# Do limit groups satisfy Howson's theorem?

Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must $A \cap B$ be finitely generated?

Recall that a limit group is a group whose existential theory (the set of true sentences in first order theory which use only the quantifier $\exists$ and not $\forall$) is the same as that of a nonabelian free group.

Howson's theorem says that the answer is positive in case that $G$ is free.