Pregroups and Lyndon Length Functions

  • I. M. Chiswell
Conference paper
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 19)


The term pregroup was introduced by Stallings in [S1] and [S2], and arose from his work on groups with infinitely many ends. The ideas in the definition go back to Baer [Ba]. A pregroup is a set with a partial multiplication having certain group-like properties, to which one can associate a group (the universal group of the pregroup), and there is a normal form for the elements of the group in terms of the pregroup. This generalises the construction of free groups, free products and HNN-extensions. The definition of pregroup is designed to enable the well-known and elegant argument of van der Waerden to be used to prove the normal form theorem (this argument was originally used in [W] for free products). The structure of arbitrary pregroups is mysterious, and the purpose of this article is to give a brief survey of recent progress in understanding them. This involves constructing actions of the universal group on simplicial trees, making use of the order tree of a pregroup, a (of a kind well-known in set theory) generalised tree with basepoint invented by Stallings and introduced in Rimlinger’s book [R2; Sect. 2].


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  1. [AB]
    R.C. Alperin and H. Bass, Length functions of group actions on A-trees, in “Combinatorial Group Theory and Topology”, ed. by S.M. Gersten and J.R. Stallings, Annals of Math. Studies 111 (Princeton, University Press 1987).Google Scholar
  2. [Ba]
    R. Baer, Free sums of groups and their generalisations, Amer. J. Math. 72 (1950), 647–670.MathSciNetMATHCrossRefGoogle Scholar
  3. [C1]
    I.M. Chiswell,Abstract length functions in groups, Math. Proc. Cambridge Philos. Soc. 80 (1976), 451–463.MathSciNetMATHCrossRefGoogle Scholar
  4. [C2]
    I.M. Chiswell Embedding theorems for groups with an integer-valued length function, Math. Proc. Cambridge Philos. Soc. 85 (1979), 417–429.MathSciNetMATHCrossRefGoogle Scholar
  5. [C3]
    I.M. Chiswell, Length functions and free products with amalgamation of groups,Proc. London Math. Soc. 42 (1981), 42–58.MathSciNetMATHGoogle Scholar
  6. [C4]
    I M. Chiswell, Length functions and pregroups, Proc Edinburgh Math. Soc. 30 (1987), 57–67.MathSciNetMATHGoogle Scholar
  7. [Ha]
    N. Harrison, Real length functions in groups, Trans. Amer. Math. Soc. 174 (1972), 77–106.MathSciNetCrossRefGoogle Scholar
  8. [Ho]
    A.H.M. Hoare, Pregroups and length functions, Math. Proc. Cambridge Philos. Soc. 104 (1988), 21–30.MathSciNetMATHCrossRefGoogle Scholar
  9. [KL]
    H. Kushner and S. Lipschutz, A generalization of Stallings’ pregroup, J. Algebra 119 (1988), 170–184.MathSciNetMATHCrossRefGoogle Scholar
  10. [Ly]
    R.C. Lyndon, Length functions in groups, Math. Scand. 12 (1963), 209–234.MathSciNetMATHGoogle Scholar
  11. [N]
    F.H. Nesayef, Groups generated by elements of length zero and one, (Ph.D. thesis, University of Birmingham, England 1983).Google Scholar
  12. [R1]
    F.S. Rimlinger, A subgroup theorem for pregroups, in “Combinatorial Group Theory and Topology”, ed. by S.M. Gersten and J.R. Stallings, Annals of Math. Studies 111 (Princeton, University Press 1987).Google Scholar
  13. [R2]
    F.S. Rimlinger, Pregroups and Bass-Serre Theory, Mem. Amer. Math. Soc., No. 361 (Providence, American Mathematical Society 1987).Google Scholar
  14. [S1]
    J. R. Stallings, Groups of cohomological dimension one, Proc. Sympos. Pure Math. Amer. Math. Soc. 17 (1970), 124–128.MathSciNetGoogle Scholar
  15. [S2]
    J.R. Stallings, Group theory and three dimensional manifolds, (New Haven and London, Yale University Press 1971).MATHGoogle Scholar
  16. [S3]
    J.R. Stallings, The cohomology of pregroups, in “Lecture Notes in Mathematics” 319 ed. by R.W. Gatterdam and K.W. Weston (Heidelberg, Springer 1973).Google Scholar
  17. [S4]
    J.R. Stallings, Adian groups and pregroups, in “Essays in group theory”, MSRI Publications 8, ed. by S.M. Gersten (New York, Springer 1987).Google Scholar
  18. [W]
    B.L. van der Waerden, Free products of groups, Amer. J. Math. 79 (1948), 527–528.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • I. M. Chiswell
    • 1
  1. 1.School of Mathematical SciencesQueen Mary and Westfield CollegeLondonEngland

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