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.
| Comments: | 35 pages, submitted |
| Subjects: | Logic (math.LO); Group Theory (math.GR) |
| MSC classes: | 03E15, 22A05 |
| Cite as: | arXiv:2605.24379 [math.LO] |
| (or arXiv:2605.24379v1 [math.LO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24379 arXiv-issued DOI via DataCite (pending registration) |
From: Longyun Ding [view email]
[v1]
Sat, 23 May 2026 03:36:08 UTC (32 KB)
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