Discrete & Computational Geometry

, Volume 14, Issue 2, pp 123–149 | Cite as

Hyperbolic and parabolic packings

  • Zheng-Xu He
  • O. Schramm
Article

Abstract

The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. LetG be the 1-skeleton of a triangulation of an open disk.G is said to be CP parabolic (resp. CP hyperbolic) if there is a locally finite disk packingP in the plane (resp. the unit disk) with contacts graphG. Several criteria for deciding whetherG is CP parabolic or CP hyperbolic are given, including a necessary and sufficient combinatorial criterion. A criterion in terms of the random walk says that if the random walk onG is recurrent, theG is CP parabolic. Conversely, ifG has bounded valence and the random walk onG is transient, thenG is CP hyperbolic.

We also give a new proof thatG is either CP parabolic or CP hyperbolic, but not both. The new proof has the advantage of being applicable to packings of more general shapes. Another new result is that ifG is CP hyperbolic andD is any simply connected proper subdomain of the plane, then there is a disk packingP with contacts graphG such thatP is contained and locally finite inD.

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References

  1. 1.
    E. M. Andreev, On convex polyhedra in Lobačevskiî spaces,Mat. Sb. (N.S.) 81 (123) (1970), 445–478; English transl.Math. USSR-Sb. 10 (1970), 413–440.MathSciNetGoogle Scholar
  2. 2.
    E. M. Andreev, On convex polyhedra of finite volume in Lobačevskiî space,Mat. Sb. (N.S.) 83 (125) (1970), 256–260; English transl.Math. USSR-Sb. 12 (1970), 255–259.MathSciNetGoogle Scholar
  3. 3.
    A. F. Beardon and K. Stephenson, The uniformization theorem for circle packings,Indiana Univ. Math. J. 39 (1990), 1383–1425.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    A. F. Beardon and K. Stephenson, The Schwarz-Pick lemma for circle packings,Illinois J. Math. 141 (1991), 577–606.MathSciNetGoogle Scholar
  5. 5.
    A. F. Beardon and K. Stephenson, Circle packings in different geometries,Tôhoku Math. J. 43 (1991), 27–36.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    P. L. Bowers, The upper Perron method for labeled complexes with applications to circle packings,Math. Proc. Cambridge Philos. Soc. 114 (1993), 321–345.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    G. R. Brightwell and E. R. Scheinerman, Representations of planar graphs,SIAM J. Discrete Math. 6 (1993), 214–229.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    J. W. Cannon, The combinatorial Riemann mapping theorem,Acta Math. 173 (1994), 155–234.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem,The Mathematical Heritage of Wilhelm Magnus—Groups, Geometry and Special Functions, Contemporary Mathematics Series, American Mathematical Society, Providence, RI, 1994.Google Scholar
  10. 10.
    Y. Colin De Verdiére, Un principe variationnel pour les empilements de cercles,Invent. Math. 104 (1991), 655–669.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    P. G. Doyle and J. L. Snell,Random Walks and Electric Networks, The Carus Mathematical Monographs, Vol. 22, Mathematical Association of America, Washington, DC, 1984.MATHGoogle Scholar
  12. 12.
    R. J. Duffin, The extremal length of a network,J. Math. Anal. Appl. 5 (1962), 200–215.MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    B. T. Garrett, Circle packings and polyhedral surfaces,Discrete Comput. Geom. 8 (1992), 429–440.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    A. A. Grigor'yan, On the existence of positive fundamental solutions of the laplace equation on Riemannian manifolds,Math. USSR-Sb. 56 (1987), 349–358.MATHCrossRefGoogle Scholar
  15. 15.
    Z.-X. He and O. Schramm, Fixed points, Koebe uniformization and circle packings,Ann. of Math. 237 (1993), 369–406.MathSciNetCrossRefGoogle Scholar
  16. 16.
    P. Koebe, Kontaktprobleme der konformen Abbildung,Ber. Ver. Sächs. Akad. Wiss. Leipzig Math.-Phys. Klasse 88 (1936), 141–164.Google Scholar
  17. 17.
    O. Lehto and K. I. Virtanen,Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, 1973.MATHCrossRefGoogle Scholar
  18. 18.
    A. Marden and B. Rodin,On Thurston's Formulation and Proof of Andreev's Theorem, Lecture Notes in Mathematics, Vol. 1435, Springer-Verlag, Berlin, 1989, pp. 103–115.Google Scholar
  19. 19.
    M. H. A. Newman,Elements of the Topology of Plane Sets of Points, 2nd, edn., Cambridge University Press, Cambridge, 1939, Dover, New York, 1992.Google Scholar
  20. 20.
    B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping,J. Differential Geom. 26 (1987), 349–360.MATHMathSciNetGoogle Scholar
  21. 21.
    O. Schramm, Existence and uniqueness of packings with specified combinatorics,Israel J. Math. 73 (1991), 321–341.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    O. Schramm, Rigidity of infinite (circle) packings,J. Amer. Math. Soc. 4 (1991), 127–149.MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    O. Schramm, How to cage an egg,Invent. Math. 107 (1992), 543–560.MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    O. Schramm, Conformal uniformization and packings,Israel J. Math. (to appear).Google Scholar
  25. 25.
    O. Schramm, Square tilings with prescribed combinatorics,Israel J. Math. 84 (1993), 97–118.MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    O. Schramm, Transboundary extremal length,J. Analyse Math. (to appear).Google Scholar
  27. 27.
    P. M. Soardi, Recurrence and transience of the edge graph of a tiling of the euclidean plane,Math. Ann. 287 (1990), 613–626.MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    W. P. Thurston,The Geometry and Topology of 3-Manifolds, Princeton University Notes, Princeton University Press, Princeton, NJ, 1982.Google Scholar
  29. 29.
    W. Woess, Random walks on infinite graphs and groups—a survey on selected topics,Bull. London Math. Soc. 26 (1994), 1–60.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • Zheng-Xu He
    • 1
  • O. Schramm
    • 2
  1. 1.University of California at San DiegoLa JollaUSA
  2. 2.Mathematics DepartmentThe Weizmann InstituteRehovotIsrael

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