connection between Riemannian geometry and group theory, and also some questions.

Supported by Grant-in-Aid for Scientific Research (No. 23244005, 15H05739).

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References

  1. Abels, H., Margulis, G.: Coarsely geodesic metrics on reductive groups. In: Modern Dynamical Systems and Applications, Cambridge University Press, Cambridge, pp. 163–183 (2004)

    Google Scholar 

  2. Breuillard, E.: Geometry of locally compact groups of polynomial growth and shape of large balls. Groups Geom. Dyn. 8(3), 669–732 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Breuillard, E., Le Donne, E.: On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry. Proc. Natl. Acad. Sci. USA 110(48), 19220–19226 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bridson, M. R., Haefliger, A.: Metric spaces of non-positive curvature. In: Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)

    Google Scholar 

  5. Burago, D. Yu.: Periodic metrics. Representation theory and dynamical systems. Adv. Soviet Math. Amer. Math. Soc., Providence, RI, 9, 205–210 (1992)

    Google Scholar 

  6. Burns, K., Katok, A.: Manifolds with nonpositive curvature. Ergodic Theor. Dynam. Syst. 5, 307–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Christopher, B.: Croke, rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65(1), 150–169 (1990)

    Article  MathSciNet  Google Scholar 

  8. Culler, M., Morgan, J.: Group actions on \(\mathbb{R}\)-trees. Proc. London Math. Soc. 55(3), 571–604 (1987)

    Google Scholar 

  9. Eberlein, P.: Geometry of \(2\)-step nilpotent groups with a left invariant metric. In: Annales Scientifiques de l’École Normale Sup rieure, Sér. 4, vol. 27 no. 5, pp. 611–660 (1994)

    Google Scholar 

  10. Fujiwara, K.: Asymptotically isometric metrics on relatively hyperbolic groups and marked length spectrum. J. Topol. Anal. 7(2), 345–359 (2015). doi:10.1142/S1793525315500132

    Google Scholar 

  11. Fujiwara, K.: Jason Fox Manning, CAT(0) and CAT(1) fillings of hyperbolic manifolds. J. Differential Geom. 85(2), 229–269 (2010)

    Google Scholar 

  12. Furman, A.: Coarse-geometric perspective on negatively curved manifolds and groups. In: Rigidity in dynamics and geometry (Cambridge (2000)), pp. 149–166. Springer, Berlin (2002)

    Google Scholar 

  13. Gornet, R.: The marked length spectrum vs. the Laplace spectrum on forms on Riemannian nilmanifolds. Comment. Math. Helvetici 71, 297–329 (1996)

    Google Scholar 

  14. Gornet, R.: A new construction of isospectral Riemannian nilmanifolds with examples. Michigan Math. J. 43(1), 159–188 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hamenstädt, U.: Cocycles, symplectic structures and intersection. Geom. Funct. Anal. 9(1), 90–140 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Krat, S.A.: On pairs of metrics invariant under a cocompact action of a group. Electron. Res. Announc. Amer. Math. Soc. 7, 79–86 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leeb, Bernhard: 3-manifolds with(out) metrics of nonpositive curvature. Invent. Math. 122(2), 277–289 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Margulis, G.: Metrics on reductive groups. In: Geometric group theory, hyperbolic dynamics and symplectic geometry. Oberwolfach Reports, European Mathematical Society, vol. 3, no. 3., pp. 1991–2057,(2006)

    Google Scholar 

  19. Otal, J.P.: Le spectre marquè des longueurs des surfaces à courbure negative. Ann. of Math. (2) 131(1), 151–162 (1990)

    Google Scholar 

  20. Spatzier, R. J.: An invitation to rigidity theory. In: Modern dynamical systems and applications, Cambridge University Press, Cambridge, pp. 211–231 (2004)

    Google Scholar 

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Acknowledgments

I’d like to thank Emmanuel Breuillard for discussions.

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Koji Fujiwara

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  1. Koji Fujiwara
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Correspondence to Koji Fujiwara .

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Editors and Affiliations

  1. The University of Tokyo, Meguro-ku, Tokyo, Japan

    Akito Futaki

  2. Mathematical Institute, Tohoku University, Sendai,Aoba-ku, Japan

    Reiko Miyaoka

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, China

    Zizhou Tang

  4. Chern Institute of Mathematics, Nankai University, Tianjin, China

    Weiping Zhang

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Fujiwara, K. (2016). Can One Hear the Shape of a Group?. In: Futaki, A., Miyaoka, R., Tang, Z., Zhang, W. (eds) Geometry and Topology of Manifolds. Springer Proceedings in Mathematics & Statistics, vol 154. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56021-0_7

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