Finiteness properties of generalized Thompson groups via expansion sets

Abstract

We outline a general procedure that builds classifying spaces for generalized Thompson groups $\Gamma$. The construction depends on a small number of choices: (1) an inverse semigroup $S$ of partial transformations that ``locally determine" $\Gamma$; (2) an equivalence relation on certain pairs $(f,D)$, and (3) an ``expansion" rule $\mathcal{E}$. These choices determine an \emph{expansion set} $\mathcal{B}$, which is a combinatorial device that outputs a simplicial complex $\Delta^{f}_{\mathcal{B}}$ upon which $\Gamma$ acts. Under favorable conditions, often achieved in practice, $\Delta^{f}_{\mathcal{B}}$ is contractible, and the action of $\Gamma$ has small stabilizers. The definition of $\Delta^{f}_{\mathcal{B}}$ is such that ascending and descending links in $\Delta^{f}_{\mathcal{B}}$ can be described via formulas that depend only on the expansion rule $\mathcal{E}$. The result is to facilitate the usual computations of the connectivity of the descending link. Under natural hypotheses, one can prove that the acting group has type $F_{\infty}$. The net effect of our results is to automate results of this kind. Several applications are given; in particular, we sketch unified proofs that $V$, $nV$, R\"{o}ver's group $G$, and the Lodha-Moore group have type $F_{\infty}$.