On the complexity of extensions of non-archimedean Polish groups admitting a compatible complete left-invariant metric
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866911711177474048 |
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| author | Ding, Longyun Wang, Xu |
| author_facet | Ding, Longyun Wang, Xu |
| contents | 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 $α$-CLI and L-$α$-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 $α$-CLI and $β$-CLI, respectively. Is $G$ always $(α+β)$-CLI?
We provide a positive answer under a certain additional assumption. We then construct two examples yielding negative answers:
for each countably infinite ordinal $α$, there exists a group $G$ that is not $α$-CLI, but $G$ has a $1$-CLI normal subgroup $N$ such that $G/N$ is proper $α$-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 $α$-CLI and $β$-CLI with $β>0$, respectively, then $G$ is $β\cdot(ω\cdotα+1)$-CLI, which gives an upper bound on the complexity of the extended group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24379 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the complexity of extensions of non-archimedean Polish groups admitting a compatible complete left-invariant metric Ding, Longyun Wang, Xu Logic Group Theory 03E15, 22A05 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 $α$-CLI and L-$α$-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 $α$-CLI and $β$-CLI, respectively. Is $G$ always $(α+β)$-CLI? We provide a positive answer under a certain additional assumption. We then construct two examples yielding negative answers: for each countably infinite ordinal $α$, there exists a group $G$ that is not $α$-CLI, but $G$ has a $1$-CLI normal subgroup $N$ such that $G/N$ is proper $α$-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 $α$-CLI and $β$-CLI with $β>0$, respectively, then $G$ is $β\cdot(ω\cdotα+1)$-CLI, which gives an upper bound on the complexity of the extended group. |
| title | On the complexity of extensions of non-archimedean Polish groups admitting a compatible complete left-invariant metric |
| topic | Logic Group Theory 03E15, 22A05 |
| url | https://arxiv.org/abs/2605.24379 |