## Expansion Complexes for Finite Subdivision Rules I

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

### Abstract

This paper develops the basic theory of conformal structures on finite
subdivision rules. The work depends heavily on the use of expansion
complexes, which are defined and discussed in detail. It is proved that
a
finite subdivision rule with bounded valence and mesh approaching 0 is
conformal (in the combinatorial sense) if there is a conformal structure
on
the model subdivision complex with respect to which the subdivision map
is
conformal. This gives a new approach to the difficult combinatorial
problem
of determining when a finite subdivision rule is conformal.
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