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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The discrete Schwarz-Pick lemma for overlapping circles
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by Jeff Van Eeuwen PDF
Proc. Amer. Math. Soc. 121 (1994), 1087-1091 Request permission

Abstract:

Let P and $P’$ be circle packings in the hyperbolic plane such that they are combinatorically equivalent, neighboring circles in P overlap one another at some fixed angle between 0 and $\pi /2$ and the corresponding pairs of circles in $P’$ overlap at the same angle, and the radius for any boundary circle of P is less than or equal to that of the corresponding boundary circle of $P’$. In this paper we show that the radius of any interior circle of P is less than or equal to that of the corresponding circle in $P’$, and the hyperbolic distance between the centers of circles in P is less than or equal to the distance between the corresponding circles in $P’$. Furthermore, a single instance of finite equality in either of the above implies equality for all.
References
  • Alan F. Beardon and Kenneth Stephenson, The Schwarz-Pick lemma for circle packings, Illinois J. Math. 35 (1991), no. 4, 577–606. MR 1115988
  • William Thurston, The geometry and topology of 3-manifolds, preprint, Princeton University Notes.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 121 (1994), 1087-1091
  • MSC: Primary 30C80; Secondary 51M10, 52C15, 57M50
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1191873-3
  • MathSciNet review: 1191873