All finite subdivision rules are combinatorially equivalent to three-dimensional subdivision rules

TL;DR

This paper demonstrates that any finite subdivision rule in arbitrary dimensions can be represented as a three-dimensional subdivision rule, simplifying visualization and analysis of high-dimensional subdivisions.

Contribution

It introduces a method to convert finite subdivision rules of any dimension into equivalent three-dimensional rules using history graphs.

Findings

Finite subdivision rules are equivalent to three-dimensional rules.

The history graph characterizes the subdivision rule.

Gromov boundaries of certain hyperbolic groups relate to 3D space.

Abstract

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.