Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

On finitely generated subgroups which are of finite index in generalized free products


Authors: A. Karrass and D. Solitar
Journal: Proc. Amer. Math. Soc. 37 (1973), 22-28
MSC: Primary 20F05; Secondary 20E30
DOI: https://doi.org/10.1090/S0002-9939-1973-0320152-5
MathSciNet review: 0320152
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G = (A \ast B;\;U)$ be the free product of A and B with the subgroup U amalgamated. Various conditions are given which imply that every finitely generated subgroup H containing a (nontrivial) normal subgroup of G has finite index in G (in such a case we say G has the f.g.c.n. property). In particular, if A is a noncyclic free group and U is cyclic, then G has the f.g.c.n. property. We use this last result to give a combinatorial proof that Fuchsian groups have the f.g.c.n. property; this was first proved by Greenberg using non-Euclidean geometry.


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

  • [1] B. Baumslag, Intersections of finitely generated subgroups in free products, J. London Math. Soc. 41 (1966), 673-679. MR 33 #7396. MR 0199247 (33:7396)
  • [2] R. Burns, On the finitely generated subgroups of an amalgamated product of two groups, Trans. Amer. Math. Soc. 169 (1972), 293-306. MR 0372043 (51:8260)
  • [3] L. Greenberg, Discrete groups of motions, Canad. J. Math. 12 (1960), 414-426. MR 22 #5932. MR 0115130 (22:5932)
  • [4] H. B. Griffiths, A covering-space approach to theorems of Greenberg in Fuchsian, Kleinian, and other groups, Comm. Pure Appl. Math. 20 (1967), 365-399; Correction, ibid. 21 (1968), 521-522. MR 35 #2282; MR 38 #1671. MR 0211401 (35:2282)
  • [5] G. Higman, Some problems and results in the theory of groups. II, Notes of a Mini-Conference, Oxford, 12th and 13th August 1966, pp. 15-16.
  • [6] A. H. M. Hoare, A. Karrass and D. Solitar, Subgroups of infinite index in Fuchsian groups, Math. Z. 125 (1972), 59-69. MR 0292948 (45:2029)
  • [7] A. Karrass and D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227-255. MR 41 #5499. MR 0260879 (41:5499)
  • [8] B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236-248. MR 15, 931. MR 0062122 (15:931b)
  • [9] J. Schreier and S. Ulam, Über die Permutationsgruppe der natürlichen Zahlenfolge, Studia Math. 4 (1933), 134-141.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20F05, 20E30

Retrieve articles in all journals with MSC: 20F05, 20E30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0320152-5
Keywords: Generalized free products, amalgamated products, small subgroups
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society