On free discrete subgroups of $\mathrm{Diff}(I)$

Abstract

We prove that the free group ${\mathbb{F}}_2$ admits a faithful discrete representation into ${Diff}_{+}^1\left[0,1\right]$. We also prove that ${\mathbb{F}}_2$ admits a faithful discrete representation of bi-Lipschitz class into ${Homeo}_{+}\left[0,1\right]$. Some properties of these representations are studied.