Siberian Mathematical Journal

, Volume 60, Issue 1, pp 10–19 | Cite as

On Urysohn’s ℝ-Tree

  • V. N. BerestovskiiEmail author


In the short note of 1927, Urysohn constructed the metric space R that is nowhere locally separable. There is no publication with indications that R is a (noncomplete) ℝ-tree that has valency c at each point. The author in 1989, as well as Polterovich and Shnirelman in 1997, constructed ℝ-trees isometric to R unaware of the paper by Urysohn. In this paper the author considers various constructions of the ℝ-tree R and of the minimal complete ℝ-tree of valency c including R, as well as the characterizations of ℝ-trees, their properties, and connections with ultrametric spaces.


boundary four-point property injective hull left-invariant geodesic metric ℝ-tree submetry ultrametric 


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  1. 1.
    Urysohn P. S., “Beispiel eines nirgends separablen metrischen Raumes,” Fund. Math., vol. 9, 119–121 (1927).CrossRefzbMATHGoogle Scholar
  2. 2.
    Urysohn P. S., “An example of a metric space nowhere satisfying the second countability axiom,” in: Urysohn P. S., Works on Topology and Other Areas of Mathematics. Vol. 2 [Russian], GITTL, Moscow and Leningrad, 1951, 778–780.Google Scholar
  3. 3.
    Kuratowski K., Topology. Vol. 1, Academic Press and Polish Scientific Publishers, New York, London, and Warszawa (1966).zbMATHGoogle Scholar
  4. 4.
    Engelking R., General Topology, Heldermann Verlag, Berlin (1989).zbMATHGoogle Scholar
  5. 5.
    Dyubina A. G. and Polterovich I. V., “The structure of hyperbolic spaces at infinity,” Russian Math. Surveys, vol. 53, No. 5, 1093–1094 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berestovskii V. N., “Quasicones of Lobachevskii spaces at infinity,” in: Abstracts: Proceedings of the International Conference on Algebra Dedicated to the Memory of A. I. Malcev (1909–1967) (Novosibirsk, 21–26 August 1989). Algebraic Geometry. Algebraic Methods in Geometry, Analysis, and Theoretical Physics. Applied and Computer Algebra, Novosibirsk, 1989, 9.Google Scholar
  7. 7.
    Polterovich I. V. and Shnirelman A. I., “An asymptotic subcone of the Lobachevskii plane as a space of functions,” Russian Math. Surveys, vol. 52, No. 4, 842–843 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Urysohn P. S., “The universal metric space,” in: Urysohn P. S., Works on Topology and Other Areas of Mathematics. Vol. 2 [Russian], GITTL, Moscow and Leningrad, 1951, 747–777.Google Scholar
  9. 9.
    Urysohn P. S., “Sur un espace métrique universel,” Bull. Sci. Math., Ser. 2, vol. 51, No. 4, 1–38 (1927).zbMATHGoogle Scholar
  10. 10.
    Tits J., “A theorem of Lie–Kolchin’ for trees,” in: Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin (H. Bass, P. Cassidy, J. Kovacic, eds.)., Academic Press, New York, 1977, 377–388.CrossRefGoogle Scholar
  11. 11.
    Alexandrov A. D. and Zalgaller V. A., “Two–dimensional manifolds of bounded curvature,” Proc. Steklov Inst. Math., vol. 76, 1–183 (1965).MathSciNetGoogle Scholar
  12. 12.
    Kuratowski K. and Mostowski A., Set Theory: With an Introduction to Descriptive Set Theory, PWN—Polish Sci. Publ., Warszawa and North–Holland Publ. Co., Amsterdam, New York, and Oxford (1976).zbMATHGoogle Scholar
  13. 13.
    Dyubina A. and Polterovich I., “Explicit constructions of universal ℝ–trees and asymptotic geometry of hyperbolic spaces,” Bull. Lond. Math. Soc., vol. 33, 727–734 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Berestovskii V. N. and Plaut C. P., “Covering ℝ–trees, ℝ–free groups, and dendrites,” Adv. Math., vol. 224, No. 5, 1765–1783 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mayer J., Nikiel J., and Oversteegen L., “Universal spaces for ℝ–trees,” Trans. Amer.Math. Soc., vol. 334, 411–432 (1992).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Berestovskii V. N., “Submetries of space–forms of negative curvature,” Sib. Math. J., vol. 28, No. 4, 552–562 (1987).CrossRefGoogle Scholar
  17. 17.
    Berestovskii V. N. and Guijarro L., “A metric characterization of Riemannian submersions,” Ann. Global Anal. Geom., vol. 18, 577–588 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Alexandrov A. D., On a Generalization of Riemannian Geometry, Jahresber. Humb. Univ., Berlin (1955).Google Scholar
  19. 19.
    Alexandrov A. D., “Über eine Verallgemeinerung der Riemannschen Geometrie,” in: Der Begriff des Räumes in der Geometrie. Bericht von der Riemann–Tagung des Forschungs–Instituts für Mathematik, Schriftenreihe Inst. Math., Berlin, 1957, Heft I, 33–84.Google Scholar
  20. 20.
    Alexandrov A. D., “On a generalization of Riemannian geometry,” in: A. D. Alexandrov. Selected Works. Vol. 3. Articles of Various Years [Russian], Nauka, Novosibirsk, 2008, 188–242.Google Scholar
  21. 21.
    Mayer J. C. and Oversteegen L. G., “A topological characterization of ℝ–trees,” Trans. Amer. Math. Soc., vol. 320, No. 1, 395–415 (1990).MathSciNetzbMATHGoogle Scholar
  22. 22.
    Morgan J. W. and Shalen P., “Valuations, trees, and degenerations of hyperbolic structures. I,” Ann. Math., vol. 120, 401–476 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gromov M., “Hyperbolic groups,” in: Essays in Group Theory (S. M. Gersten, ed.), Springer, New York, 1987, 75–263 (Math. Sci. Inst. Publ.; vol. 8).Google Scholar
  24. 24.
    Gromov M., Geometric Group Theory. Vol. 2: Asymptotic Invariants of Infinite Groups, Cambridge Univ. Press, Cambridge (1993) (Lond. Math. Soc. Lect. Note Ser.; vol. 182).zbMATHGoogle Scholar
  25. 25.
    Morgan J. W. and Shalen P., “Free actions of surface groups on R–trees,” Topology, vol. 30, No. 2, 143–154 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Morgan J. W., “Λ–trees and their applications,” Bull. Amer. Math. Soc. (N.S.), vol. 26, No. 1, 87–112 (1992).MathSciNetCrossRefGoogle Scholar
  27. 27.
    de la Harpe P. and Ghys E. (eds.), Sur les groupes hyperboliques, d’apr`es Mikhael Gromov, Birkhäuser, Basel etc. (1990).Google Scholar
  28. 28.
    Gaboriau D., Levitt G., and Paulin F., “Pseudogroups of isometries of ℝ and Rips’ theorem on free actions on R–trees,” Israel J. Math., vol. 87, 403–428 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Bestvina M., “R–Trees in topology, geometry and group theory,” in: Handbook of Geometric Topology (R. Daverman, R. Sher, eds.), Elsevier, Amsterdam, 2002, 55–91.Google Scholar
  30. 30.
    Nikiel J., “Topologies on pseudo–trees and applications,” Mem. Amer. Math. Soc., vol. 416, 116 (1989).MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kirk W. A., “Hyperconvexity of ℝ–trees,” Fund. Math., vol. 156, 67–72 (1998).MathSciNetzbMATHGoogle Scholar
  32. 32.
    Aronszajn N. and Panitchpakdi P., “Extensions of uniformly continuous transformations and hyperconvex metric spaces,” Pacific J. Math., vol. 6, 405–439 (1956).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kuratowski K., Topology. Vol. 2, Academic Press and Polish Scientific Publishers, New York, London, Toronto, Sydney, San Francisco, and Warszawa (1968).Google Scholar
  34. 34.
    Buneman P., “A note on the metric properties of trees,” J. Comb. Theory, Ser. B, vol. 17, 48–50 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Isbell J. R., “Six theorems about injective metric spaces,” Comment. Math. Helv., vol. 39, 439–447 (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Aksoy A. G. and Maurizi B., “Metric trees, hyperconvex hulls and extensions,” Turkish J. Math., vol. 32, 219–234 (2008).MathSciNetzbMATHGoogle Scholar
  37. 37.
    Borkowski M., Bugajewski D., and Phulara D., “On some properties of hyperconvex spaces,” Fixed Point Theory Appl. 2010. 2010:213812.Google Scholar
  38. 38.
    Berestovskii V. N., “Ultrametric spaces,” in: Studies on Analysis and Geometry (Novosibirsk, Akademgorodok, 1999), Sobolev Institute of Mathematics, Novosibirsk, 2000, vol. 32, 47–72.zbMATHGoogle Scholar
  39. 39.
    Hughes B., “Trees and ultrametric spaces: a categorial equivalence,” Adv. Math., vol. 189, No. 1, 148–191 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Hughes B., “Trees, ultrametrics, and noncommutative geometry,” Pure Appl. Math. Q., vol. 8, No. 1, 221–312 (2012).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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