On the Intersection of Double Cosets in Free Groups, with an Application to Amalgamated Products

It is shown that ifH, Kare any finitely generated subgroups of a free groupFandUis any cyclic subgroup ofF, then any intersectionHg1U ∩ Kg2Uof double cosets contains only a finite number of double cosets (H ∩ K)gU, and an explicit upper bound for this number is given in terms of the ranks ofHandKand a generator ofU. This result is then applied to the intersection of finitely generated subgroupsH, Kof a free product with amalgamation [formula] withAfree andUmaximal cyclic inA. Under the assumption thatHandKintersect all conjugates ofUtrivially, an upper estimate is established for the “Karrass–Solitar rank” ofH ∩ Kin terms of the KS-ranks ofHandK, a generator ofU, and [equation] Here theKarrass–Solitar rankof [formula] is defined to be the size of a natural set of generating subgroups ofH, afforded by the Karrass–Solitar subgroup theorem for amalgamated products [formula].