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 . 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.

**[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*, , 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