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