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Remarks on the Intersection of Finitely Generated Subgroups of a Free Group

Published online by Cambridge University Press:  20 November 2018

R. G. Burns ,
Wilfried Imrich  and
Brigitte Servatius
R. G. Burns
Affiliation:
R. G. Burns, Department Of Mathematics, York University, North York, Toronto, Ontario, CanadaM3J 1P3
Wilfried Imrich
Affiliation:
Wilfried Imrich, Institute For Mathematics and Applied Geometry, Montanuniversität Leoben, A-8700, Leoben, Austria
Brigitte Servatius
Affiliation:
Brigitte Servatius, Department of Mathematics, Syracuse University, Syracuse, N.Y. 13210, U.S.A.
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Abstract

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The first result gives a (modest) improvement of the best general bound known to date for the rank of the intersection U ∩ V of two finite-rank subgroups of a free group F in terms of the ranks of U and V. In the second result it is deduced from that bound that if A is a finite-rank subgroup of F and B < F is non-cyclic, then the index of A ∩ B in B, if finite, is less than 2(rank(A) - 1), whence in particular if rank (A) = 2, then B ≤ A. (This strengthens a lemma of Gersten.) Finally a short proof is given of Stallings' result that if U, V (as above) are such that U ∩ V has finite index in both U and V, then it has finite index in their join 〈U, V〉.

Keywords

20E0520E07
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 2 , 01 June 1986 , pp. 204 - 207
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Bums, Robert G., On the intersection of finitely generated subgroups of a free group, Math. Zeitschr. 119(1971), pp. 121–130.Google Scholar
2. Bums, R. G., A note on free groups, Proc. Amer. Math. Soc. 23 (1969), pp. 14—17.Google Scholar
3. Gersten, S.M., Intersections of finitely generated subgroups of free groups and resolutions of graphs, Invent. Math. 71 (1983), pp. 567–591.Google Scholar
4. Greenberg, L., Discrete groups of motions, Canad. J. Math. 12 (1960), pp. 414–425.Google Scholar
5. Howson, A.G., On the intersection of finitely generated free groups, J. London Math. Soc. 29 (1954), pp. 428–434.Google Scholar
6. Karrass, A. and Solitar, D., On finitely generated subgroups of a free group, Proc. Amer. Math. Soc. 22 (1969), pp. 209–213.Google Scholar
7. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, (Interscience, New York, 1966).Google Scholar
8. Neumann, Hanna, On the intersection of finitely generated free groups, Publ. Math. Debrecen 4 (1956), 186–189. Addendum, Publ. Math. Debrecen 5 (1957/58), p. 128.Google Scholar
9. Nickolas, Peter, Intersections of finitely generated free groups. Bull. Austral. Math. Soc, 31 (1985), pp. 339–348.Google Scholar
10. Servatius, Brigitte, A short proof of a theorem of Burns, Math. Zeitschr. 184 (1983), pp. 133—137.Google Scholar
11. Stallings, John R., Topology of finite graphs, Invent. Math. 71 (1983), pp. 551–565.Google Scholar