The group of boundary fixing homeomorphisms of the disc is not left-orderable

Abstract

A left-order on a group $G$ is a total order $\lt$ on $G$ such that for any $f$, $g$ and $h$ in $G$ we have $f \lt g \Leftrightarrow hf \lt hg$. We construct a finitely generated subgroup $H$ of $\mathrm{Homeo} (I^2;\delta I^2)$, the group of those homeomorphisms of the disc that fix the boundary pointwise, and show $H$ does not admit a left-order. Since any left-order on $\mathrm{Homeo} (I^2;\delta I^2)$ would restrict to a left-order on $H$, this shows that $\mathrm{Homeo} (I^2;\delta I^2)$ does not admit a left-order. Since $\mathrm{Homeo}(I;\delta I)$ admits a left-order, it follows that neither $H$ nor $\mathrm{Homeo} (I^2;\delta I^2)$ embed in $\mathrm{Homeo}(I;\delta I)$.