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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The subgroups of a tree product of groups

Author: J. Fischer
Journal: Trans. Amer. Math. Soc. 210 (1975), 27-50
MSC: Primary 20F30
MathSciNet review: 0376868
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Abstract: Let $ G = {\Pi ^ \ast }({A_i};{U_{jk}} = {U_{kj}})$ be a tree product with $ H$ a subgroup of $ G$. By extending the technique of using a rewriting process we show that $ H$ is an HNN group whose base is a tree product with vertices of the form $ x{A_i}{x^{ - 1}} \cap H$. The associated subgroups are contained in vertices of the base, and both the associated subgroups of $ H$ and the edges of its base are of the form $ y{U_{jk}}{y^{ - 1}} \cap H$. The $ x$ and $ y$ are certain double coset representatives for $ G\bmod (H,{A_i})$ and $ G\bmod (H,{U_{jk}})$, respectively, and the elements defined by the free part of $ H$ are specified. More precise information about $ H$ is given when $ H$ is either indecomposable or $ H$ satisfies a nontrivial law. Introducing direct tree products, we use our subgroup theorem to prove that if each edge of $ G$ is contained in the center of its two vertices then the cartesian subgoup of $ G$ is a free group. We also use our subgroup theorem in proving that if each edge of $ G$ is a finitely generated subgroup of finite index in both of its vertices and some edge is a proper subgroup of both its vertices then $ G$ is a finite extension of a free group iff the orders of the $ {A_i}$ are uniformly bounded.

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Keywords: Presentations, Kuroš subgroup theorem, Karrass-Solitar subgroup theorem, subgroup structure, HNN group, level function, Reidemeister-Schreier theory, extended Schreier system, compatible Kuroš rewriting process, compatible regular enlarged Schreier system for trees, indecomposable subgroup, nontrivial law, direct tree product, cartesian subgroup, finite extensions of free groups
Article copyright: © Copyright 1975 American Mathematical Society