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 $f < g \Leftrightarrow hf < hg$. 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 $\mathrm {Homeo}(I;\delta I)$.
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@MISC{Calegari,
author = {Calegari, D},
title = {Orderability, and groups of homeomorphisms of the disk},
note = {Geometry and the imagination [blog]},
url = {https://lamington.wordpress.com/2009/07/04/orderability-and-groups-of-homeomorphisms-of-the-disk/},
zblnumber = {},
} -
[Navas1] B. Deroin, A. Navas, and C. Rivas, Groups, Orders, and Dynamics, 2014.
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[CalegariRolfsen]
D. Calegari and D. Rolfsen, "Groups of PL homeomorphisms of cubes," Ann. Fac. Sci. Toulouse Math. (6), vol. 24, iss. 5, pp. 1261-1292, 2015.
@ARTICLE{CalegariRolfsen,
author = {Calegari, Danny and Rolfsen, Dale},
title = {Groups of {PL} homeomorphisms of cubes},
journal = {Ann. Fac. Sci. Toulouse Math. (6)},
fjournal = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
volume = {24},
year = {2015},
number = {5},
pages = {1261--1292},
issn = {0240-2963},
mrclass = {20F38 (20F60 20F65 57M07 57Q99)},
mrnumber = {3485335},
mrreviewer = {Gilbert Levitt},
doi = {10.5802/afst.1484},
url = {https://doi.org/10.5802/afst.1484},
zblnumber = {1355.57025},
} -
[ClayRolfsen]
A. Clay and D. Rolfsen, Ordered Groups and Topology, Amer. Math. Soc., Providence, RI, 2016, vol. 176.
@BOOK{ClayRolfsen,
author = {Clay, Adam and Rolfsen, Dale},
title = {Ordered Groups and Topology},
series = {Grad. Stud. Math.},
volume = {176},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2016},
pages = {x+154},
isbn = {978-1-4704-3106-8},
mrclass = {57-02 (20F34 20F36 20F60 57M07 57M50)},
mrnumber = {3560661},
mrreviewer = {Sebastian Wolfgang Hensel},
zblnumber = {1362.20001},
doi = {10.1090/gsm/176},
} -
[Navas2] A. Navas, Group actions on 1-manifolds: a list of very concrete open questions, 2017.
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[Kourovka] V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version), 2014.
@MISC{Kourovka,
author = {Mazurov, V. D. and Khukhro, E. I.},
title = {Unsolved Problems in Group Theory. {T}he {K}ourovka Notebook. {N}o. 18 ({E}nglish version)},
arxiv = {1401.0300v14},
year = {2014},
zblnumber = {},
}