Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T20:34:45.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Pregroups and length functions

Published online by Cambridge University Press:  24 October 2008

A. H. M. Hoare
A. H. M. Hoare
Affiliation:
Department of Mathematics, University of Birmingham
Get access Rights & Permissions [Opens in a new window]

Extract

Pregroups were defined by Stallings[7] who showed that the elements of the group they define have a normal form up to an equivalence called interleaving. Recently Rimlinger[5] has shown that subject to a discreteness and a boundedness condition any pregroup P defines a graph of groups. We show here that closer analysis of P makes the boundedness condition superfluous. In § 1 we give results of Stallings and Rimlinger and prove some key lemmas. In §2 we show that the discreteness condition gives an integer-valued length function in the sense of Lyndon [4]. It follows from the work of Chiswell [2] and Serre [6] that this defines a graph of groups. I would like to thank the referee for his careful reading and useful comments on this paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 1 , July 1988 , pp. 21 - 30
Copyright
Copyright © Cambridge Philosophical Society 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alperin, R. C. and Bass, H.. Length functions of group actions on Λ-trees. Ann. of Math. Stud. no. 111 (Princeton University Press, 1987), pp. 265–378.Google Scholar
[2]Chiswell, I. M.. Abstract length functions in groups. Math. Proc. Cambridge Philos. Soc. 85 (1976), 417–429.CrossRefGoogle Scholar
[3]Imrich, W.. On metric properties of tree-like spaces. In Beiträge zur Graphentheorie und deren Anwendungen (Sektion MAROK der Technischen Hochschule Ilmenmau, 1979). pp. 129–156.Google Scholar
[4]Lyndon, R. C.. Length functions in groups. Math. Scand. 12 (1963), 209–234.CrossRefGoogle Scholar
[5]Rimlinger, F. S.. Pregroups and Bass-Serre Theory. Mem. Amer. Math. Soc. no. 361 (American Mathematical Society, 1987).Google Scholar
[6]Serre, J. P.. Trees (Springer-Verlag. 1980).CrossRefGoogle Scholar
[7]Stallings, J. R.. Group Theory and Three-dimensional Manifolds. Yale Math. Monographs no. 4 (Yale University Press, 1971).Google Scholar