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
MathSciNet review:
4163821
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Abstract | References | Similar Articles | Additional Information
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
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
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
