Strongly bounded groups and infinite powers of finite groups

Abstract

We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the so-called uncountable strong cofinality, recently introduced by G. Bergman. Our main result is that G^I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that omega_1-existentially closed groups are strongly bounded.

10 pages, no figure. Versions 1-3 were entitled "Uncountable groups with Property (FH)". To appear in Comm. Algebra

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• Strong distortion in transformation groups
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• Strong and uniform boundedness of groups
• Steep uncountable groups
• Metric groups, unitary representations and continuous logic
• Wreath products of groups acting with bounded orbits