Abstract

We study the relationship between subdivision rules, 3-dimensional manifolds, and circle packings. We find explicit subdivision rules for closed right-angled hyperbolic manifolds, a large family of hyperbolic manifolds with boundary, and all 3-manifolds of the E^3,H^2 x R, S^2 x R, SL_2(R), and S^3 geometries (up to finite covers). We define subdivision rules in all dimensions and find explicit subdivision rules for the n-dimensional torus as an example in each dimension. We define a graph and space at infinity for all subdivision rules, and use that to show that all subdivision rules for non-hyperbolic manifolds have mesh not going to 0. We provide an alternate proof of the Combinatorial Riemann Mapping Theorem using circle packings (although this has been done before). We provide a new definition of conformal for subdivision rules of unbounded valence, show that the subdivision rules for the Borromean rings complement are conformal and show that barycentric subdivision is almost conformal. Finally, we show that subdivision rules can be degenerate on a dense set, while still having convergent circle packings.

Degree

PhD

College and Department

Physical and Mathematical Sciences; Mathematics

Rights

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