Classification of subdivision rules for geometric groups of low dimension (Q2922927)

The author extends the notion of subdivision rule \(R\) for a finite \(2\)-dimensional CW complex due to Cannon, Floyd and Parry [\textit{J. W. Cannon} et al., Conform. Geom. Dyn. 5, 153--196 (2001; Zbl 1060.20037)] to a finite \(n\)-dimensional CW complex \(S_R\), \(n\) arbitrary. The tiles of \(S_R\) are partitioned into \textit{ideal} and \textit{non-ideal} tiles. If \(X\) is a finite \(R\)-complex, one obtains a sequence \(R(X), R^2(X), \ldots\) of \(R\)-complexes by applying the subdivision rule \(R\). One has the notion of \(R\) being hyperbolic with respect to an \(R\)-complex \(X\).NEWLINENEWLINEDenote by \(\Lambda_k\subset R^k(X)\) the union of all non-ideal tiles in the \(k\)th level subdivision. The space \(\Lambda:=\cup_{k\geq 0} \Lambda_k\) is called the limit set. The history graph \(\Gamma=\Gamma(R,X)\) is defined as follows: The vertex set consists of (i) a vertex \(O\) called the origin, (ii) a vertex for each cell in \(\Lambda_k\), for \(k\geq 0\), which is thought of as being located at level \(k\). Two vertices at the same level are joined by a \textit{horizontal} edge if there is a containment relation between the corresponding (closed) cells; a vertex \(v\) in level \(k\) is joined by a \textit{vertical} edge to a vertex \(w\) in level \(k+1\) if the corresponding cells \(T_w\) is contained in the interior of \(R(T_v)\). Finally every vertex at level \(0\) is joined to the origin \(O\). The following are among the main results of the paper. Theorem 5. \textit{Let \(R\) be a subdivision rule and let \(X\) be a finite \(R\)-complex. If \(R\) is hyperbolic with respect to \(X\), then the history graph \(\Gamma(R,X)\) is Gromov hyperbolic. } Theorem 7. \textit{History graphs of subdivision rules are combable.}NEWLINENEWLINEIn particular, any group which is quasi-isometric to a history graph satisfies a quadratic isoperimetric inequality. The main results are applied to classification of subdivision rules, in terms of their limit sets, when their history graph is quasi-isometric to the group of integers or a Fuchsian group. Results for three-dimensional geometric groups are also obtained.