Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T16:56:30.022Z Has data issue: false hasContentIssue false
  • English
  • Français

Inherited group actions on ℝ-trees

Part of: Permutation groups

Published online by Cambridge University Press:  26 February 2010

Zad Khan  and
David L. Wilkens
Zad Khan
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Birmingham, B15 2TT
David L. Wilkens
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Birmingham, B15 2TT
Get access

Extract

Group actions on ℝ-trees may be split into different types, and in Section 1 of this paper five distinct types are defined, with one type splitting into two sub-types. For a group G acting as a group of isometries on an ℝ-tree, conditions are considered under which a subgroup or a factor group may inherit the same type of action as G. In Section 2 subgroups of finite index are considered, and in Section 3 normal subgroups and also factor groups are considered. The results obtained here, Theorems 2.1 and 3.4, allow restrictions on possible types of actions for hypercentral, hypercyclic and hyperabelian groups to be given in Theorem 3.6. In Section 4 finitely generated subgroups are considered, and this gives rise to restrictions on possible actions for groups with certain local properties. The results throughout are stated in terms of group actions on trees. Using Chiswell's construction in [3], they could equally be stated in terms of restrictions on possible types of Lyndon length functions.

MSC classification

Secondary: 20B27: Infinite automorphism groups
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 206 - 213
Copyright
Copyright © University College London 1995

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

1.Roger Alperin and Hyman Bass. Length functions of group actions of A-trees. In ‘Combinatorial group theory and topology’, Annals of Math. Studies III (edited by Gersten, S. M. and Stallings, J. R.) (Princeton Univ. Press, 1987), 265–378.Google Scholar
2. Brown, Kenneth S.. Trees, valuations, and the Bieri-Neumann-Strebel invariant. Invent. Math., 90 (1987), 479–504.CrossRefGoogle Scholar
3. Chiswell, I. M.. Abstract length functions in groups. Math. Proc. Cambridge Phil. Soc, 80 (1976), 451–463.Google Scholar
4. Culler, M. and Morgan, J. W.. Group actions on ℝ-trees. Proc. London Math. Soc. (3), 55 (1987), 571–604.Google Scholar
5. Hoare, A. H. M. and Wilkens, D. L.. On groups with unbounded non-archimedean elements. In ‘Groups–St Andrews 1981’ (edited by Campbell, and Robertson, ), London Math. Soc. Lecture Note Series 71 (Cambridge Univ. Press, 1982), 228–236.Google Scholar
6. Khan, Zad. Lyndon length functions, R-trees and hyperbolic length functions. Ph.D. thesis, University of Birmingham (1993).Google Scholar
7. Lyndon, Roger C.. Length functions in groups. Math. Scand., 12 (1963), 209–234.Google Scholar
8. Tits, J.. A theorem of Lie-Kolchin for trees. Contributions to Algebra (A collection of papers dedicated to Ellis Kolchin) (Academic Press, New York, 1977), 377–388.Google Scholar
9. Wilkens, David L.. Group actions on trees and length functions. Michigan Math. J., 35 (1988), 141–150.Google Scholar
10. Wilkens, David L.. Bounded group actions on trees and hyperbolic and Lyndon length functions. Michigan Math. J., 36 (1989), 303–308.CrossRefGoogle Scholar
11. Wilkens, David L.. Invariants and examples of group actions on trees and length functions. Proc. Edinburgh Math. Soc, 34 (1991), 313–320.Google Scholar
12. Wilkens, David L.. Group actions on trees with and without fixed points. In ‘Discrete groups and geometry’, London Math. Soc. Lecture Note Series, 173 (Cambridge University Press, 1992), 243–248.Google Scholar