Skip to main content
SpringerLink
Log in

Proportionality principle for the simplicial volume of families of \(\mathbb Q \)-rank 1 locally symmetric spaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for \(\mathbb Q \)-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the proportionality principle for the locally finite simplicial volume and the relative simplicial volume of \(\mathbb Q \)-rank \(1\) locally symmetric spaces with amenable cusp groups established by Löh and Sauer [26].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borel, A.: Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strabourg, XV. Actualités Scientifiques et Industrielles, No. 1341. Hermann, Paris (1969)

  2. Borel, A., Ji, L.: Compactifications of Symmetric and Locally Symmetric Spaces. Mathematics: Theory and Applications. Birkhäuser, Boston, MA (2006)

    Google Scholar 

  3. Borel, A., Moore, J.C.: Homology theory for locally compact spaces. Michigan Math. J. 7(2), 137–159 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bredon, G.: Sheaf Theory. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  5. Bucher-Karlsson, M.: Simplicial volume of locally symmetric spaces covered by \(\text{ SL }(3,\mathbb{R})/\text{ SO }(3)\). Geom. Dedicata 125(1), 203–224 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bucher, M.: The proportionality constant for the simplicial volume of locally symmetric spaces. Colloq. Math. 111(2), 183–198 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bucher, M.: The simplicial volume of closed manifolds covered by \(\mathbb{H}^2 \times \mathbb{H}^2\). J. Topol. 1(3), 584–602 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bucher, M., Burger, M., Iozzi, A.: A dual interpretation of the Gromov–Thurston proof of Mostow rigidity and volume rigidity for representations of hyperbolic lattices. To appear in Trends in Harmonic Analysis, Indam-Springer (2012)

  9. Bucher, M., Burger, M., Frigerio, R., Iozzi, A., Pagliantini, C., Pozzetti, M.B.: Isometric properties of relative bounded cohomology. arXiv:1205.1022 (2012)

  10. Bucher, M., Monod N.: The norm of the Euler class. Math. Ann. 353(2), 523–544 (2012)

    Google Scholar 

  11. de Rham, G.: Differentiable Manifolds. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  12. Demaily, J.-P.: Complex analysis and differential geometry. http://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf (2009)

  13. Frigerio, R., Pagliantini, C.: The simplicial volume of hyperbolic manifolds with geodesic boundary. Algebra. Geom. Topol. 10, 979–1001 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gromov, M.: Volume and bounded cohomology. Inst. Hautes Éudes Sci. Publ. Math. 56, (1982), 5–99 (1983)

  15. Guichardet, A.: Cohomologie des groupes topologiques et des algebres de Lie, Textes Mathématiques [Mathematical Texts], vol. 2. CEDIC, Paris (1980)

  16. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  17. Jungreis, D.: Chains that realize the Gromov invariant of hyperbolic manifolds. Ergod. Th. Dyn. Syst. 17, 643–648 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kim, S., Kim, I.: Simplicial volume of \(\mathbb{Q}\)-rank one locally symmetric spaces covered by the product of \(\mathbb{R}\)-rank one symmetric spaces. Algebra. Geome. Topol. 12, 1165–1181 (2012)

    Article  MATH  Google Scholar 

  19. Kim, S., Kuessner, T.: Simplicial volume of compact manifolds with amenable boundary. arXiv:1205.1375 (2012)

  20. Kuessner, T.: Efficient fundamental cycles of cusped hyperbolic manifolds. Pacific J. Math. 211, 283–313 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lafont, J.-F., Schmidt, B.: Simplicial volume of closed locally symetric spaces of noncompact type. Acta Math. 197(1), 129–143 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lee, J.M.: Introduction to smooth manifolds. Springer, Berlin (2006)

    Google Scholar 

  23. Leuzinger, E.: An exhaustion of locally symmetric spaces by compact submanifolds with corners. Invent. math. 121, 389–410 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Löh, C.: \(\ell ^1\)-homology and simplicial volume. PhD thesis, WWU Münster, (2007)

  25. Löh, C., Sauer, R.: Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volume. J. Topol. 2(1), 193–225 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Löh, C., Sauer, R.: Simplicial volume of Hilbert modular varieties. Comment. Math. Helvetici 84(3), 457–470 (2009)

    Article  MATH  Google Scholar 

  27. Monod, N.: Continuous Bounded Cohomology of Locally Compact Groups, Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001)

    Book  Google Scholar 

  28. Savage, R.P.: The space of positive definite matrices and Gromov’s invariant. Trans. A. M. S. 274(1), 239–263 (1982)

    MATH  MathSciNet  Google Scholar 

  29. Thurston, W.: Geometry and Topology of 3-Manifolds, Lecture Notes. Princeton (1978)

Download references

Acknowledgments

M. Bucher gratefully acknowledges support from the Swiss National Science Foundation (Grant No. PP00P2-128309/1). I. Kim gratefully acknowledges the partial support of NRF Grant (R01-2008-000-10052-0) and a warm support of IHES and IHP during his stay. M. Bucher and I. Kim are thankful to the Mittag-Leffler Institute (Djursholm, Sweden) for its hospitality during the preparation of this paper.

Author information

Authors and Affiliations

  1. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case Postale 64, 1211 , Genève, Switzerland

    Michelle Bucher

  2. School of Mathematics, KIAS, Hoegiro 85, Dongdaemun-gu, Seoul, 130-722, Korea

    Inkang Kim & Sungwoon Kim

Authors
  1. Michelle Bucher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Inkang Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sungwoon Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Inkang Kim.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bucher, M., Kim, I. & Kim, S. Proportionality principle for the simplicial volume of families of \(\mathbb Q \)-rank 1 locally symmetric spaces. Math. Z. 276, 153–172 (2014). https://doi.org/10.1007/s00209-013-1191-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-013-1191-4

Keywords

Search

Navigation