Strongly bounded groups of various cardinalities

Corson, Samuel; Shelah, Saharon

doi:10.1090/proc/14998

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.

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