Skip to main content

Fundamental groups and Laplacians

  • Conference paper
  • First Online:
Geometry and Analysis on Manifolds

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1339))

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.F. Atiyah, Elliptic operators, discrete groups and von Neumann algebra, Astérisque, 32–33 (1976), 43–72.

    MathSciNet  MATH  Google Scholar 

  2. R. Brooks, The fundamental groups and the spectrum of the Laplacian, Comment. Math. Helvetici 56 (1981), 581–598.

    Article  MathSciNet  MATH  Google Scholar 

  3. R.Brooks, Combinatorial problems in spectral geometry, in the Proceedings of the Taniguchi Symposium "Curvature and topology of Riemannian manifolds" 1985, Springer Lect. Note 1201, 14–32.

    Google Scholar 

  4. R. Brooks, The spectral geometry of tower of coverings, J.Diff.Geom. 23 (1986), 97–107.

    MathSciNet  MATH  Google Scholar 

  5. M. Burger, Estimation de petites valeurs propres du Laplacien d'un revetément de variétés Riemanniennes compactes, C.R.Acad.Sci. Paris 302 (1986), 191–194.

    MathSciNet  MATH  Google Scholar 

  6. P.Buser, On Cheeger's inequality λ1≧h2/4, in Geometry of the Laplace operator, (Proc.Symp. Pure Math., Hawaii (1979), 29–77.

    Google Scholar 

  7. J.M.G. Fell, Weak containment and induced representations of groups, Canadian J.Math. 14 (1962), 237–268.

    Article  MathSciNet  MATH  Google Scholar 

  8. F.P. Greenleaf, Invariant Means on Topological Groups and Their Applications, von Nostrand, Reinhald 1969.

    MATH  Google Scholar 

  9. Y. Ihara, On discrete subgroups of the two-by-two projective linear group over p-adic field, J.Math.Soc. Japan 18 (1966), 219–235.

    Article  MathSciNet  MATH  Google Scholar 

  10. A.Katsuda and T.Sunada, Homology and closed geodesics in a compact Riemann surface, to appear in Amer.J.Math.

    Google Scholar 

  11. D.A. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Funct.Anal.Appl. 1 (1967), 63–65.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Phillips and P. Sarnak, Geodesics in homology classes, Duke Math.J. 55 (1987), 287–297.

    Article  MathSciNet  MATH  Google Scholar 

  13. K.Ono, T.Kobayashi and T.Sunada, Spectrum of the Laplacian on a non-compact Riemannian manifold with compact quotient, in preparation.

    Google Scholar 

  14. B. Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull.Amer.Math.Soc. 80 (1974), 996–1000.

    Article  MathSciNet  MATH  Google Scholar 

  15. M.C.Reed and B.Simon, Methods of Modern Mathematical Physics, Vol. IV, Academic Press, 1978.

    Google Scholar 

  16. P. Sarnak, Entropy estimates for geodesic flows, Ergod.Th. and Synam.Sys. 2 (1982), 513–524.

    MathSciNet  MATH  Google Scholar 

  17. A. Selberg, Harmonic analysis and discontinuous subgroups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J.Indian Math.Soc. 20 (1956), 47–87.

    MathSciNet  MATH  Google Scholar 

  18. J.P. Serre, Tree, Springer, New York, 1980.

    Book  Google Scholar 

  19. D. Sullivan, Related aspects of positivity in Riemannian geometry, J.Diff.Geom. 25 (1987), 327–351.

    MathSciNet  MATH  Google Scholar 

  20. T. Sunada, Trace formula for Hill's operators, Duke Math.J. 47 (1980), 529–546.

    Article  MathSciNet  MATH  Google Scholar 

  21. T.Sunada, Trace formula, Wiener integrals and asymptotics, Proc. Japan-France Seminar (Spectra of Riemannian Manifolds), Kaigai Publ. Tokyo 1983, 159–169.

    Google Scholar 

  22. T. Sunada, Geodesic flows and geodesic random walks, Advanced Studies in Pure Math. (Geometry of Geodesics and Related Topics) Vol.3 (1984), 47–85.

    MathSciNet  MATH  Google Scholar 

  23. T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985), 169–186.

    Article  MathSciNet  MATH  Google Scholar 

  24. T.Sunada, L2-functions in geometry and some applications, Proc. Taniguchi Symp. 1985 (Curvature and Topology of Riemannian Manifolds) 266–284, Springer Lect. Note 1201.

    Google Scholar 

  25. T.Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, preprint. (to appear in Topology)

    Google Scholar 

  26. T.Sunada, Spectrum of symmetric random walks on a graph, in preparation.

    Google Scholar 

  27. R.J. Zimmer, Ergodic Theory and Semi-simple Groups, Birkhäuser, Boston, 1984.

    Book  MATH  Google Scholar 

  28. H. Donnelly and P. Li, Pure point spectrum and negative curvature for non-compact manifolds, Duke Math.J. 46 (1979), 497–503.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Kesten, Full Banach mean values of countable groups, Math. Scand. 7 (1959), 146–156.

    MathSciNet  MATH  Google Scholar 

  30. A. Selberg, On the estimation of Fourier coefficients of modular forms, in Proc.Sym. in Pure Math. Vol.8, A.M.S., Providence, RI 1965.

    MATH  Google Scholar 

  31. H. Donnelly, On L2-Betti numbers for abelian groups, Canad.Math. Bull. 24 (1981), 91–95.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, preprint.

    Google Scholar 

Download references

Authors

Editor information

Toshikazu Sunada

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag

About this paper

Cite this paper

Sunada, T. (1988). Fundamental groups and Laplacians. In: Sunada, T. (eds) Geometry and Analysis on Manifolds. Lecture Notes in Mathematics, vol 1339. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083059

Download citation

  • DOI: https://doi.org/10.1007/BFb0083059

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50113-8

  • Online ISBN: 978-3-540-45930-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics