
> 
> 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. 



Do limit groups satisfy Howson's theorem?
The answer is 'yes'. A geometric proof, showing in fact that every finitely generated subgroup of a limit group is relatively quasiconvex, was given by [Dahmani](http://arxiv.org/abs/math/0203258).


