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Minimal length functions on groups

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  26 February 2010

David L. Wilkens
David L. Wilkens
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Birmingham. B15 2TT
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In Theorem 7.13 of [1], Proposition 3.1 of [5], and Theorem 1 of [10], minimal group actions on R-trees are considered. If a group G acts on a tree T, then a Lyndon length function lu is associated with each point u∈T. Abstract minimal length functions are defined in Section 2 of this paper by a simple reduction process, where lengths of elements are reduced by a fixed amount (except that any length must remain non-negative). It is shown in Theorem 2.3 that minimal length functions correspond to minimal actions by following Chiswell's construction of actions on trees from length functions, given in [4]. A parallel result to Theorem 1 of [10] is given for minimal length functions in Theorem 2.2. One outcome of these results is that to determine which length functions can arise from an action of a group on the same tree, it suffices to consider only minimal length functions. Section 1 is concerned with some preparatory properties on lengths of products of elements. These lead in Proposition 1.6 to an alternative description of the maximal trivializable subgroup associated with a length function, defined in [3].

MSC classification

Secondary: 20E08: Groups acting on trees
Type
Research Article
Information
Mathematika , Volume 39 , Issue 1 , June 1992 , pp. 56 - 61
Copyright
Copyright © University College London 1992

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