Strongly bounded groups and infinite powers of finite groups. (English) Zbl 1125.20023

The author defines a group \(G\) to be strongly bounded if every action of \(G\) on a metric space has bounded orbits. This property is shown to be equivalent to \(G\) having uncountable cofinality and being Cayley bounded. Here, \(G\) has uncountable cofinality iff \(G\) is not the union of some countable ascending chain of subgroups, and \(G\) is Cayley bounded if for every generating subset \(U\subseteq G\) there exists some \(n\) such that every element of \(G\) is a product of at most \(n\) elements from \(U\cup U^{-1}\).
The latter property was established recently by G. M. Bergman [Bull. Lond. Math. Soc. 38, No. 3, 429-440 (2006; Zbl 1103.20003)] for the infinite symmetric groups. The main result here is that if \(G\) is a finite perfect group and \(I\) is any set, then \(G^I\) is strongly bounded. Also, \(\omega_1\)-existentially closed groups are shown to be strongly bounded.


20E15 Chains and lattices of subgroups, subnormal subgroups
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
20A15 Applications of logic to group theory
20E22 Extensions, wreath products, and other compositions of groups


Zbl 1103.20003
Full Text: DOI arXiv


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