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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

REFERENCES

[1]Alperin, R. C. and Moss, K. N.Complete trees for groups with a real-valued length function. J. London Math. Soc. 31 (1985), 55–68.CrossRefGoogle Scholar
[2]Chiswell, I. M.Introduction to Λ-Trees. (World Scientific, 2001).CrossRefGoogle Scholar
[3]Chiswell, I. M. and Müller, T. W.A Class of Groups Universal for Free ℝ-Tree Actions. (Submitted to Cambridge University Press).Google Scholar
[4]Cohn, P. M.Algebra (second edition), Volume 3. (John Wiley & Sons, 1991).Google Scholar
[5]Holt, D. F., Eick, B. and O'Brien, E.Handbook of Computational Group Theory. Discrete Mathematics and its Applications (Boca Raton) (Chapman & Hall/CRC, Boca Raton, 2005), (Birkhäuser, Basel und Stuttgart, 1975).Google Scholar
[6]Mayer, J. C., Nikiel, J. and Oversteegen, L. G.Universal spaces for ℝ-trees. Trans. Amer. Math. Soc. 334 (1992), 411–432.Google Scholar
[7]Myasnikov, A. G., Remeslennikov, V. N. and Serbin, D.Regular free length functions on Lyndon's free ℤ[t]-group F ℤ[t]. Groups, Languages, Algorithms (ed. A. V. Borovik), Contemp. Math. 378 (2005), 33–77.Google Scholar
[8]Morgan, J. W. and Shalen, P. B.Valuations, trees, and degenerations of hyperbolic structures, I. Ann. of Math. 120 (1984), 401–476.CrossRefGoogle Scholar
[9]Morgan, J. W. and Shalen, P. B.Free actions of surface groups on ℝ-trees. Topology 30 (1991), 143–154.CrossRefGoogle Scholar
[10]Newman, M. H. A.On theories with a combinatorial definition of “equivalence”. Ann. Math. 43 (1942), 223–243.CrossRefGoogle Scholar
[11]Stallings, J. R.Adian groups and pregroups. Essays in Group Theory (ed. S. M. Gersten). Math. Sci. Res. Inst. Publ. 8 (1987), 321–342.Google Scholar