A finite subdivision rule for the n-dimensional torus

Abstract

Cannon, Floyd, and Parry have studied subdivisions of the 2-sphere extensively, especially those corresponding to 3-manifolds, in an attempt to prove Cannon's conjecture. There has been a recent interest in generalizing some of their tools, such as extremal length, to higher dimensions. We define finite subdivision rules of dimension n, and find an n-1-dimensional finite subdivision rule for the n-dimensional torus, using a well-known simplicial decomposition of the hypercube. We hope to expand on this and find finite subdivision rules for many higher-dimensional manifolds, including hyperbolic n-manifolds.