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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
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
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References:

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