Modulus of unbounded valence subdivision rules

TL;DR

This paper examines the conformality of subdivision rules with unbounded valence, focusing on linear and exponential growth, and demonstrates how existing criteria apply differently depending on growth type.

Contribution

It introduces new insights into the conformality of subdivision rules with unbounded valence, especially for exponential growth, and extends the applicability of the 1,2,3-tile criterion.

Findings

Borromean rings subdivision rule is conformal

1,2,3-tile criterion suffices for linear growth

Criterion is weaker for exponential growth

Abstract

Cannon, Floyd and Parry have studied the modulus of finite subdivision rules extensively. We investigate the properties of the modulus of subdivision rules with linear and exponential growth at every vertex, using barycentric subdivision and a subdivision rule for the Borromean rings as examples. We show that the subdivision rule arising from the Borromean rings is conformal, and conjecture that the subdivision rules for all alternating links are conformal. We show that the 1,2,3-tile criterion of Cannon, Floyd, and Parry is sufficient to prove conformality for linear growth, but not exponential growth. We show that the criterion gives a weaker form of conformality for subdivision rules of exponential growth at each vertex. We contrast this with the known, bounded-valence case, and illustrate our results with circle packings using Ken Stephenson's Circlepack.