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Brian Rushton

Publications and source records attributed to Brian Rushton.

10 recordsLinked to original sources
arXiv2017

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 cell complexes which carry all quasi-isometry information about the group. This extends a result from Cannon and Swenson in 1998 that hyperbolic groups can be described by a recursive sequence of overlapping coverings by possibly wild sets, and demonstrates the existence of non-cubulated groups that can be represented by subdivision rules.

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arXiv2015

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 is needed. We characterize the history graph of a subdivision rule, and define a combinatorial subdivision rule based on such graphs. We use this to show that a finite subdivision rule of arbitrary dimension is combinatorially equivalent to a three-dimensional subdivision rule. We use this to show that the Gromov boundary of special cubulated hyperbolic groups is a quotient of a compact subset of three-dimensional space, with connected preimages at each point.

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arXiv2015

Enumeration of m-Endomorphisms

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 conjugate under an automorphism of the semigroup. In this paper, we specialize an argument of N. G. de Bruijn to produce a formula for the number of combinatorial equivalence classes of m-endomorphisms on a rank-n semigroup. From this formula, we derive several little-known integer sequences.

math.CO↗
arXiv2014

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 give a criterion for a subdivision rule to represent a Gromov hyperbolic space, and show that a subdivision rule for a hyperbolic group determines the Gromov boundary. We give a criterion for a subdivision rule to represent a Euclidean space of dimension less than 4. We also show that Nil and Sol geometries can not be modeled by subdivision rules. We use these tools and previous theorems to classify the geometry of subdivision rules for low-dimensional geometric groups by the combinatorial properties of their subdivision rules.

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arXiv2013

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 a group. We show how these tilings detect properties such as growth, ends, divergence, etc. We include figures of several worked out examples.

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arXiv2012

A finite subdivision rule for the n-dimensional torus

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.

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arXiv2012

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 the sphere. Explicit subdivision rules have also been found for some closed and finite-volume hyperbolic manifolds, as well as a few non-hyperbolic knot complements. We extend these results by finding explicit finite subdivision rules for closed manifolds of the E3, H2XR, S2XR, S3, and SL2(R) manifolds by examining model manifolds. Because all manifolds in these geometries are the same up to finite covers, the subdivision rules for these model manifolds will be very similar to subdivision rules for all other manifolds in their respective geometries. We also discuss the existence of subdivision rules for Nil and Sol geometries. We use Ken Stephenson's Circlepack to visualize the subdivision rules and the resulting space at infinity.

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arXiv2012

Creating subdivision rules from polyhedra with identifications

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 known. We construct an explicit subdivision rule for many 3-manifolds from polyhedral gluings. The manifolds that satisfy the conditions include all closed manifolds created from right-angled hyperbolic polyhedra, as well as many 3-manifolds with toral or hyperbolic boundary.

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arXiv2011

Modulus of unbounded valence subdivision rules

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 exponential growth at every vertex, using barycentric subdivision and a subdivision rule for the Borromean rings as examples. We show that the subdivision rule arising from the Borromean rings is conformal, and conjecture that the subdivision rules for all alternating links are conformal. We show that the 1,2,3-tile criterion of Cannon, Floyd, and Parry is sufficient to prove conformality for linear growth, but not exponential growth. We show that the criterion gives a weaker form of conformality for subdivision rules of exponential growth at each vertex. We contrast this with the known, bounded-valence case, and illustrate our results with circle packings using Ken Stephenson's Circlepack.

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