Pregroups and length functions

Source abstract

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.