On the complexity of extensions of non-archimedean Polish groups admitting a compatible complete left-invariant metric

Abstract

In this article, motivated by a problem asked by Allison and Panagiotopoulos, we study a problem concerning the complexity of group extensions within a hierarchy (denoted by $\alpha$-CLI and L-$\alpha$-CLI) on the class of non-archimedean CLI Polish groups: Given a non-archimedean Polish group $G$ and one of its closed normal subgroup $N$, suppose $N$ and $G/N$ are $\alpha$-CLI and $\beta$-CLI, respectively. Is $G$ always $(\alpha+\beta)$-CLI? We provide a positive answer under a certain additional assumption. We then construct two examples yielding negative answers: for each countably infinite ordinal $\alpha$, there exists a group $G$ that is not $\alpha$-CLI, but $G$ has a $1$-CLI normal subgroup $N$ such that $G/N$ is proper $\alpha$-CLI; there exists a proper $3$-CLI group $U$ that has an abelian normal subgroup $N$ such that $U/N$ is also abelian. These examples also provide negative answers to the original problem raised by Allison and Panagiotopoulos. Finally, we show that if $N$ and $G/N$ are $\alpha$-CLI and $\beta$-CLI with $\beta>0$, respectively, then $G$ is $\beta\cdot(\omega\cdot\alpha+1)$-CLI, which gives an upper bound on the complexity of the extended group.