The Group of Boundary Fixing Homeomorphisms of the Disc is Not\n Left-Orderable

2018/10/30 by James Hyde, Hyde, James
Mathematics · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Group Theory (math.GR) #Mathematics and Applications #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1810.12851

openalex publication_date 2018/10/30 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

A left-order on a group G is a total order < on G such that for any\nf, g and h in G we have f < g \⇔ hf < hg. We construct a\nfinitely generated subgroup G of \Homeo (I²;\δ I²), the\ngroup of those homeomorphisms of the disc that fix the boundary pointwise, and\nshow G does not admit a left-order. Since any left-order on\n\Homeo (I²;\δ I²) would restrict to a left-order on G\nthis shows that \Homeo (I²;\δ I²) does not admit a\nleft-order. Since \Homeo (I;\δ I) admits a left-order it\nfollows that neither G nor \Homeo (I²;\δ I²) embed in\n\Homeo (I;\δ I).\n

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