Strongly Bounded Groups and Infinite Powers of Finite Groups

By Yves Cornulier. 2004

54 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/46b13cc1936cc183f7497fa7086a7bd64a452fb2

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Property (T) and rigidity for actions on Banach spaces (2005, background): “As pointed out by Y. de Cornulier [ Cr ], uncountable discrete groups that have G. Bergman’s cofinality property [Bn] have (FH) but fail (T).) Some of the above applications use this latter characterization.”
Turbulence, amalgamation, and generic automorphisms of homogeneous structures (2004, result): “It is not hard to see, and has indeed been noticed independently by other authors (e.g., Cornulier [ 11 ]) that the Bergman property also implies that any action of the group by isometries on a metric space has bounded orbits.”
A topological version of the Bergman property (2005, unknown): “We express this by saying that the group is Cayley bounded, since it corresponds to every Cayley graph being of bounded diameter with respect to the word metric.”