Stallings showed that free products with amalgamation and HNN extensions
also have pregroup structures. It is a fact that if (P,D) is a pregroup, QCP,
QxQcD, and (x,y)eQxQ =^xyeQ, then Q inherits a group structure from P.
In this event we say that Q is a subgroup of P. For example, in Stallings's original
constructions of pregroup structures for A *
B and v _ " )
, the maximal subgroups
of the pregroup structures are A and B in the case of the free product with amalga-
mation and A in the HNN case.
Two natural questions arise at this point. Given a pregroup P, does U(P) act
on a tree? Conversely, given a group which acts on a tree, does this group have a
pregroup structure which reflects the structure of the action. We answer both these
questions in the affirmative, provided (i) the tree in question is an ordinary simpli-
cial tree, (ii) the quotient graph of the tree in question has geodesies of finite
bounded length, and (iii) the pregroup in question satisfies pregroup theoretic cri-
teria analogous to (i) and (ii). The exact results are stated here. The summary
below refers the pregroup theoretic notions to the main body of the paper.
Theorem A: Let P be a pregroup of finite height . Let G be the union of the fun-
damental group systems for each Ul{P\ i=0, . . . ,d. Then P is a pregroup struc-
ture for the fundamental group of a graph of groups ^(F^F). The base groups of Y
are in 1-1 correspondence with the elements of G, and the corresponding groups are
isomorphic. The graph Y is such that the oriented edges of the complement of a
maximal tree in Y are in 1-1 correspondence with the union of the spanning sets of
the fundamental groups of G.
Theorem B: Let (H,7) be a graph of groups with bases V, edges E, basepoint v0,
and maximal tree T. Suppose the graph Y is of finite diameter and (H,7) is proper.
Let XcE be the edges of Y not in T, and let
be an orientation of X. Then
has a pregroup structure Q satisfying the following conditions:
(i) for some d 0, Q has depth d, and a good Q-sequence
((Gd,0), . . . , (G]?0)5(Go, X)) satisfying (ii), (iii), and (iv):
(ii) for i = \, . . . ,d, the groups of G, are in 1-1 correspondence with the maxi-
mal bases of depth i9 and corresponding fundamental groups and base
groups are isomorphic,
(iii) X
is in 1-1 correspondence with
(iv) G0 is in 1-1 correspondence with
{veB | depth(v)=0 }(J B, where ^ C { veB \ depth(v)0} .
If G eG
corresponds to Gy for some v eB of depth 0, then G is isomorphic
to Hv eH. If G G G
corresponds to some veB, then G is isomorphic to a
subgroup of Hv.
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