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Conformal Geometry and Dynamics

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Classification of subdivision rules for geometric groups of low dimension

Author: Brian Rushton
Journal: Conform. Geom. Dyn. 18 (2014), 171-191
MSC (2010): Primary 20F65, 05C25, 57M50
Published electronically: October 7, 2014
MathSciNet review: 3266238
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Abstract: 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 cannot 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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Additional Information

Brian Rushton
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Received by editor(s): April 7, 2014
Received by editor(s) in revised form: May 1, 2014, May 19, 2014, and June 22, 2014
Published electronically: October 7, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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