Skip to main content
SpringerLink
Log in

Translation Lengths of Parabolic Isometries of CAT(0) Spaces and Their Applications

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this article, we provide a sufficient and necessary condition on parabolic isometries of positive translation lengths on complete visibility CAT(0) spaces. One of the consequences is that each parabolic isometry of a complete simply connected visibility manifold of nonpositive sectional curvature has zero translation length. Applications on the geometry of open negatively curved manifolds will also be discussed.

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. Abresch, U., Schroeder, V.: Graph manifolds, ends of negatively curved spaces and the hyperbolic 120-cell space. J. Differ. Geom. 35(2), 299–336 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Belegradek, I.: Topology of Open Nonpositively Curved Manifolds, ArXiv e-prints (2013)

  3. Brock, J., Farb, B.: Curvature and rank of Teichmüller space. Am. J. Math. 128(1), 1–22 (2006)

    Article  MATH  Google Scholar 

  4. Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of Nonpositive Curvature, Progress in Mathematics, vol. 61. Birkhäuser Boston Inc., Boston (1985)

    Book  MATH  Google Scholar 

  5. Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)

  6. Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bridson, M.R.: Semisimple actions of mapping class groups on \({{\rm CAT}}(0)\) spaces. Geometry of Riemann surfaces, London Math. Soc. Lecture Note Ser., vol. 368, pp. 1–14

  8. Buyalo, S.V.: An example of a four-dimensional manifold of negative curvature. Algebra Anal. 5(1), 193–199 (1993)

    MathSciNet  MATH  Google Scholar 

  9. Buyalo, S.V.: Geodesics in Hadamard spaces. Algebra Anal 10(2), 93–123 (1998)

    MathSciNet  Google Scholar 

  10. Eberlein, P.: Geodesic flows on negatively curved manifolds. II. Trans. Am. Math. Soc. 178, 57–82 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eberlein, P.: Lattices in spaces of nonpositive curvature. Ann. Math. (2) 111(3), 435–476 (1980)

  12. Eberlein, P., O’Neill, B.: Visibility manifolds. Pac. J. Math. 46, 45–109 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Farb, B.: Some problems on mapping class groups and moduli space. Problems on Mapping Class Groups and Related Topics. Proceedings of Symposia in Pure Mathematics, vol. 74, pp. 11–55. American Mathematical Society, Providence (2006)

  14. Farb, B., Margalit, D.: A Primer on Mapping Class Groups, Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  15. Fujiwara, K.: A construction of negatively curved manifolds. Proc. Jpn. Acad. Ser. A. Math. Sci. 64(9), 352–355 (1988)

    MathSciNet  Google Scholar 

  16. Gromov, M.: Hyperbolic Groups. Essays in Group Theory. Mathematical Sciences Research Institute Publications, New York (1987)

    Book  MATH  Google Scholar 

  17. Heintze, E., Im Hof, H.C.: Geometry of horospheres. J. Differ. Geom. 12 (1977), no. 4, 481–491 (1978)

  18. Ivanov, N.V.: Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, vol. 115, American Mathematical Society, Providence, RI: Translated from the Russian by E. J. F, Primrose and revised by the author (1992)

  19. Karlsson, A., Margulis, G.A.: A multiplicative ergodic theorem and nonpositively curved spaces. Commun. Math. Phys. 208(1), 107–123 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Phan, T.N.: On Finite Volume. Negatively Curved Manifolds, ArXiv e-prints (2011)

    Google Scholar 

  21. Raghunathan, M. S.: Discrete subgroups of Lie groups, Springer-Verlag, New York-Heidelberg, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68

  22. Schroeder, V.: Finite volume and fundamental group on manifolds of negative curvature. J. Differ. Geom. 20(1), 175–183 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, Y.: A note on Riemannian metrics on the moduli space of Riemann surfaces. Proc. Am. Math. Soc. 144(6), 2513–2519 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Most of the work is from part of the author’s thesis work. The author is greatly indebted to his advisor, Jeffrey Brock, for his consistent encouragement and support. Part of this work was finished when the author was a G. C. Evans instructor at the Department of Mathematics of Rice University. He would like to thank the department for all the support in the past several years. The author would also like to thank the anonymous referee whose comments and suggestions are very helpful to improve the paper.

Author information

Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, China

    Yunhui Wu

Authors
  1. Yunhui Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunhui Wu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Y. Translation Lengths of Parabolic Isometries of CAT(0) Spaces and Their Applications. J Geom Anal 28, 375–392 (2018). https://doi.org/10.1007/s12220-017-9824-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-017-9824-1

Keywords

Mathematics Subject Classification

Search

Navigation