Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-22T12:55:27.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Actions of locally nilpotent groups on Λ-trees

Published online by Cambridge University Press:  14 November 2011

I. M. Chiswell
I. M. Chiswell
Affiliation:
School of Mathematical Sciences Queen Mary and Westfield College, University of London Mile End Road. London E1 4NS, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

For certain classes of groups, it is shown that there are restrictions on the type of action a group in the class can have on a Λ-tree, where Λ is an arbitrary ordered abelian group, generalizing results by other authors in the case Λ = ℝ. The main classes considered are locally nilpotent, polycyclic by finite, locally (polycyclic by finite) and locally (hyperabelian by finite). The arguments involve an investigation of the relation between the type of action a group has on a Λ-tree and the type of action of its subgroups by restriction.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 5 , 1999 , pp. 913 - 924
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alperin, R. C. and Bass, H.. Length functions of group actions on Λ-trees. In Combinatorial group theory and topology (ed. Gersten, S. M. and Stallings, J. R.). Annals of Mathematics Studies, vol. 111, pp. 265–378 (Princeton University Press, 1987).Google Scholar
2Bass, H.. Group actions on non-Archimedean trees. In Arboreal group theory (ed. Alperin, R. C.). MSRI Publications, vol. 19, pp. 69–131 (New York: Springer. 1991).CrossRefGoogle Scholar
3Chiswell, I. M.. Harrison's theorem for Λ-trees. Q. J. Math Oxford (2) 45 (1994), 1–12.CrossRefGoogle Scholar
4Chiswell, I. M.. Properly discontinuous actions on Λ-trees. Proc. Edinb. Math. Soc. 37 (1994), 423–444.CrossRefGoogle Scholar
5Chiswell, I. M.. Minimal group actions on Λ-trees. Proc. R. Soc. Edinb. A 128 (1998). 23–36.CrossRefGoogle Scholar
6Culler, M. and Morgan, J.. Group actions on ℝ-trees. Proc. Lond. Math. Soc. (3) 55 (1987), 571–604.CrossRefGoogle Scholar
7Khan, Z. and Wilkens, D. L.. Inherited group actions on ℝ-trees. Mathematika 42 (1995), 206–213.CrossRefGoogle Scholar
8Serre, J.-P.. Trees (New York: Springer. 1980).CrossRefGoogle Scholar
9Shalen, P. B.. Dendrology of groups: an introduction. In Essays in group theory (ed. Gersten, S. M.). MSRI Publications, vol. 8, pp. 265–319 (New York: Springer, 1987).CrossRefGoogle Scholar
10Tits, J.. A ‘Theorem of Lie–Kolchin’ for trees. In Contributions to algebra: a collection of papers dedicated to Ellis Kolchin (New York: Academic Press. 1977).Google Scholar
11Wilkens, D. L.. On non-archimedean lengths in groups. Mathematica 23 (1976), 57–61.Google Scholar