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 $P’$ 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.
Alan F. Beardon and Kenneth R. Stephenson, The Schwarz-Pick Lemma for circle packings, Illinois J. Math. 35 (1991), 577-606.
William Thurston, The geometry and topology of 3-manifolds, preprint, Princeton University Notes.