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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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.


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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