On the Howson property of descending HNN-extensions of groups

Abstract: A group $G$ is said to have the Howson property (or to be a Howson group) if the intersection of any two finitely generated subgroups of $G$ is finitely generated subgroup. It is proved that descending HNN-extension is not a Howson group under some assumptions satisfied by the base group of HNN-extension. In particular, a result of the paper joined with a Burns - Brunner result (received in 1979) implies that any descending HNN-extension of non-cyclic free group does not have the Howson property.