On Dense Subgroups of Homeo(I)

Abstract

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

This is the version that has existed in author's webpage since 2014 with a further modification/addition in the last section to extend the result (by a short argument) to all compact manifolds