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].
KeywordsManifold Eter Summing
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