A finite subdivision rule for the n-dimensional torus

  title={A finite subdivision rule for the n-dimensional torus},
  author={Brian Rushton},
  journal={Geometriae Dedicata},
  • Brian Rushton
  • Published 14 October 2011
  • Mathematics
  • Geometriae Dedicata
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. 
All finite subdivision rules are combinatorially equivalent to three-dimensional subdivision rules
Finite subdivision rules in high dimensions can be difficult to visualize and require complex topological structures to be constructed explicitly. In many applications, only the history graph isExpand
Subdivision rules for all Gromov hyperbolic groups
This paper shows that every Gromov hyperbolic group can be described by a finite subdivision rule acting on the 3-sphere. This gives a boundary-like sequence of increasingly refined finite cellExpand
Classification of subdivision rules for geometric groups of low dimension
Subdivision rules create sequences of nested cell structures on CW-complexes, and they frequently arise from groups. In this paper, we develop several tools for classifying subdivision rules. We giveExpand
Subdivision rules for special cubulated groups
We find explicit subdivision rules for all special cubulated groups. A subdivision rule for a group produces a sequence of tilings on a sphere which encode all quasi-isometric information for aExpand


Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube
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 theExpand
Subdivision rules and the eight model geometries
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 theExpand
Alternating Links and Subdivision Rules
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 tilingsExpand
Lack of Sphere Packing of Graphs via Non-Linear Potential Theory
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 isExpand
On limits of graphs sphere packed in Euclidean space and applications
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 extendedExpand
Finite subdivision rules
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) → SRExpand
The combinatorial structure of cocompact discrete hyperbolic groups
Principles of mathematical analysis
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: BasicExpand
Empilements de cercles et modules combinatoires
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
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 spaceExpand