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Embedding theorems for tree-free groups

Published online by Cambridge University Press:  05 May 2010

IAN CHISWELL  and
THOMAS MÜLLER
IAN CHISWELL
Affiliation:
School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London E1 4NS, UK.
THOMAS MÜLLER
Affiliation:
School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London E1 4NS, UK.
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Abstract

We establish two embedding theorems for tree-free groups. The first result embeds a group G acting freely and without inversions on a Λ-tree X into a group acting freely, without inversions, and transitively on a Λ-tree in such a way that X embeds into by means of a G-equivariant isometry. The second result embeds a group G acting freely and transitively on an ℝ-tree X into (H) for some suitable group H, again in such a way that X embeds G-equivariantly into the ℝ-tree XH associated with (H). The group (H) referred to here belongs to a class of groups introduced and studied by the present authors in [3]. As a consequence of these two theorems, we find that -groups and their associated ℝ-trees are in fact universal for free ℝ-tree actions. Moreover, our first embedding theorem throws light on the question, arising from the results of [7], whether a group endowed with a Lyndon length function L can always be embedded in a length-preserving way into a group with a regular Lyndon length function; modulo an obvious necessary restriction we show that this is indeed the case if L is free.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 1 , July 2010 , pp. 127 - 146
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

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