PREGROUPS
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From theorems A and B and result of Karrass, Pietrowski, and Solitar [1973]
generalizing Stagings' work on the ends of a group [1968], we deduce the following
corollary.
Corollary: A group G is free by finite if and only if G is the universal group of a
finite pregroup.
The paper is organized as follows:
Part I: We review Stallings' work on pregroups and the connection between pre-
groups and groups defined by reduced words. We make an initial investigation of
the subgroups of a pregroup. We provide a geometric interpretation of the pregroup
axioms, which serves as a convenient computational tool for part II.
We exploit a tree ordering, discovered by Stallings, on the elements of a pre-
group in order to define the subcategory of pregroups of finite height. We define the
full subpregroup of units, U(P), of a pregroup P, and show how this leads to a des-
cending sequence PDU(P)DU\P) * D
Ud(P)
of subpregroups of given pregroup
P of finite depth d.
Part II: We define the notion of a pregroup action on a set. We show that the
subpregroup of units U(P) acts on the maximal vertices of P. We exploit this action
to define a generating set for P , consisting of a fundamental group system and a
spanning set for P. We prove that in a certain sense U(P)\J{ fundamental group
system } U{ spanning sets } is a subset of P which minimally generates U(P),
(theorem 4.24). This result pares the way for theorem 5.3, in which we give a
presentation of P in terms of a generating set of P. We give an example which illus-
trates this presentation and shows how to get from a pregroup to a graph of groups.
As an easy application of this presentation, we prove the the universal group of a
finite pregroup with no nontrivial subgroups is free.
Part III: We give an example which indicates the major technical problem of the
proof of theorem A. This example motivates an inductive argument based on the
presentation of part II. To prove theorem B, we first review chapter one of Serre
[1980] from a pregroup theoretic point of view. We construct a pregroup structure,
due to Stallings, for F{H,Y). This group is defined in Serre [1980] and is a large
group containing ^^11,7,v0), the fundamental group of a graph of groups. We
define, for each aeF(H,Y\ a sequence of paths 7(a) in 7 reflecting the reduced
word structure for a, which in turn is derived from the pregroup structure for
F(H,7). Using this idea we define a subset Q of F(H,Y) and prove that Q is in
fact a subpregroup of F(H,7). Finally, by reference to the proof of theorem A, we
show that U(Q) is isomorphic to 7r{{H,Y,V0), establishing theorem B. We end the
paper with examples showing how to go from a free product with amalgamation or
an HNN extension to a pregroup. These examples resemble some of the original
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