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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 $f < g \Leftrightarrow hf < hg$. 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 $\mathrm {Homeo}(I;\delta I)$.
@article{10_4007_annals_2019_190_2_5,
author = {James Hyde},
title = {The group of boundary fixing homeomorphisms of the disc is not left-orderable},
journal = {Annals of mathematics},
pages = {657--661},
publisher = {mathdoc},
volume = {190},
number = {2},
year = {2019},
doi = {10.4007/annals.2019.190.2.5},
mrnumber = {3997131},
zbl = {07107184},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.2.5/}
}
TY - JOUR AU - James Hyde TI - The group of boundary fixing homeomorphisms of the disc is not left-orderable JO - Annals of mathematics PY - 2019 SP - 657 EP - 661 VL - 190 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.2.5/ DO - 10.4007/annals.2019.190.2.5 LA - en ID - 10_4007_annals_2019_190_2_5 ER -
%0 Journal Article %A James Hyde %T The group of boundary fixing homeomorphisms of the disc is not left-orderable %J Annals of mathematics %D 2019 %P 657-661 %V 190 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.2.5/ %R 10.4007/annals.2019.190.2.5 %G en %F 10_4007_annals_2019_190_2_5
James Hyde. The group of boundary fixing homeomorphisms of the disc is not left-orderable. Annals of mathematics, Tome 190 (2019) no. 2, pp. 657-661. doi: 10.4007/annals.2019.190.2.5
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