Strongly bounded groups of various cardinalities

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.