Classification of subdivision rules for geometric groups of low dimension

Abstract

Abstract. Subdivision rules create sequences of nested cell structures on CWcomplexes, and they frequently arise from groups. In this paper, we develop several tools for classifying subdivision rules. We give a criterion for a subdivision rule to represent a Gromov hyperbolic space, and show that a subdivision rule for a hyperbolic group determines the Gromov boundary. We give a criterion for a subdivision rule to represent a Euclidean space of dimension less than 4. We also show that Nil and Sol geometries cannot be modeled by subdivision rules. We use these tools and previous theorems to classify the geometry of subdivision rules for low-dimensional geometric groups by the combinatorial properties of their subdivision rules.

Expansion properties for finite subdivision rules I

“…Because Γ n might not be connected, δ n might take the value ∞. Following Rushton [23], we define a transition function f m,n : Γ n → Γ m for all integers m, n ≥ −1 such that m ≤ n. Let t be a tile of R n (S 2 ). Then f (v(t)) = v(s), where s is the tile of R m (S 2 ) which contains t. We extend f m,n to the edges of Γ n in the straightforward way, so that f m,n is a cellular map.…”

“…For the proof, we use some general results of Rushton [23]. These lead to a condition on R which is necessary and sufficient for the fat path subdivision graph to be Gromov hyperbolic.…”

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

“…(Note: In [11], we used an alternative definition for history graph that had a vertex for every cell of any dimension in Λ n , with edges being induced by inclusion. However, the two history graphs are quasi-isometric when any top-dimensional cells that intersect do so in a codimension 1 subset.…”

“…Proof. In [11], we showed that the limit set Λ of a subdivision pair whose history graph is quasi-isometric to a hyperbolic group G has a canonical quotient onto the Gromov boundary ∂G, with connected preimages.…”

“…(1) captures all of the quasi-isometry information of the group via a graph called the history graph, and (2) has simple combinatorial tests for many quasi-isometry properties [11].…”