Geometriae Dedicata

, Volume 120, Issue 1, pp 19–35 | Cite as

Coarse Equivalences Between Warped Cones

  • Hyun Jeong Kim


Warped cones were introduced by J. Roe in Geometry Topol. 9 (2005) 163–178 where he discussed Property A of these spaces. In this paper, we discuss the coarse equivalence of warped cones on the circle with the \(\mathbb{Z}\)-action by irrational rotations. First, we prove that two irrational numbers related by PSL(2, \(\mathbb{Z}\)) give coarsely equivalent warped cones. Second, we prove that there are at least countably many warped cones that are not coarsely equivalent to each other by using a ‘secondary growth function’.


coarse geometry warped cones irrational rotations coarse invariants rational approximations growth functions asymptotic dimension 

Mathematics Subject Classifications (2000)

Primary: 51K05 Secondary: 53C20 53C12 20F69 


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  1. 1.
    Bell G., Dranishnikov A. (2001) On asymptotic dimension of groups. Algebraic Geom. Topol. 1, 57–71MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Gromov M.: Asymptotic Invariants of Infinite Groups Vol. 2, Geometric Group Theory, Cambridge University Press, 1993Google Scholar
  3. 3.
    Hardy G. and Wright E.: An Introduction to the Theory of Numbers, 5th edition, Clarencdon Press, Oxford, 1979Google Scholar
  4. 4.
    Higson N., Roe J. (2000) Amenable group actions and the Novikov conjecture. J. Reine Angew. Math. 519 143–153MATHMathSciNetGoogle Scholar
  5. 5.
    Hurewicz W., Wallman H., (1948) Dimension Theory, Rev. edn, Princeton University Press, PrincetonGoogle Scholar
  6. 6.
    Kim H.: Coarse geometry of Warped cones, Ph.D Thesis, Penn State University, 2005.Google Scholar
  7. 7.
    Lorentzen L. and Waadeland H.: Continued Fractions, Stud. Comput. Math 3, North-Holland, Amsterdam.Google Scholar
  8. 8.
    Rieffel M. A.: C*-algebras associated with irrational rotations, Pacific. J. Math. 93(2) (1981).Google Scholar
  9. 9.
    Roe J. Coarse cohomology and index theory for complete Riemanninan manifold, Mem. Amer. Math. Soc. (497) (1993).Google Scholar
  10. 10.
    Roe J. From foliations to coarse geometry and back. In: Masa X., Macias-Virgos E., Alvarez Lopez J.A., (eds), Analysis and Geometry in Foliated Manifolds, World Scientific, Singapore, 1995 195–206Google Scholar
  11. 11.
    Roe J. (1996). Index Theory, Coarse Geometry, and the Topology of Manifolds, Vol 90 CBMS Conference Proceedings. Amer. Math. Soc. Providence R.IGoogle Scholar
  12. 12.
    Roe, J.: Lectures on Coarse Geometry, University Lecture Series, 31, Amer. Math. Soc., 2003.Google Scholar
  13. 13.
    Roe J. (2005), Warped Cones and Property A. Geometry Topol. 9: 163–178MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Yu G. (1998), The Novikov conjecture for groups with finite asymptotic dimension. Ann. Math. 147(2): 325–355MATHCrossRefGoogle Scholar
  15. 15.
    Yu G. (2000), The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math., 139: 201–240MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Hyun Jeong Kim
    • 1
  1. 1.Penn State UniversityUniversity ParkUSA

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