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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, Canada M3J 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

Information

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

References

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