On Dense Subgroups of Homeo(I)

We prove that a dense subgroup of ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_{+}(I)$ is not elementary amenable. We also show that the topological group ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_{+}(I)$ does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_{+}(I)$ admits a faithful discrete representation in it. In the last section, we demonstrate that finitely generated dense subgroups have infinite girth.