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.
- George M. Bergman, Generating infinite symmetric groups, Bull. London Math. Soc. 38 (2006), no. 3, 429–440. MR 2239037, DOI https://doi.org/10.1112/S0024609305018308
- Tomek Bartoszyński and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295
- Krzysztof Ciesielski and Janusz Pawlikowski, On the cofinalities of Boolean algebras and the ideal of null sets, Algebra Universalis 47 (2002), no. 2, 139–143. MR 1916611, DOI https://doi.org/10.1007/s00012-002-8179-y
- Yves de Cornulier, Strongly bounded groups and infinite powers of finite groups, Comm. Algebra 34 (2006), no. 7, 2337–2345. MR 2240370, DOI https://doi.org/10.1080/00927870600550194
- Manfred Droste and Rüdiger Göbel, Uncountable cofinalities of permutation groups, J. London Math. Soc. (2) 71 (2005), no. 2, 335–344. MR 2122432, DOI https://doi.org/10.1112/S0024610704006167
- P. Erdős, A. Hajnal, and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196. MR 202613, DOI https://doi.org/10.1007/BF01886396
- Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
- Winfried Just and Piotr Koszmider, Remarks on confinalities and homomorphism types of Boolean algebras, Algebra Universalis 28 (1991), no. 1, 138–149. MR 1083827, DOI https://doi.org/10.1007/BF01190417
- Sabine Koppelberg, Boolean algebras as unions of chains of subalgebras, Algebra Universalis 7 (1977), no. 2, 195–203. MR 434914, DOI https://doi.org/10.1007/BF02485429
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064
- Saharon Shelah, On a problem of Kurosh, Jónsson groups, and applications, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Stud. Logic Foundations Math., vol. 95, North-Holland, Amsterdam-New York, 1980, pp. 373–394. MR 579953
- Saharon Shelah, Was Sierpiński right? I, Israel J. Math. 62 (1988), no. 3, 355–380. MR 955139, DOI https://doi.org/10.1007/BF02783304
- Stevo Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), no. 3-4, 261–294. MR 908147, DOI https://doi.org/10.1007/BF02392561
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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