On the intersection of double cosets in free groups, with an application to amalgamated products

It is shown that if H, K are any finitely generated subgroups of a free gro up F and U is any cyclic subgroup of F, then any intersection Hg1U boolean AND Kg(2)U of double cosets contains only a finite number of double cosets (H boolean AND K)gU, and an explicit upper bound for this number is given i n terms of the ranks of H and K and a generator of U. This result is then a pplied to the intersection of finitely generated subgroups H, K of a free p roduct with amalgamation G = A (U)* B with A free and U maximal cyclic in A . Under the assumption that H and K intersect all conjugates of U trivially , an upper estimate is established for the "Karrass-Solitar rank" of H bool ean AND K in terms of the KS-ranks of H and K, a generator of U, and max(g is an element of G){rank(g(-1)Hg boolean AND A)}, max(g is an element of G){rank(g(-1)Kg boolean AND A)}. Here the KnMnss-Solirarlank of H less than or equal to A (U)* B is defined to be the size of a natural set of generating subgroups of H, afforded by t he Karrass-Solitar subgroup theorem for amalgamated products A (U)* B.