Finiteness properties of stabilisers of oligomorphic actions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915822445789184 |
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| author | Fournier-Facio, Francesco Kropholler, Peter H. Lyman, Robert Alonzo Zaremsky, Matthew C. B. |
| author_facet | Fournier-Facio, Francesco Kropholler, Peter H. Lyman, Robert Alonzo Zaremsky, Matthew C. B. |
| contents | An action of a group on a set is oligomorphic if it has finitely many orbits of $n$-element subsets for all $n$. We prove that for a large class of groups (including all groups of finite virtual cohomological dimension and all countable linear groups), for any oligomorphic action of such a group on an infinite set there exists a finite subset whose stabiliser is not of type $\mathrm{FP}_\infty$. This leads to obstructions on finiteness properties for permutational wreath products and twisted Brin-Thompson groups. We also prove a version for actions on flag complexes, and discuss connections to the Boone-Higman conjecture. In the appendix, we improve on the criterion of Bartholdi-Cornulier-Kochloukova for finiteness properties of wreath products, and the criterion of Kropholler-Martino for finiteness properties of graph-wreath products. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_02319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness properties of stabilisers of oligomorphic actions Fournier-Facio, Francesco Kropholler, Peter H. Lyman, Robert Alonzo Zaremsky, Matthew C. B. Group Theory Geometric Topology 20F65, 20B07 An action of a group on a set is oligomorphic if it has finitely many orbits of $n$-element subsets for all $n$. We prove that for a large class of groups (including all groups of finite virtual cohomological dimension and all countable linear groups), for any oligomorphic action of such a group on an infinite set there exists a finite subset whose stabiliser is not of type $\mathrm{FP}_\infty$. This leads to obstructions on finiteness properties for permutational wreath products and twisted Brin-Thompson groups. We also prove a version for actions on flag complexes, and discuss connections to the Boone-Higman conjecture. In the appendix, we improve on the criterion of Bartholdi-Cornulier-Kochloukova for finiteness properties of wreath products, and the criterion of Kropholler-Martino for finiteness properties of graph-wreath products. |
| title | Finiteness properties of stabilisers of oligomorphic actions |
| topic | Group Theory Geometric Topology 20F65, 20B07 |
| url | https://arxiv.org/abs/2506.02319 |