×

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.

MSC:

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

Citations:

Zbl 1103.20003
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bekka , P. Report of the workshopGeometrization of Kazhdan’s Property(T) (organizers: B. Bekka, P. de la Harpe, A. Valette; 2001). Unpublished; currently available athttp://www.mfo.de/cgi-bin/tagungsdb?type=21&tnr=0128a .
[2] Bekka , B. , de la Harpe , P. , Valette , A. ( 2004 ).Kazhdan’s Property(T). Forthcoming book, currently available athttp://poncelet.sciences.univ-metz.fr/bekka/ .
[3] Bergman , G. M. ( 2005 ). Generating infinite symmetric groups. To appear inBull. London Math. Soc. currently available athttp://math.berkeley.edu/gbergman/papers/ .
[4] Bridson M. R., Metric Spaces of Non-Positive Curvature (1999) · Zbl 0988.53001
[5] DOI: 10.1112/S0024610704006167 · Zbl 1070.20001 · doi:10.1112/S0024610704006167
[6] DOI: 10.1515/form.2005.17.4.699 · Zbl 1093.20016 · doi:10.1515/form.2005.17.4.699
[7] de la Harpe P., Astérisque 175 (1989)
[8] Lyndon R. C., Combinatorial Group Theory (1977) · Zbl 0368.20023
[9] Koppelberg S., C. R. Math. Acad. Sci. Paris, Sér. A 279 pp 583– (1974)
[10] Sabbagh G., C. R. Math. Acad. Sci. Paris, Sér. A 280 pp 763– (1975)
[11] DOI: 10.1090/S0002-9939-1951-0040299-6 · doi:10.1090/S0002-9939-1951-0040299-6
[12] Serre J.-P, SL2. Astérisque. 46 (1977)
[13] DOI: 10.1016/S0049-237X(08)71346-6 · doi:10.1016/S0049-237X(08)71346-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.