Simple totally disconnected locally compact groups separated by finiteness properties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908902447120384 |
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| author | Bonn, Laura Giersbach, Sebastian |
| author_facet | Bonn, Laura Giersbach, Sebastian |
| contents | We construct a sequence of simple non-discrete totally disconnected locally compact (tdlc) groups separated by finiteness properties; that is, for every positive integer $n$ there exists a simple non-discrete tdlc group that is of type $F_{n-1}$ but not of type $F_n$. This generalizes a result for discrete groups of Skipper--Witzel--Zaremsky. Furthermore, we construct a simple non-discrete tdlc group that is of type $FP_2$ over $\mathbb{Z}$ but not compactly presented. Our examples arise as Smith universal groups $\mathcal{U}(M, N)$ associated to permutation groups $M$ and $N$. We generalize a theorem of Haglund--Wise to tdlc groups and show that under mild conditions on $M$ and $N$ the finiteness properties of $\mathcal{U}(M, N)$ reflect those of its local actions $M$ and $N$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_05101 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple totally disconnected locally compact groups separated by finiteness properties Bonn, Laura Giersbach, Sebastian Group Theory 22D05 (Primary) 20E06, 20E08, 20E32, 20F05, 20F65 (Secondary) We construct a sequence of simple non-discrete totally disconnected locally compact (tdlc) groups separated by finiteness properties; that is, for every positive integer $n$ there exists a simple non-discrete tdlc group that is of type $F_{n-1}$ but not of type $F_n$. This generalizes a result for discrete groups of Skipper--Witzel--Zaremsky. Furthermore, we construct a simple non-discrete tdlc group that is of type $FP_2$ over $\mathbb{Z}$ but not compactly presented. Our examples arise as Smith universal groups $\mathcal{U}(M, N)$ associated to permutation groups $M$ and $N$. We generalize a theorem of Haglund--Wise to tdlc groups and show that under mild conditions on $M$ and $N$ the finiteness properties of $\mathcal{U}(M, N)$ reflect those of its local actions $M$ and $N$. |
| title | Simple totally disconnected locally compact groups separated by finiteness properties |
| topic | Group Theory 22D05 (Primary) 20E06, 20E08, 20E32, 20F05, 20F65 (Secondary) |
| url | https://arxiv.org/abs/2509.05101 |