A hierarchy on non-archimedean Polish groups admitting a compatible complete left-invariant metric

2023/01/27 by Ding, Longyun, Wang, Xu
#03E15 #22A05 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2301.12014

Abstract

In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by α-CLI and L-α-CLI where α is a countable ordinal. We establish three results: \beginenumerate \item G is 0-CLI iff G=\1_G\; \item G is 1-CLI iff G admits a compatible complete two-sided invariant metric; and \item G is L-α-CLI iff G is locally α-CLI, i.e., G contains an open subgroup that is α-CLI. \endenumerate Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups G_α and H_α for α<ω₁, such that \beginenumerate \item H_α is α-CLI but not L-β-CLI for β<α; and \item G_α is (α+1)-CLI but not L-α-CLI. \endenumerate

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