Subdivision rules for special cubulated groups

Abstract

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.

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

“…In [10], we showed that every compact special cube complex has a fundamental group that is quasi-isometric to the history graph of some subdivision pair.…”