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 initiated by 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 ω₁-existentially closed groups are strongly bounded.

Keywords:

• Bergman Property
• Infinite directs products
• Strongly bounded groups
• Uncountable cofinality

2000 Mathematics Subject Classification:

• Primary 20E15
• Secondary 06E10, 20A15

ACKNOWLEDGMENTS

I thank Bachir Bekka, who suggested to me to show that the groups studied in Koppelberg and Tits (1974) have Property (FH). I am grateful to David Madore, Romain Tessera, and Alain Valette for their useful corrections and comments.

Notes

²In the literature, Cayley boundedness is sometimes referred to “Bergman's Property”.

Communicated by A. Olshanskii.