## Strongly bounded groups of various cardinalities

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- by Samuel M. Corson and Saharon Shelah
- Proc. Amer. Math. Soc.
**148**(2020), 5045-5057 - DOI: https://doi.org/10.1090/proc/14998
- Published electronically: September 24, 2020
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## Abstract:

Strongly bounded groups are those groups for which every action by isometries on a metric space has orbits of finite diameter. Many groups have been shown to have this property, and all the known infinite examples so far have cardinality at least $2^{\aleph _0}$. We produce examples of strongly bounded groups of many cardinalities, including $\aleph _1$, answering a question of Yves de Cornulier [Comm. Algebra 34 (2006), no. 7, 2337–2345]. In fact, any infinite group embeds as a subgroup of a strongly bounded group which is, at most, two cardinalities larger.## References

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## Bibliographic Information

**Samuel M. Corson**- Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, 28049 Madrid, Spain
- MR Author ID: 1133429
- ORCID: 0000-0003-0050-2724
- Email: sammyc973@gmail.com
**Saharon Shelah**- Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904 Israel; Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- Email: shelah@math.huji.ac.il
- Received by editor(s): June 28, 2019
- Published electronically: September 24, 2020
- Additional Notes: The first author’s work was supported by the European Research Council grant PCG-336983 and by the Severo Ochoa Programme for Centres of Excellence in R&D SEV-20150554.

The second author’s work was supported by the European Research Council grant 338821. Paper number 1169 on Shelah’s archive. A new 2019 version of the second author’s paper number 1098 will in some respect continue this paper on other problems and cardinals. - Communicated by: Martin Liebeck
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5045-5057 - MSC (2010): Primary 20A15, 20E15; Secondary 03E05, 03E17
- DOI: https://doi.org/10.1090/proc/14998
- MathSciNet review: 4163821