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 $α$-CLI and L-$α$-CLI where $α$ 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-$α$-CLI iff $G$ is locally $α$-CLI, i.e., $G$ contains an open subgroup that is $α$-CLI. \end{enumerate} Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_α$ and $H_α$ for $α<ω_1$, such that \begin{enumerate} \item $H_α$ is $α$-CLI but not L-$β$-CLI for $β<α$; and \item $G_α$ is $(α+1)$-CLI but not L-$α$-CLI. \end{enumerate}

Theorems & Definitions (66)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Definition 3.1