## Conformal modulus the graph paper invariant

or

The conformal shape of an algorithm

###
J. W. Cannon, W. J. Floyd, and W. R. Parry

December 19, 1996

### Abstract

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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