The discrete Schwarz-Pick lemma for overlapping circles

Author:
Jeff Van Eeuwen

Journal:
Proc. Amer. Math. Soc. **121** (1994), 1087-1091

MSC:
Primary 30C80; Secondary 51M10, 52C15, 57M50

MathSciNet review:
1191873

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Abstract: Let *P* and 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 and the corresponding pairs of circles in 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 . 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 , and the hyperbolic distance between the centers of circles in *P* is less than or equal to the distance between the corresponding circles in . Furthermore, a single instance of finite equality in either of the above implies equality for all.

**[1]**Alan F. Beardon and Kenneth Stephenson,*The Schwarz-Pick lemma for circle packings*, Illinois J. Math.**35**(1991), no. 4, 577–606. MR**1115988****[2]**William Thurston,*The geometry and topology of 3-manifolds*, preprint, Princeton University Notes.

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1191873-3

Article copyright:
© Copyright 1994
American Mathematical Society