Finite number of double cosets in a free product with amalgamation
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 . This generalizes a well-known result of Karrass and Solitar. If H is a finitely generated subgroup of the free product with amalgamation 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.
-  Wilfried Imrich, Subgroup theorems and graphs, Combinatorial mathematics, V (Proc. Fifth Austral. Conf., Roy. Melbourne Inst. Tech., Melbourne, 1976) Springer, Berlin, 1977, pp. 1–27. Lecture Notes in Math., Vol. 622. MR 0463016
-  A. Karrass and D. Solitar, Note on a theorem of Schreier, Proc. Amer. Math. Soc. 8 (1957), 696–697. MR 0086813, https://doi.org/10.1090/S0002-9939-1957-0086813-1
-  J.-P. Serre, Arbres, amalgames, , Astérisque, No. 46 (1977).
- 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)
- A. Karass and D. Solitar, Note on a theorem of Schreier, Proc. Amer. Math. Soc. 8 (1957), 696-697. MR 0086813 (19:248a)
- J.-P. Serre, Arbres, amalgames, , Astérisque, No. 46 (1977).
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Keywords: Free group, free product with amalgamation, cyclically reduced words, tree, cycle, fundamental group of a graph
Article copyright: © Copyright 1979 American Mathematical Society