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We outline a general procedure that builds classifying spaces for generalizedThompson groups $\Gamma$. The construction depends on a small number ofchoices: (1) an inverse semigroup $S$ of partial transformations that ``locallydetermine" $\Gamma$; (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 thatoutputs 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 $\Gamma$ hassmall stabilizers.  The definition of $\Delta^{f}_{\mathcal{B}}$ is such that ascending anddescending links in $\Delta^{f}_{\mathcal{B}}$ can be described via formulasthat depend only on the expansion rule $\mathcal{E}$. The result is tofacilitate 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 thiskind.  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$F_{\infty}$.

Finiteness properties of generalized Thompson groups via expansion sets