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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2312.10211 |
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| _version_ | 1866913496409571328 |
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| author | Farley, Daniel |
| author_facet | Farley, Daniel |
| contents | We outline a general procedure that builds classifying spaces for generalized Thompson groups $Γ$. The construction depends on a small number of choices: (1) an inverse semigroup $S$ of partial transformations that ``locally determine" $Γ$; (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 $Δ^{f}_{\mathcal{B}}$ upon which $Γ$ acts. Under favorable conditions, often achieved in practice, $Δ^{f}_{\mathcal{B}}$ is contractible, and the action of $Γ$ has small stabilizers.
The definition of $Δ^{f}_{\mathcal{B}}$ is such that ascending and descending links in $Δ^{f}_{\mathcal{B}}$ 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 $V$, $nV$, Röver's group $G$, and the Lodha-Moore group have type $F_{\infty}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10211 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finiteness properties of generalized Thompson groups via expansion sets Farley, Daniel Group Theory 20F65 We outline a general procedure that builds classifying spaces for generalized Thompson groups $Γ$. The construction depends on a small number of choices: (1) an inverse semigroup $S$ of partial transformations that ``locally determine" $Γ$; (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 $Δ^{f}_{\mathcal{B}}$ upon which $Γ$ acts. Under favorable conditions, often achieved in practice, $Δ^{f}_{\mathcal{B}}$ is contractible, and the action of $Γ$ has small stabilizers. The definition of $Δ^{f}_{\mathcal{B}}$ is such that ascending and descending links in $Δ^{f}_{\mathcal{B}}$ 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 $V$, $nV$, Röver's group $G$, and the Lodha-Moore group have type $F_{\infty}$. |
| title | Finiteness properties of generalized Thompson groups via expansion sets |
| topic | Group Theory 20F65 |
| url | https://arxiv.org/abs/2312.10211 |