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The Schwarz-Pick lemma for circle packings. (English) Zbl 0753.30016

It was suggested by W. Thurston that the circle packing isomorphism of Andreev could be used to approximate the conformal mapping of a simply connected domain \(\Omega\) onto the disk \(\Delta\): Given a certain packing \(P\) of the domain, \(P_ a\) is a combinatorially equivalent packing of \(\Delta\). The isomorphism can be used to define a piecewise affine mapping and as the sizes of the circles go to zero, the mapping should converge to the conformal mapping of \(\Omega\) onto \(\Delta\), as was subsequently proved by B. Rodin and D. Sullivan [J. Differ. Geom. 26, 349-360 (1987; Zbl 0694.30006)].
Here the authors develop the analogy of the discrete with the continuous cases by studying the isomorphism of a fixed \(P\) with \(P_ a\). The main result is the “Discrete Schwarz-Pick Lemma” (DSPL), which parallels the classical result as set in the hyperbolic plane. The informal statement of the result is this:
Let \(P\) be a circle packing in the hyperbolic plane, \(P_ a\) its Andreev packing. Then: (a) Each circle of \(P\) has a hyperbolic radius which is less than or equal to that of the corresponding circle in \(P_ a\). (b) The hyperbolic distance between (centers of) circles in \(P\) is less than or equal to the hyperbolic distance between the corresponding circles in \(P_ a\). Moreover, a single instance of finite equality in either (a) or (b) implies equality in every instance, i.e., \(P\) and \(P_ a\) are hyperbolically congruent.
In proving this result, the authors develop the analogs of several classical ideas. A circle packing is viewed as a simplicial complex which is then endowed with a hyperbolic structure. In fact, the hyperbolic complex is first defined and then certain of these are recognized as packings. Andreev’s theorem is then stated in this setting. (In an appendix, a new, inductive, proof of Andreev’s theorem is presented.) Next, an analog of Perron’s method is used to solve a boundary value problem, and this result is used to prove the DSPL. The paper is intended to begin the development of a discrete complex analysis.
Two of the figures in the paper were left unlabelled by the printer. Corrected versions of the figures appear in the Summer 1992 issue of the journal (Vol. 36, p. 177).

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
51M99 Real and complex geometry
31C20 Discrete potential theory
52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)

Citations:

Zbl 0694.30006
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