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

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