Pregroups and Lyndon Length Functions
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].
KeywordsInitial Segment Free Product Length Function Simplicial Tree Order Tree
Unable to display preview. Download preview PDF.
- [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
- [N]F.H. Nesayef, Groups generated by elements of length zero and one, (Ph.D. thesis, University of Birmingham, England 1983).Google Scholar
- [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
- [R2]F.S. Rimlinger, Pregroups and Bass-Serre Theory, Mem. Amer. Math. Soc., No. 361 (Providence, American Mathematical Society 1987).Google Scholar
- [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
- [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