Conformal modulus the graph paper invariant
The conformal shape of an algorithm

J. W. Cannon, W. J. Floyd, and W. R. Parry
December 19, 1996


This paper is an expository paper about our joint work, which the first author presented in a series of lectures at the University of Auckland (New Zealand), the University of Melbourne (Australia), and the Australian National University in Canberra (Australia). The first section, which is our own nonproof of the Riemann Mapping Theorem, can be used as a good intuitive introduction to the long and fussy proof of our own combinatorial Riemann mapping theorem [CRMT]. In particular, it demonstrates the geometry underlying the classical conformal modulus of a quadrilateral or annulus. The second section shows how the classical conformal modulus is applied to combinatorics, with the intent of preparing for the exposition of sections 3 and 4. The third section shows that, under subdivision, a topological quadrilateral can develop wildly oscillating conformal modulus, a behavior which was perhaps not expected. The final section, section 4, reviews how combinatorial moduli apply to the study of negatively curved or Gromov word hyperbolic groups and shows by example how our work might be used to recognize a Kleinian group combinatorially.

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