A finite subdivision rule for the n-dimensional torus

Abstract

Abstract. Cannon, Floyd, and Parry have studied subdivisions of the 2-sphere extensively, especially those corresponding to 3-manifolds, in an attempt to prove Cannon's conjecture. There has been a recent interest in generalizing some of their tools, such as extremal length, to higher dimensions. We define finite subdivision rules of dimension n, and find an n − 1-dimensional finite subdivision rule for the n-dimensional torus, using a well-known simplicial decomposition of the hypercube.

Classification of subdivision rules for geometric groups of low dimension

“…The most commonly studied type of subdivision rule is a finite subdivision rule [2,27]. Intuitively, a finite subdivision rule takes a CW-complex where each cell is labelled and refines each cell into finitely many smaller labelled cells according to a recursive rule.…”

“…There are numerous explicit examples of finite subdivision rules on the 2-sphere that represent hyperbolic 3-manifolds, as well as subdivision rules representing hyperbolic knot complements [2,[25][26][27].…”

Subdivision rules for special cubulated groups

“…In this paper, we find subdivision rules for RAAG's that subdivide or act on the n-sphere. In [23], we defined a subdivision rule in higher dimensions in a way analogous to subdivision rules in dimension 2. We repeat that definition here.…”

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

“…Cannon and Swenson and shown that two-dimensional subdivision rules for a three-dimensional hyperbolic manifold group contain enough information to reconstruct the group itself [2,5]. This was later generalized to show that many groups can be associated to a subdivision rule of some dimension, and that these subdivision rules [9,8]:…”