Invariable generation of certain groups of piecewise projective homeomorphisms of the real line

TL;DR

This paper proves that various groups of piecewise projective homeomorphisms and certain Thompson groups are invariably generated, meaning they can be generated by a set that remains generating under any conjugation.

Contribution

It establishes the invariable generation property for several classes of groups, including piecewise projective homeomorphisms and Thompson groups, expanding understanding of their algebraic structure.

Findings

Groups of piecewise projective homeomorphisms are invariably generated.

Thompson groups $F_n$ and $F_{\tau}$ are invariably generated.

The property holds for groups with rational or projective breakpoints.

Abstract

We show that the following groups are invariably generated; the group of piecewise projective homeomorphisms of the real line, the group of piecewise $\mathrm{PSL}(2,\mathbb{Z})$ homeomorphisms of the real line, Monod's group $H(\mathbb{Z})$, the group of piecewise $\mathrm{PSL}(2,\mathbb{Q})$ homeomorphisms of the real line with rational breakpoints. We also show that the Higman--Thompson group $F_n$ for every $n \in \mathbb{Z}_{\geq 3}$ and the golden ratio Thompson group $F_{\tau}$ are invariably generated.