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Conformal modulus: The graph paper invariant or the conformal shape of an algorithm. (English) Zbl 1020.30020
Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14-19, 1996. Berlin: de Gruyter. 71-102 (1999).
This paper gives an exposition of previous and present work of the authors on combinatorial conformal moduli. A quadrilateral or a ring $$Q$$ is given, with no Riemannian metric. Instead, $$Q$$ is equipped with a tiling by disks. There is a function $$\rho:T\to[0,\infty[$$, which is defined on the set $$T$$ of tiles, which is not identically $$0$$, and which is called a weight function on $$T$$. The function $$\rho$$ is thought of as a local scaling function, and it plays the role of a conformal change of metric. It is used to define approximate distances and areas in $$Q$$, and this in turn allows the authors to define height, width or circumference, area, and combinatorial modulus. A conformal invariant is then defined, by taking the supremum of the conformal moduli for all weight functions $$\rho$$. One beautiful result of the authors is the existence and uniqueness of combinatorial optimal weight functions, that is, weight functions that realize the supremum of conformal modulus (like in the classical analytical case). The authors give an exposition of what they call the graph paper theorem, which describes geometrically the optimal weight functions (a fact which has been proven independently by the authors [Contemp. Math. 169, 133-212 (1994; Zbl 0818.20043)] and by O. Schramm [Isr. J. Math. 84, 97-118 (1993; Zbl 0788.05019)]). As a corollary to this theorem, the authors obtained their finite Riemann mapping theorem. In the present paper, the authors describe and discuss several algorithms for finding the combinatorial optimal weight functions. They review then how combinatorial moduli apply to the study of negatively curved or Gromov word hyperbolic groups, and they give examples to show how their work might be used to recognise a Kleinian group combinatorially. The paper ends with a list of open questions.
For the entire collection see [Zbl 0910.00040].

##### MSC:
 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods 30F99 Riemann surfaces