A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC

Abstract

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.