The classical Riemann mapping theorem asserts that any topological quadrilateral in the complex plane can be mapped conformally onto a rectangle. The finite Riemann mapping theorem asserts that anyâ€¦ (More)

It has been conjectured that if G is a negatively curved discrete group with space at infinity âˆ‚G the 2 -sphere, then G has a properly discontinuous, cocompact, isometric action on hyperbolic 3â€¦ (More)

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 isâ€¦ (More)

Let G be a finitely generated group, and let Z be a finite generating set of G . The growth function of (G, Z) is the generating function f(z) = 12T=oa"z" > where aâ€ž is the number of elements of Gâ€¦ (More)

We give a mechanical recipe for creating simple face-pairing descriptions of closed 3-manifolds. We call the technique twisted face-pairing. Among the simpler twisted face-pairings we have studied,â€¦ (More)

This paper is an enriched version of our introductory paper on twisted face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also everyâ€¦ (More)

We take an in-depth look at Thurstonâ€™s combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston maps. These are orientation-preservingâ€¦ (More)

We introduce and study finite subdivision rules. A finite subdivision rule is a finite list of instructions which determines a subdivision of a given planar tiling. Given a finite subdivision ruleâ€¦ (More)

Suppose R is a finite subdivision rule with an edge pairing. Then the subdivision map ÏƒR is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2-sphere. If Râ€¦ (More)

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 criticalâ€¦ (More)