2015/12/01 by Brian Rushton, Rushton, Brian
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.1512.00367
6 pages
arxiv created 2015/12/01 · arxiv updated 2015/12/02
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.