On rank of the join of two subgroups in a free group

Abstract

Let $H,K$ be two finitely generated subgroups of a free group, let $⟨H,K⟩$ denote the subgroup generated by $H,K$, called the join of $H,K$, and let neither of $H$, $K$ have finite index in $⟨H,K⟩$. We prove the existence of an epimorphism $\zeta :⟨H,K⟩\to F_2$, where $F_2$ is a free group of rank 2, such that the restriction of $\zeta$ on both $H$ and $K$ is injective and the restriction ${\zeta }_0:H\cap K\to \zeta (H)\cap \zeta (K)$ of $\zeta$ on $H\cap K$ to $\zeta (H)\cap \zeta (K)$ is surjective. This is obtained as a corollary of an analogous result on rank of the generalized join of two finitely generated subgroups in a free group.

14 pages, 1 figure