This paper investigates the structure of pregroups, Stallings generalization of free
products with amalgamation. Stallings discovered an order relation on pregroups
which gives rise to the concept of finite height. We show that the universal group of
a pregroup P of finite height may be realized as the fundamental group of a graph of
groups. The vertex groups of this graph of groups correspond to the P-conjugacy
classes of the maximal subgroups of P. Conversely, we construct a pregroup struc-
ture for the fundamental group of any graph of groups whose geodesies are of finite
bounded length. Via a theorem of Karrass, Pietrowski, and Solitar, we deduce that a
group is a finite extension of a finitely generated free group if and only if it is the
universal group of a finite pregroup.
1980 Mathematics Subject Classification. Primary 20E07, 20E34, 20F10; Secondary 20E06.
Keywords and phrases. Pregroup, word problem, reduced word, graph of groups, HNN extension,
free by finite.
Library of Congress Cataloging-in-Publication Data
Rimlinger, Frank, 1957-
Pregroups and Biss-Serre theory.
(Memoirs of the American Mathematical Society, 0065-9266;
no. 361 (Jan. 19S7»
"January 1987, volume 65, number 361 (fourth of 5 numbers).'
1. Pregroups. I. Title. II. Title: Bass-Serre theory. III. Series.
QA3.A57 no. 361 [QA171] 510s [512'.22] 86-32112