2 citations · 6 across the 11 of their papers we have counts for
11 papers
Subdivision rules for all Gromov hyperbolic groups
Brian Rushton
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 refine…
All finite subdivision rules are combinatorially equivalent to three-dimensional subdivision rules
Brian Rushton
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 hi…
Enumeration of m-Endomorphisms
Louis Rubin, Brian Rushton
An m-endomorphism of a free semigroup is an endomorphism that sends every generator to a word of length at most m. Two m-endomorphisms are combinatorially equivalent if they are co…
Classification of subdivision rules for geometric groups of low dimension
Brian Rushton
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 subdivi…
Subdivision rules for special cubulated groups
Brian Rushton
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 info…
Subdivision rules and the eight model geometries
Brian Rushton
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 t…
Creating subdivision rules from polyhedra with identifications
Brian Rushton
Cannon, Swenson, and others have proved numerous theorems about subdivision rules associated to hyperbolic groups with a 2-sphere at infinity. However, few explicit examples are kn…
A finite subdivision rule for the n-dimensional torus
Brian Rushton
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…
Modulus of unbounded valence subdivision rules
Brian Rushton
Cannon, Floyd and Parry have studied the modulus of finite subdivision rules extensively. We investigate the properties of the modulus of subdivision rules with linear and exponent…
A circle packing proof of the Combinatorial Riemann Mapping Theorem
Brian Rushton
The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle pa…
Constructing subdivision rules from alternating links
Brian C. Rushton
The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construc…