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

  • R. H. Crowell
  • N. Smythe
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 372)


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.


Disjoint Union Free Product Homotopy Type Graph Product Vertex Group 
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  1. [1]
    R.H. Crowell and N. Smythe, The theory cf grcNlonets (in preparation, partial preprint available).Google Scholar
  2. [2]
    Philip J. Higgins, Notes on categories and groupoids (Van Nostrand Reinhold Mathematical Studies, 32. Van Nostrand, London, 1971 ). Zb1.226.20054.Google Scholar
  3. [3]
    A. Karrass and D. Solitar, “The subgroups of a free product of two groups with an amalgamated subgroup”, Trans. Amer. Math. Soc. 150 (1970), 227–255. MR41#5499.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    A. Karrass and D. Solitar, “Subgroups of HNJJ groups and groups with one defining relation”, Canad. J. Math. 23 (1971), 627–643. Zb1.232.20051.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Hanna Neumann, “Generalized free products with amalgamated subgroups. Part I. Definitions and general properties”, Amer. J. Math. 70 (1948), 590–625. MR10,233.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Hanna Neumann, “Generalized free products with amalgamated subgroups. Part II. The subgroups of generalized free products”, Amer. J. Math. 71 (1949), 491–540. MR11,8.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Edward T. Ordman, “On subgroups of amalgamated free products”, Proc. Cambridge Philos. Soc. 69 (1971), 13–23. Zb1.203,323.MathSciNetCrossRefGoogle Scholar
  8. [8]
    John R. Stallings, “A topological proof of Grushko’s theorem on free products”, Math. Z. 90 (1965), 1–8. MR32#5723.MathSciNetCrossRefGoogle Scholar
  9. [9]
    F. Waldhausen, “Whitehead groups of generalized free products”, preprint.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • R. H. Crowell
    • 1
    • 2
  • N. Smythe
    • 1
    • 2
  1. 1.Dartmouth CollegeNew HampshireUSA
  2. 2.Australian National UniversityCanberraAustralia

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