Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-19T12:42:58.035Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal group actions 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 El 4NS, U.K.
Get access Rights & Permissions [Opens in a new window]

Extract

We consider the existence and uniqueness of minimal invariant subtrees for abelian actions of groups on Λ-trees, and whether or not a minimal action is determined up to isomorphism by the hyperbolic length function. The main emphasis is on actions of end type. For a trivial action of end type, there is no minimal invariant subtree. However, if a finitely generated group has an action of end type, the action is nontrivial and there is a unique minimal invariant subtree. There are examples of infinitely generated groups with a nontrivial action of end type for which there is no minimal invariant subtree. These results can be used to study actions of cut type.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 1 , 1998 , pp. 23 - 36
Copyright
Copyright © Royal Society of Edinburgh 1998

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, Annals of Mathematics Studies 111, eds Gersten, S. M. and Stallings, J. R., 265–378 (Princeton: University Press, 1987).CrossRefGoogle Scholar
2Bass, H.. Group actions on non-archimedean trees. In Arboreal Group Theory, MRSI Publications 19, ed. Alperin, R. C., 69–131 (New York: Springer, 1991).CrossRefGoogle Scholar
3Brown, K. S.. Trees, valuations and the Bieri–Neumann–Strebel invariant. Invent. Math. 90 (1987), 479–504.CrossRefGoogle Scholar
4Chiswell, I. M.. Introduction to Λ-trees. In Semigroups, Formal Languages and Groups, ed. Fountain, J., 255–93 (Dordrecht: Kluwer, 1995).CrossRefGoogle Scholar
5Culler, M. and Morgan, J.. Group actions on ℝ-trees. Proc. London Math. Soc. (3) 55 (1987), 571–604.CrossRefGoogle Scholar
6Morgan, J. W. and Shalen, P. B.. Valuations, trees and degenerations of hyperbolic structures: I. Ann. Math. (2) 122 (1985), 398–476.Google Scholar
7Serre, J.-P.. Trees (New York: Springer, 1980).CrossRefGoogle Scholar