# A finite subdivision rule for the n-dimensional torus

@article{Rushton2013AFS, title={A finite subdivision rule for the n-dimensional torus}, author={Brian Rushton}, journal={Geometriae Dedicata}, year={2013}, volume={167}, pages={23-34} }

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.

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#### References

SHOWING 1-10 OF 10 REFERENCES

Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube

- Mathematics
- 2010

Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the… Expand

Subdivision rules and the eight model geometries

- Mathematics
- 2012

Cannon and Swenson have shown that each hyperbolic 3-manifold group has a natural subdivision rule on the space at infinity, and that this subdivision rule captures the action of the group on the… Expand

Alternating Links and Subdivision Rules

- Mathematics
- 2009

ALTERNATING LINKS AND SUBDIVISION RULES Brian Rushton Department of Mathematics Master of Science The study of geometric group theory has suggested several theorems related to subdivision tilings… Expand

Lack of Sphere Packing of Graphs via Non-Linear Potential Theory

- Mathematics
- 2009

It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1} or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z, in R^d, for all d. A similar result is… Expand

On limits of graphs sphere packed in Euclidean space and applications

- Mathematics, Computer Science
- Eur. J. Comb.
- 2011

The core of this note is the observation that links between circle packings of graphs and potential theory developed in Benjamini and Schramm (2001) [4] and He and Schramm (1995) [11] can be extended… Expand

Finite subdivision rules

- Mathematics
- 2001

We introduce and study finite subdivision rules. A finite subdivision rule R consists of a finite 2-dimensional CW complex SR, a subdivision R(SR) of SR, and a continuous cellular map φR : R(SR) → SR… Expand

Principles of mathematical analysis

- Mathematics
- 1964

Chapter 1: The Real and Complex Number Systems Introduction Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix Exercises Chapter 2: Basic… Expand

Empilements de cercles et modules combinatoires

- Mathematics
- 2006

Le but cette note est de tenter d'expliquer les liens etroits qui unissent la theorie des empilements de cercles et des modules combinatoires, et de comparer les approches a la conjecture de J.W.… Expand

RECOGNIZING CONSTANT CURVATURE DISCRETE GROUPS IN DIMENSION 3

- 1997

We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space H3 in terms of the combinatorics of the action of G on its space… Expand