On the intersection of free subgroups in free products of groups

Let (Gi | i ∈ I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi.Let $\mathcal{F}$ denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define We are interested in precise bounds for $\upp$. In the special case where I is empty, Hanna Neumann proved that $\upp$ ∈ [1,2], and conjectured that $\upp$ = 1; fifty years later, this interval has not been reduced.With the understanding that ∞/(∞ − 2) is 1, we define Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.