2 problems

Let $F$ be a free group, and let $H,K\le F$ be subgroups with equal rank $m\ge 2$. If … then Guzman's group-theoretic conjecture. … Here $H\vee K$ is the subgroup generated by $H$…

Let $F$ be a free group, and let $h,k,v,c$ be integers satisfying … For subgroups $H,K\le F$, the quantities $\operatorname{rk}(H)$, $\operatorname{rk}(K)$,…