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Geometry of ideal boundaries of geodesic spaces with nonpositive curvature in the sense of Busemann

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Abstract

We establish relations between different approaches to the ideal closure of a geodesic metric space with nonpositive curvature in the sense of Busemann. We construct the counterexample showing that the Busemann ideal closure can differ from the geodesic closure.

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References

  1. Ballmann W., “The Martin boundary of certain Hadamard manifolds” in Proc. on Analysis and Geometry (Ed. S. K. Vodop’yanov) (Sobolev Institute Press, Novosibirsk, 2000), pp. 36–46.

    Google Scholar 

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

    Google Scholar 

  3. Bowditch B. H., “Minkowskian subspaces of nonpositively curved metric spaces,” Bull. London Math. Soc. 27(6), 575–584 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  4. Bridson M. R. and Haefliger A., Metric Spaces of Nonpositive Curvature in vol. 319 of Fundamental Principles of Mathematical Sciences (Springer-Verlag, Berlin, 1999).

    Google Scholar 

  5. Burago D, Burago Y, and Ivanov S., A Course in Metric Geometry in vol. 33 of Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2001).

    Google Scholar 

  6. Busemann H., “Spaces with nonpositive curvature,” Acta Math. 80, 259–310 (1948).

    Article  MathSciNet  Google Scholar 

  7. Busemann H., The Geometry of Geodesics (Academic Press Inc., New York, N. Y., 1955).

    MATH  Google Scholar 

  8. Busemann H. and Phadke B. B., Spaces with Distinguished Geodesics (Marsel Dekker Inc., New-York; Basel, 1987).

    MATH  Google Scholar 

  9. Eberlein P. and O’Neill B., “Visibility manifolds,” Pacific J. Math. 46, 45–109 (1973.).

    MATH  MathSciNet  Google Scholar 

  10. Gromov M., “Hyperbolic groups,” Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, 75–263 (1987).

    MathSciNet  Google Scholar 

  11. Hosaka T., “Limit sets of geometrically finite groups acting on Busemann spaces,” Topology Appl. 122(3), 565–580 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  12. Hotchkiss Ph. K., “The boundary of Busemann space,” Proc. Amer. Math. Soc. 125(7), 1903–1912 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  13. Kapovich I. and Benakli N., “Boundaries of hyperbolic groups: Combinatorial and geometric group theory,” Contemp. Math. 296, 39–94 (2004).

    MathSciNet  Google Scholar 

  14. Kuratovskiĭ K., Topology, Vol. I (Mir, Moscow, 1966) [in Russian].

    Google Scholar 

  15. Martin R. S., “Minimal positive harmonic functions,” Trans. Amer. Math. Soc. 49, 137–172 (1941).

    Article  MATH  MathSciNet  Google Scholar 

  16. Rieffel M., “Group C*-algebras as compact quantum metric spaces,” Doc. Math. 7, 605–651 (2002).

    MATH  MathSciNet  Google Scholar 

  17. Rinow W., Die Innere Geometrie der Metrischen Raume in vol. 105 of Fundamental Principles of Mathematical Sciences (Springer-Verlag, Berlin; Gottingen; Heidelberg, 1961).

    Google Scholar 

  18. Webster C. and Winchester A., “Boundaries of hyperbolic metric spaces,” Pacific J. Math. 221(1), 147–158 (2005).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Pomor State University, Arkhangelsk, 163002, Russia

    P. D. Andreev

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  1. P. D. Andreev
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Correspondence to P. D. Andreev.

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Original Russian Text © P.D. Andreev, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 1, pp. 16–28.

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Andreev, P.D. Geometry of ideal boundaries of geodesic spaces with nonpositive curvature in the sense of Busemann. Sib. Adv. Math. 18, 95–102 (2008). https://doi.org/10.3103/S105513440802003X

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  • DOI: https://doi.org/10.3103/S105513440802003X

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