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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.04219 |
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| _version_ | 1866929376439828480 |
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| author | Bodart, Corentin |
| author_facet | Bodart, Corentin |
| contents | We provide new examples of groups without rational cross-sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Specifically, our examples are extensions of infinite torsion groups, groups of Grigorchuk type, wreath products similar to $C_2\wr(C_2\wr \mathbb Z)$ and $\mathbb Z\wr F_2$, a group of permutations of $\mathbb Z$, and a finitely presented HNN extension of the first Grigorchuk group. This last group is the first example of finitely presented group with solvable word problem and without rational cross-sections. It is also not autostackable, and has no left-regular complete rewriting system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04219 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rational cross-sections, bounded generation and orders on groups Bodart, Corentin Group Theory Formal Languages and Automata Theory 06F15, 20F05, 20F10 We provide new examples of groups without rational cross-sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Specifically, our examples are extensions of infinite torsion groups, groups of Grigorchuk type, wreath products similar to $C_2\wr(C_2\wr \mathbb Z)$ and $\mathbb Z\wr F_2$, a group of permutations of $\mathbb Z$, and a finitely presented HNN extension of the first Grigorchuk group. This last group is the first example of finitely presented group with solvable word problem and without rational cross-sections. It is also not autostackable, and has no left-regular complete rewriting system. |
| title | Rational cross-sections, bounded generation and orders on groups |
| topic | Group Theory Formal Languages and Automata Theory 06F15, 20F05, 20F10 |
| url | https://arxiv.org/abs/2210.04219 |