A finite subdivision rule for the n-dimensional torus

TL;DR

This paper introduces a finite subdivision rule for the n-dimensional torus, extending subdivision concepts from 2-spheres to higher dimensions, with potential applications to hyperbolic n-manifolds.

Contribution

It defines finite subdivision rules for n-dimensional spaces and constructs an explicit rule for the n-torus using a known hypercube decomposition.

Findings

Established an n-1-dimensional subdivision rule for the n-torus

Utilized a simplicial decomposition of the hypercube

Lays groundwork for subdivision rules of higher-dimensional manifolds

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.