Classification of subdivision rules for geometric groups of low dimension

TL;DR

This paper develops tools to classify subdivision rules associated with low-dimensional geometric groups, linking their combinatorial properties to the underlying geometric structures such as hyperbolic and Euclidean spaces.

Contribution

It introduces criteria for subdivision rules to represent specific geometries and classifies low-dimensional geometric groups based on these rules' properties.

Findings

Subdivision rules can represent hyperbolic and Euclidean geometries under certain criteria.

Nil and Sol geometries cannot be modeled by subdivision rules.

Classification of subdivision rules correlates with the geometric type of the associated groups.

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 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.