Skip to main content
SpringerLink
Log in

Uniformly continuous maps between ends of \({\mathbb{R}}\)-trees

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even classes of metrically proper Lipschitz maps) between the trees.

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. Baues H.J., Quintero A.: Infinite Homotopy Theory. K-Monographs in Mathematics, no. 6. Kluwer, Boston (2001)

    Google Scholar 

  2. Bestvina, M.: \({\mathbb{R}}\)-trees in Topology, Geometry and Group Theory. Handbook of geometric topology, pp. 55–91. North-Holland, Amsterdam (2002)

  3. Borsuk K.: On some metrization of the hyperspace of compact sets. Fund. Math. 41, 168–202 (1954)

    MathSciNet  Google Scholar 

  4. Hughes B.: Trees and ultrametric spaces: a categorical equivalence. Adv. Math. 189, 148–191 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hughes, B., Ranicki: Ends of Complexes. Cambridge Tracts in Mathematics, vol. 123. Cambridge University Press, Cambridge (1996)

  6. Mac Lane S.: Categories for the Working Mathematician. Springer, New York (1971)

    MATH  Google Scholar 

  7. Morgan J.W.: Λ-trees and their applications. Bull. Am. Math. Soc. 26(1), 87–112 (1992)

    Article  MATH  Google Scholar 

  8. Morón M.A., Ruiz del Portal F.R.: Shape as a Cantor completion process. Mathematische Zeitschrift 225, 67–86 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Robert, A.M.: A Course in p-adic Analysis. Grad. Text Math., vol. 198. Springer, New York (2000)

  10. Roe, J.: Lectures on Coarse Geometry. University Lecture Series, vol. 31. American Mathematical Society, Providence (2003)

  11. Roe, J.: Coarse Cohomology and Index Theory on Complete Riemannian Manifolds. Memoirs of the American Mathematical Society, vol. 104, no. 497 (1993)

  12. Serre J.P.: Trees. Springer, New York (1980)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Álvaro Martínez-Pérez & Manuel A. Morón

Authors
  1. Álvaro Martínez-Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel A. Morón
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel A. Morón.

Additional information

M. A. Morón was partially supported by MTM 2006-00825.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martínez-Pérez, Á., Morón, M.A. Uniformly continuous maps between ends of \({\mathbb{R}}\)-trees. Math. Z. 263, 583–606 (2009). https://doi.org/10.1007/s00209-008-0431-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-008-0431-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation