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A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC
Published online by Cambridge University Press: 06 February 2024
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 $\alpha $-CLI and L-$\alpha $-CLI where $\alpha $ is a countable ordinal. We establish three results:
(1) G is $0$-CLI iff $G=\{1_G\}$;
(2) G is $1$-CLI iff G admits a compatible complete two-sided invariant metric; and
(3) G is L-$\alpha $-CLI iff G is locally $\alpha $-CLI, i.e., G contains an open subgroup that is $\alpha $-CLI.
Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_\alpha $ and $H_\alpha $ for $\alpha <\omega _1$, such that:
(1) $H_\alpha $ is $\alpha $-CLI but not L-$\beta $-CLI for $\beta <\alpha $; and
(2) $G_\alpha $ is $(\alpha +1)$-CLI but not L-$\alpha $-CLI.
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic