zbMATH — the first resource for mathematics

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 \(\mathrm{Homeo}(I;\delta I)\).
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
Full Text: DOI arXiv
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.