On the intersection of finitely generated subgroups in free products of groups

A subgroup H of a free product Π^*αεI Gα of groups Gα, α ε I, is called factor free if for every S ε Π^*αεI Gα and β ε I one has SHS^-1 ∩ Gβ = {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote γγ̄(K) = max(γ(K) -1,0), where γ(K) is the rank of K. It is proven that if H, K are finitely generated factor free subgroups of a free product Π^*αεI Gα then γ̄(H∩K) ≤ 6γ̄(H)γ̄(K). It is also shown that the inequality γ̄(H∩K) ≤ γ̄(H)γ̄(K) of Hanna Neumann conjecture on subgroups of free groups does not hold for factor free subgroups of free products.