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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
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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

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References

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