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A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 06 February 2024, pp. 1-19
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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.
