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A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC

Part of: Set theory Topological and differentiable algebraic systems

Published online by Cambridge University Press:  06 February 2024

LONGYUN DING Open the ORCID record for LONGYUN DING [Opens in a new window]  and
XU WANG
LONGYUN DING*
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN 300071 P.R.CHINA E-mail: 1120210025@mail.nankai.edu.cn
XU WANG
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN 300071 P.R.CHINA E-mail: 1120210025@mail.nankai.edu.cn
*
E-mail: dingly@nankai.edu.cn
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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:

  1. (1) G is $0$-CLI iff $G=\{1_G\}$;

  2. (2) G is $1$-CLI iff G admits a compatible complete two-sided invariant metric; and

  3. (3) G is L-$\alpha $-CLI iff G is locally $\alpha $-CLI, i.e., G contains an open subgroup that is $\alpha $-CLI.

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:

  1. (1) $H_\alpha $ is $\alpha $-CLI but not L-$\beta $-CLI for $\beta <\alpha $; and

  2. (2) $G_\alpha $ is $(\alpha +1)$-CLI but not L-$\alpha $-CLI.

Keywords

hierarchynon-archimedean Polish grouptree

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 22A05: Structure of general topological groups
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 19
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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