Skip to main content
SpringerLink
Log in

Extending partial isometries

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We show that a finite metric spaceA admits an extension to a finite metric spaceB so that each partial isometry ofA extends to an isometry ofB. We also prove a more precise result on extending a single partial isometry of a finite metric space. Both these results have consequences for the structure of the isometry groups of the rational Urysohn metric space and the Urysohn metric space.

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. H. Becker and A. S. Kechris,The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, 232, Cambridge University Press, 1996.

  2. B. Herwig and D. Lascar,Extending partial automorphisms and the profinite topology on free groups, Transactions of the American Mathematical Society352 (1999), 1985–2021.

    Article  MathSciNet  Google Scholar 

  3. E. Hrushovski,Extending partial isomorphisms of graphs, Combinatorica12 (1992), 411–416.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. S. Kechris, V. Pestov and S. Todorcevic,Fraisse limits, Ramsey theory, and topological dynamics of automorphism groups, Geometric and Functional Analysis15 (2005), 106–189.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. S. Kechris and C. Rosendal,Turbulence, amalgamation and generic automorphisms of homogeneous structures, to appear.

  6. G. W. Mackey,Ergodic theory and virtual groups, Mathematische Annalen166 (1966), 187–207.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Marker,Model Theory: An Introduction, Graduate Texts in Mathematics, Springer, Berlin, 2002.

    MATH  Google Scholar 

  8. S. A. Morris and V. Pestov,A topological generalization of the Higman-Neumann-Neumann theorem, Journal of Graph Theory1 (1998), 181–187.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. Pestov,Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Israel Journal of Mathematics,127 (2002), 317–357;A corrigendum to the article: Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Israel Journal of Mathematics145 (2005), 375–379.

    MATH  MathSciNet  Google Scholar 

  10. P. Urysohn,Sur en espace métrique universel, Bulletin des Sciences Mathématiques51 (1927), 43–64, 74–90.

    Google Scholar 

  11. V. V. Uspenskij,On the group of isometries of the Urysohn universal metric space, Commentationes Mathematicae Universitatis Carolinae31 (1990), 181–182.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 W. Green St., 61801, Urbana, IL, USA

    Sławomir Solecki

Authors
  1. Sławomir Solecki
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by NSF grant DMS-0400931. I would like to thank Ward Henson and Alekos Kechris for conversations and emails regarding the paper. Part of this work was done when I visited Caltech in May 2004. I thank the mathematics department there for support

Rights and permissions

Reprints and permissions

About this article

Cite this article

Solecki, S. Extending partial isometries. Isr. J. Math. 150, 315–331 (2005). https://doi.org/10.1007/BF02762385

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02762385

Keywords

Search

Navigation