On the rank of intersection of subgroups of a free product of groups Original Research Article

Let A * B be the free product of groups A, B. We define in the natural way the Kurosh rank (Krk H) of any subgroup H ≤ A * B, essentially as the number of obvious free factors in the decomposition of H as a free product in accordance with the Kurosh subgroup theorem. We establish the following analogue of the Howson-Hanna Neumann formula for free groups: For any two subgroups H, K ≤ A * B, one has Krk(H∩K)−1≤2(Krk H−1)(Krk−1)+2min{Krk H −, Krk−1}. If A has the property that for all m,n ≥ 1 there is an integer ƒA(m,n) bounding the rank of the intersection of any two subgroups of ranks ≤ m, n, respectively, and B has the same property relative to a corresponding function ƒB(m, n), then it follows via an easy application of the Grushko-Neumann theorem that for any subgroups H, K of A * B of ranks ≤ m, ≤ n, respectively, one has rk(H∩K)≤[2(m−1)(n−1)+2min {m,n}]max{fA(m,n), fB(m,n)}. This improves significantly on the bound of Soma (1990).