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 ∗ ∗ i∈I Gi, the free product of F and all the Gi. Let 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∩Ggi = {1}. By the Kurosh Subgroup Theorem, every element of F is a free group. For each free group H, the reduced rank of H, denoted r̄(H), is defined as max{rank(H)− 1, 0} ∈ N∪{∞} ⊆ [0,∞]. To avoid the vacuous case, we make the additional assumption that F contains a non-cyclic group, and we define σ := sup{ r̄(H∩K) r̄(H)· r̄(K) : H, K ∈ F and r̄(H)· r̄(K) 6= 0} ∈ [1,∞]. We are interested in precise bounds for σ. In the special case where I is empty, Hanna Neumann proved that σ ∈ [1, 2], and conjectured that σ = 1; almost fifty years later, this interval has not been reduced. With the understanding that ∞ ∞−2 is 1, we define θ := max{ |L| |L|−2 : L is a subgroup of G and |L| 6= 2} ∈ [1, 3]. Generalizing Hanna Neumann’s theorem, we prove that σ ∈ [θ, 2 θ], and, moreover, σ = 2 θ whenever G has 2-torsion. Since σ is finite, F is closed under finite intersections. Generalizing Hanna Neumann’s conjecture, we conjecture that σ = θ whenever G does not have 2torsion.