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Finite number of double cosets in a free product with amalgamation


Author: Dragomir Ž. Djoković
Journal: Proc. Amer. Math. Soc. 75 (1979), 19-22
MSC: Primary 20E05
DOI: https://doi.org/10.1090/S0002-9939-1979-0529204-3
MathSciNet review: 529204
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Abstract: If H is a finitely generated subgroup of a free group G such that every conjugate of H contains a cyclically reduced word then $ (G:H) < \infty $. This generalizes a well-known result of Karrass and Solitar. If H is a finitely generated subgroup of the free product with amalgamation $ G = A\;{ \ast _U}B$ such that every conjugate of H meets A and B trivially and contains a cyclically reduced word then G has only finitely many (H, U)-double cosets. Both theorems are proved by defining an action of G on a tree such that H acts freely.


References [Enhancements On Off] (What's this?)

  • [1] W. Imrich, Subgroup theorems and graphs, Proc. Fifth Australian Conf. on Combinatorial Math., Lecture Notes in Math., vol. 622, Springer-Verlag, Berlin and New York, 1977, pp. 1-27. MR 0463016 (57:2980)
  • [2] A. Karass and D. Solitar, Note on a theorem of Schreier, Proc. Amer. Math. Soc. 8 (1957), 696-697. MR 0086813 (19:248a)
  • [3] J.-P. Serre, Arbres, amalgames, $ {\text{SL}_2}$, Astérisque, No. 46 (1977).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0529204-3
Keywords: Free group, free product with amalgamation, cyclically reduced words, tree, cycle, fundamental group of a graph
Article copyright: © Copyright 1979 American Mathematical Society

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