On finitely generated subgroups which are of finite index in generalized free products
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- by A. Karrass and D. Solitar
- Proc. Amer. Math. Soc. 37 (1973), 22-28
- DOI: https://doi.org/10.1090/S0002-9939-1973-0320152-5
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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
- Benjamin Baumslag, Intersections of finitely generated subgroups in free products, J. London Math. Soc. 41 (1966), 673–679. MR 199247, DOI 10.1112/jlms/s1-41.1.673
- R. G. Burns, On the finitely generated subgroups of an amalgamated product of two groups, Trans. Amer. Math. Soc. 169 (1972), 293–306. MR 372043, DOI 10.1090/S0002-9947-1972-0372043-5
- Leon Greenberg, Discrete groups of motions, Canadian J. Math. 12 (1960), 415–426. MR 115130, DOI 10.4153/CJM-1960-036-8
- 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. MR 211401, DOI 10.1002/cpa.3160200207 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.
- A. Howard M. Hoare, Abraham Karrass, and Donald Solitar, Subgroups of infinite index in Fuchsian groups, Math. Z. 125 (1972), 59–69. MR 292948, DOI 10.1007/BF01111114
- 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 260879, DOI 10.1090/S0002-9947-1970-0260879-9
- B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248. MR 62122, DOI 10.1112/jlms/s1-29.2.236 J. Schreier and S. Ulam, Über die Permutationsgruppe der natürlichen Zahlenfolge, Studia Math. 4 (1933), 134-141.
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- 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