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Strongly bounded groups of various cardinalities


Authors: Samuel M. Corson and Saharon Shelah
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
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.


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Additional Information

Samuel M. Corson
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, 28049 Madrid, Spain
MR Author ID: 1133429
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
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/proc/14998
Keywords: Strongly bounded group, strong uncountable cofinality, Bergman property, isometric action, small cancellation over free product
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
Article copyright: © Copyright 2020 American Mathematical Society