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
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, . . .
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
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.
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