Invariable generation of Thompson groups

TL;DR

This paper investigates the invariable generation properties of Thompson groups, showing that F is finitely invariable generated while T and V are not, revealing structural differences among these groups.

Contribution

It establishes the invariable generation status of Thompson groups F, T, and V, providing new insights into their algebraic structure.

Findings

Thompson group F is finitely invariable generated.

Thompson groups T and V are not invariable generated.

Differentiates structural properties of Thompson groups.

Abstract

A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated.