Invariable generation of certain branch groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915309001113600 |
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| author | Cox, Charles Garnet Thillaisundaram, Anitha |
| author_facet | Cox, Charles Garnet Thillaisundaram, Anitha |
| contents | Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S'$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key examples of branch groups. In particular, we prove that all generating sets of the torsion Grigorchuk groups, of the branch Grigorchuk-Gupta-Sidki groups and of the torsion multi-EGS groups (which are natural generalisations of the Grigorchuk-Gupta-Sidki groups) are invariable generating sets. Furthermore, for the first Grigorchuk group and the torsion Grigorchuk-Gupta-Sidki groups, every finitely generated subgroup has a finite invariable generating set. Our results apply to finitely generated groups in $\mathcal{MN}$, the class of groups whose maximal subgroups are all normal. We then obtain that any $2$-generated group in $\mathcal{MN}$ is almost $\frac{3}{2}$-generated, and end by applying this observation to generating graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14022 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invariable generation of certain branch groups Cox, Charles Garnet Thillaisundaram, Anitha Group Theory Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S'$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key examples of branch groups. In particular, we prove that all generating sets of the torsion Grigorchuk groups, of the branch Grigorchuk-Gupta-Sidki groups and of the torsion multi-EGS groups (which are natural generalisations of the Grigorchuk-Gupta-Sidki groups) are invariable generating sets. Furthermore, for the first Grigorchuk group and the torsion Grigorchuk-Gupta-Sidki groups, every finitely generated subgroup has a finite invariable generating set. Our results apply to finitely generated groups in $\mathcal{MN}$, the class of groups whose maximal subgroups are all normal. We then obtain that any $2$-generated group in $\mathcal{MN}$ is almost $\frac{3}{2}$-generated, and end by applying this observation to generating graphs. |
| title | Invariable generation of certain branch groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2311.14022 |