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Invariants and examples of group actions on trees and length functions

Published online by Cambridge University Press:  20 January 2009

David L. Wilkens
David L. Wilkens
Affiliation:
School of Mathematics and StatisticsUniversity of BirminghamBirminghamB15 2TTEngland
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An action of a group G on a tree, and an associated Lyndon length function l, give rise to a hyperbolic length function L and a normal subgroup K having bounded action. The Theorem in Section 1 shows that for two Lyndon length functions l, l′ to arise from the same action of G on some tree, L = L′ and K = K′. Moreover for L non-abelian L = L′ implies K = K′. That this is not so for abelian L is shown in Section 2 where two examples of Lyndon length functions l, l′ on an H.N.N. group are given, with their associated actions on trees, for which L = L′ is abelian but K≠K′.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 2 , June 1991 , pp. 313 - 320
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Alperin, Roger and Bass, Hyman, Length functions of group actions on Λ-trees, in Combinatorial group theory and topology (Alta, Utah, 1984, Princeton Univ. Press, Princeton, N.J., 1987), 265–378.Google Scholar
2.Chiswell, I. M., Abstract length functions in groups, Math. Proc. Cambridge Philos. Soc. 80 (1976), 451–463.CrossRefGoogle Scholar
3.Hoare, A. H. M. and Wilkens, D. L., On groups with unbounded non-archimedean elements, in Groups—St. Andrews 1981 (ed. Campbell, and Robertson, ), London Math. Soc. Lecture Note Series 71, C.U.P. 1982, 228–236.CrossRefGoogle Scholar
4.Imrich, Wilfried and Schwarz, Gabriele, Trees and length functions on groups (Combinatorial Mathematics (Marseille—Luminy, 1981), North-Holland, Amsterdam, 1983), 347–359.Google Scholar
5.Morgan, John W. and Shalen, Peter B., Valuations, trees and degenerations of hyperbolic structures, I, Ann. of Math. (2) 120 (1984), 401–476.CrossRefGoogle Scholar
6.Wilkens, David L., Length functions and normal subgroups, J. London Math. Soc. (2) 22 (1980), 439–448.CrossRefGoogle Scholar
7.Wilkens, David L., Group actions on trees and length functions, Michigan Math. J. 35 (1988), 141–150.CrossRefGoogle Scholar
8.Wilkens, David L., Bounded group actions on trees and hyperbolic and Lyndon length functions, Michigan Math. J. 36 (1989), 303–308.CrossRefGoogle Scholar