Arboreal Group Theory pp 169-181 | Cite as

# Pregroups and Lyndon Length Functions

## Abstract

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].

### Keywords

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