Introduction

Free groups, free products with amalgamations, and HNN extensions may be

thought of as "groups defined by reduced words." We make this concept precise as

follows: Let G be a group, and let P c G be a subset of G such that P generates G

and

P=P~l.

Let D CPxP be a subset of pairs (x,y) of elements of P such that for

all (x,y)eD, xy eP. A word (xL, . . .

,xn)ePn

is said to be reduced if for each adja-

cent pair (xz,x/

+ 1

) it is not the case that {xhxi

+ [

)eD, (for otherwise we reduce the

word in the obvious way). We sa^ that (P,D) is a reduced word structure for G if for

all g EG all reduced words representing g are of the same length.

In Rimlinger [to appear] we proved that if (P,D) is a reduced word structure for

G, then (P,D) is a pregroup and G is isomorphic to U(P), the universal group of P.

But what is a pregroup? These objects were created by John Stallings [1971], who

traces his idea back to van der Waerden [1948] and Baer [1950]. In Stallings treat-

ment, a pregroup is a set P, together with a subset D C P x P , and a special element

1 e P , called the identity element. In addition, pregroups are endowed with a partial

multiplication rn:D-+P and an involution i:P^P. Moreover, certain axioms con-

cerning the above sets and maps must also hold. These axioms may be paraphrased

by saying that 1 is the identity element, m:D-P is "as associative a multiplication

as possible," and i:P-+P takes an element to its inverse. Additionally, if three pairs

(w,x), (x,y), and (y,z) are in Z, then either (wx,y)eD or (xy,z)eD.

Stallings defined the universal group U(P) of a pregroup (P,D) to be the free

group on the set P modulo the relations { xy=z \ x,y,zeP and z=rn(x,y) }. If

G is a group isomorphic to U(P), we say, somewhat loosely, that G has a pregroup

structure P. Stallings original theorem about pregroups, interpreted in the language

developed above, is that (P,D) is a reduced word structure for U(P). Thus pre-

groups exactly capture the notion of "groups defined by reduced words," in that for

any group G, (P,D) is a reduced word structure for G if and only if (i) (P,D) is a

pregroup, (ii) G and U(P) are isomorphic, and hence (iii) P is a pregroup structure

for (7.

For example, if P= { x,x _ 1 ,l }5

D={

(x,x-l)Xx-\xUUx)XxM(hx-l)Xx-\\WA)

}

and rn.D-^P and i:P-+P are defined in the obvious way, then (P,D) is in fact a

pregroup, and U(P) is a free group of rank 1. The reduced words with respect to D

representing elements of U(P) are obtained from arbitrary words in the alphabet P

via free reduction.

v