Expansion Complexes for Finite Subdivision Rules I

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


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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