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

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: \begin{enumerate} \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-$\alpha$-CLI iff $G$ is locally $\alpha$-CLI, i.e., $G$ contains an open subgroup that is $\alpha$-CLI. \end{enumerate} 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 \begin{enumerate} \item $H_{\alpha }$ is $\alpha$-CLI but not L-$\beta$-CLI for $\beta <\alpha$; and \item $G_{\alpha }$ is $(\alpha +1)$-CLI but not L-$\alpha$-CLI. \end{enumerate}

20 pages, submitted