Simple expansion sets and non-positive curvature
Keywords:
generalized Thompson groups, CAT(0) cubical complexesAbstract
An expansion set is a set $\mathcal{B}$ such that each $b \in \mathcal{B}$ is equipped with a set of expansions $\mathcal{E}(b)$. The theory of expansion sets offers a systematic approach to the construction of classifying spaces for generalized Thompson groups, as described in [6].
We say that $\mathcal{B}$ is simple if proper expansions are unique when they exist, or, equivalently, if $|\mathcal{E}(b)| \leqslant 2$ for all $b \in \mathcal{B}$.
We will prove that any given simple expansion set determines a cubical complex with a metric of non-positive curvature. In many cases, the cubical complex will be CAT(0). We are thus able to recover proofs that Thompson's groups $F, T$, and $V[7,8]$, Houghton's groups $H_n$, and groups defined by finite similarity structures $[9,11,14]$ all act on CAT(0) cubical complexes with finite stabilizers. We further state a sufficient condition for the cubical complex to be locally finite, and show that the latter condition is satisfied in the cases of $F, T, V$, and $H_n$.
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