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

TL;DR

This paper introduces a new hierarchy for non-archimedean Polish groups based on compatible complete left-invariant metrics, establishing key characterizations and constructing examples to demonstrate the hierarchy's properness.

Contribution

It defines the $ ext{CLI}$ hierarchy for non-archimedean Polish groups, characterizes each level, and constructs explicit examples to show the hierarchy's strictness.

Findings

$0$-CLI groups are trivial.

$1$-CLI groups admit compatible complete two-sided invariant metrics.

Constructed groups demonstrate hierarchy levels are distinct.

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}