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The group of boundary fixing homeomorphisms of the disc is not left-orderable. (English) Zbl 07107184
Summary: 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 \(H\) 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 \(\mathrm{Homeo}(I;\delta I)\).
MSC:
06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
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[1] Calegari, D., Orderability, and groups of homeomorphisms of the disk
[2] Deroin, B.; Navas, A.; Rivas, C., Groups, Orders, and Dynamics, (2014)
[3] Calegari, Danny; Rolfsen, Dale, Groups of PL homeomorphisms of cubes, Ann. Fac. Sci. Toulouse Math. (6). Annales de la Facult\'e des Sciences de Toulouse. Math\'ematiques. S\'erie 6, 24, 1261-1292, (2015) · Zbl 1355.57025
[4] Clay, Adam; Rolfsen, Dale, Ordered Groups and Topology, Grad. Stud. Math., 176, x+154 pp., (2016) · Zbl 1362.20001
[5] Navas, A., Group actions on 1-manifolds: a list of very concrete open questions, (2017)
[6] Mazurov, V. D.; Khukhro, E. I., Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version), (2014) · Zbl 1372.20001
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