A Class of Rearrangement Groups that are not Invariably Generated

TL;DR

This paper extends the understanding of rearrangement groups by proving that many subgroups with certain transitivity properties are not invariably generated, generalizing previous results known for Thompson groups.

Contribution

It introduces a broad class of rearrangement groups and demonstrates that their subgroups with specific transitive properties are not invariably generated, expanding prior findings.

Findings

Rearrangement groups with certain transitive subgroups are not invariably generated.

Generalization of non-invariable generation from Thompson groups to wider classes.

Identification of conditions under which subgroups lack invariable generation.

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 $\{ s^{g_s} \mid s \in S \}$. In [GGJ16] Gelander, Golan and Juschenko 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.