Invariable generation does not pass to finite index subgroups (Q2694790)

Let \(G\) be a group. A subset \(S \subseteq G\) invariably generates \(G\) if for every function \(S \rightarrow G, \;\; s \mapsto g_{s}\), the set of conjugates \(\{s^{g_{s}} \mid s \in S \}\) generates \(G\). A group \(G\) is invariably generated (or \textsf{IG}) if it has an invariably generating set, or equivalently, if \(G\) invariably generates itself. A group \(G\) is finitely invariably generated (or \textsf{FIG}) if it has a finite invariably generating set. The classes of \textsf{IG} groups and \textsf{FIG} groups are closed under extensions and include all finite groups, hence a group with a finite index normal \textsf{IG} (resp. \textsf{FIG}) subgroup is \textsf{IG} (resp. \textsf{FIG}). The authors prove the following slight generalization of the previous property (Theorem A): A group containing a finite index \textsf{IG} (resp. \textsf{FIG}) subgroup is \textsf{IG} (resp. \textsf{FIG}). In contrast, the main theorem (Theorem B) states that: There exists a \textsf{FIG} group with an index 2 non-\textsf{IG} subgroup. Another interesting result is Theorem C: There exists a finitely generated group that is \textsf{IG}, but not \textsf{FIG}. The proofs of Theorem B and Theorem C rely on an iterative small cancellation construction. The same results were obtained independently by \textit{A. Minasyan} [Isr. J. Math. 245, No. 1, 231--257 (2021; Zbl 07456847)] using similar methods.