Invariable generation of certain branch groups

Abstract

Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S^{\prime }$ 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 $\mathcal{M}\mathcal{N}$, the class of groups whose maximal subgroups are all normal. We then obtain that any $2$-generated group in $\mathcal{M}\mathcal{N}$ is almost $\frac{3}{2}$-generated, and end by applying this observation to generating graphs.

8 pages; revised version, with some open questions added to the introduction. The version of record of this article, published in the Bulletin of the Malaysian Mathematical Sciences Society, is available online at: https://doi.org/10.1007/s40840-025-01895-5