Geodesic
Remarks on the Intersection of Finitely Generated Subgroups of a Free Group
Canadian mathematical bulletin, Tome 29 (1986) no. 2, pp. 204-207

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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〉.
DOI : 10.4153/CMB-1986-033-x
Mots-clés : 20E05, 20E07 
Burns, R. G.; Imrich, Wilfried; Servatius, Brigitte. Remarks on the Intersection of Finitely Generated Subgroups of a Free Group. Canadian mathematical bulletin, Tome 29 (1986) no. 2, pp. 204-207. doi: 10.4153/CMB-1986-033-x
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     author = {Burns, R. G. and Imrich, Wilfried and Servatius, Brigitte},
     title = {Remarks on the {Intersection} of {Finitely} {Generated} {Subgroups} of a {Free} {Group}},
     journal = {Canadian mathematical bulletin},
     pages = {204--207},
     year = {1986},
     volume = {29},
     number = {2},
     doi = {10.4153/CMB-1986-033-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1986-033-x/}
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