  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.
