The rank three case of the Hanna Neumann conjecture (Q2711627)

Let \(G\) be a free group, \(\text{rk}(G)\) be the rank of \(G\), and for a positive integer \(n\), let \(\text{rk}_{-n}(G)\) denote \(\max\{\text{rk}(G)-n,0\}\). If \(H\) and \(K\) are finitely generated subgroups of a free group, Hanna Neumann conjectured that \(\text{rk}_{-1}(H\cap K)\leq\text{rk}_{-1}(H)\cdot\text{rk}_{-1}(K)\).NEWLINENEWLINENEWLINEThe authors prove that NEWLINE\[NEWLINE\text{rk}_{-1}(H\cap K)\leq\text{rk}_{-1}(H)\cdot\text{rk}_{-1}(K)+\text{rk}_{-3}(H)\cdot\text{rk}_{-3}(K)NEWLINE\]NEWLINE and thus if \(H\) has rank three or less, then the conjecture holds. The result is established by proving a corresponding result in graph theory, which is interesting in its own right.