# 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.

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.

• I think we have the same answer Nov 10, 2015 at 16:49
• Looks like we do!
– HJRW
Nov 10, 2015 at 19:22

They do, see theoerrm 4.7. http://arxiv.org/pdf/math/0203258.pdf by Dahmani.