Constructing Subdivision Rules from Rational Maps

This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f is the subdivision map of a finite subdivision rule. We are interested here in connections between finite subdivision rules and rational maps. Finite subdivision rules arose out of our attempt to resolve Cannon’s Conjecture: If G is a Gromov-hyperbolic group whose space at infinity is a 2-sphere, then G has a cocompact, properly discontinuous, isometric action on hyperbolic 3-space. Cannon’s Conjecture can be reduced (see, for example, the Cannon-Swenson paper [5]) to a conjecture about (combinatorial) conformality for the action of such a group G on its space at infinity, and finite subdivision rules were developed to give models for the action of a Gromov-hyperbolic group on the 2-sphere at infinity. There is also a connection between finite subdivision rules and rational maps. If R is an orientation-preserving finite subdivision rule such that the subdivision complex SR is a 2-sphere, then the subdivision map σR is a critically finite branched map of this 2-sphere. In joint work [3] with Kenyon we consider these subdivision maps under the additional hypotheses that R has bounded valence (this is equivalent to its not having periodic critical points) and mesh approaching 0. In [3, Theorem 3.1] we show that if R is conformal (in the combinatorial sense) then the subdivision map σR is equivalent to a rational map. The converse follows from [4, Theorem 4.7]. In this paper we consider the problem of when a rational map f can be equivalent to the subdivision map of a finite subdivision rule. Since a subdivision complex has only finitely many vertices, such a rational map must be critically finite. We specialize here to the case that f has no periodic critical points. Our main theorem, which has also been proved by Bonk and Meyer [1] when f has a hyperbolic orbifold (which includes all but some well-understood examples), is the following: Date: March 15, 2007. 2000 Mathematics Subject Classification. Primary 37F10, 52C20; Secondary 57M12.