A class of rearrangement groups that are not invariably generated

Abstract

A group $G$ is invariably generated if there exists a subset $S\subseteq G$ such that, for every choice $g_s\in G$ for $s\in S$, the group $G$ is generated by $\{s^{g_s}\mid s\in S\}$. Gelander, Golan, and Juschenko (J. Algebra 478 (2016), 261–270) showed that Thompson groups $T$ and $V$ are not invariably generated. Here, we generalize this result to the larger setting of rearrangement groups, proving that any subgroup of a rearrangement group that has a certain transitive property is not invariably generated.