The Subgroup Theorem for Amalgamated Free Products, HNN-Constructions and Colimits

Abstract

The purpose of this work is to demonstrate how a subgroup theorem, including Karrass’ and Solitar’s results [3, 4] for tree products and HNN constructions, may be deduced from the theory of groupoids (here called “groupnets”). It is well-known that many topological proofs of group theoretic results, such as the Nielsen theorem on subgroups of free groups, the Reidemeister-Schreier Theorem, Kurosh’ and Grushko’s Theorems, may be formalized in purely algebraic terms using groupoids (see for example Ordman [7], Stallings [8], Higgins [2], Crowell and Smythe [1]), however it seems to have been the general feeling, until Higgin’s work appeared, that the benefits of such a formalization were negligible, particularly in view of the large amount of preliminary machinery which must be assembled in order to make the theory work. We feel that this effort is in fact justified, in that the theory provides a “theorem-proving” machine for combinatorial group theory. Thus, once the basic machinery has been set up, it is completely obvious that a theorem of the Karrass and Solitar type “A subgroup of an HNN group is again an HNN group” must exist, and it merely remains to find the exact statement of the theorem. Proofs in the theory are usually (not always) entirely straightforward, often amounting to checking that some obvious construction actually satisfies all the requirements made upon it. Of course the details and notation in this present work become complex, because of the nature of the result to be proved, but we trust that the reader will find the basic idea of the argument (which is summarized in Section 8) acceptably simple.