On the generation problem in Thompson's groups $F_n$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917550704558080 |
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| author | Golan, Gili Sapir, Eytan |
| author_facet | Golan, Gili Sapir, Eytan |
| contents | We study the generation problem in the Higman-Thompson groups $F_n$ via the core and closure of subgroups of $F_n$ and associated automata. We give sufficient conditions for a subset $X \subseteq F_n$ to generate $F_n$, and provide an algorithm which verifies these conditions when $X$ is finite. As an application, we answer a question of Aiello and Nagnibeda, motivated by Savchuk's problem on maximal subgroups of Thompson's group $F$. Specifically, we show that for every $n\geq 2$, the Higman-Thompson group $F_n$ contains a maximal subgroup of infinite index which fixes no point of $(0,1)$. The subgroup we construct is isomorphic to $F_{2n-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00863 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the generation problem in Thompson's groups $F_n$ Golan, Gili Sapir, Eytan Group Theory We study the generation problem in the Higman-Thompson groups $F_n$ via the core and closure of subgroups of $F_n$ and associated automata. We give sufficient conditions for a subset $X \subseteq F_n$ to generate $F_n$, and provide an algorithm which verifies these conditions when $X$ is finite. As an application, we answer a question of Aiello and Nagnibeda, motivated by Savchuk's problem on maximal subgroups of Thompson's group $F$. Specifically, we show that for every $n\geq 2$, the Higman-Thompson group $F_n$ contains a maximal subgroup of infinite index which fixes no point of $(0,1)$. The subgroup we construct is isomorphic to $F_{2n-1}$. |
| title | On the generation problem in Thompson's groups $F_n$ |
| topic | Group Theory |
| url | https://arxiv.org/abs/2606.00863 |