Invariable generation of certain branch groups
Charles Garnet Cox, Anitha Thillaisundaram
TL;DR
The paper investigates invariable generation (IG) and the finite variant FIG within branch groups, linking IG properties to the maximal-subgroup structure. It proves that generating sets are IG-sets for broad families of branch groups (including torsion Grigorchuk groups, branch Grigorchuk-Gupta-Sidki groups, and torsion multi-EGS groups) and that, in the first Grigorchuk group and torsion GGS groups, every finitely generated subgroup has a finite IG-generating set; these results extend to groups in the MN class, yielding almost $\frac{3}{2}$-generation for $2$-generated MN groups and informing generating-graph properties. A key methodological thread is leveraging maximal-subgroup analysis and subgroup-induction properties to deduce IG/FIG behavior, complemented by abelianization arguments. The work also clarifies how weak generation interacts with generating graphs, establishing precise diameter and domination results for $\Delta(G)$ in the MN setting, with concrete implications for branch GGS- and Šunić-type groups.
Abstract
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.